Crossing interfacial conduction in nanometer-sized graphitic carbon layers

Abstract

Graphitic carbon layers (GCLs), exemplified by graphene, have been proposed for potential application in high-performance functional devices due to superior electrical properties, e.g., high electron mobility. In state-of-the-art electronics, it is required that GCLs are miniaturized to nanometer scales and incorporated into the integrated circuits to exhibit novel functions at nanometer scales. However, the implementation of nanometer-scale GCLs is suspended; the function in devices is deteriorated by increasing contact resistance in miniaturized GCL/electrode interfaces. In this study, nanometer-sized GCL/gold (Au) interfaces were fabricated via atomistic visualization of nanomanipulation, and simultaneously their contact resistance was measured. We showed that the contact resistivity of the interfaces was decreased to the order of 10⁻¹⁰ Ω cm², which was 10⁴ times smaller than that of micrometer-sized or larger GCL/metal interfaces. In addition, it was revealed that peculiar electrical conduction at the nanometer-sized GCL/Au interfaces emerged; current flows throughout the entire area of the interfaces unlike micrometer-sized or larger GCL/metal interfaces. These results directly contribute to actual application of GCLs in advanced nanodevices.

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New concepts

The large contact resistance between graphitic carbon layers (GCLs) and connecting electrodes or substrates affects the excellent electrical properties of GCLs. As the miniaturization of integrated circuits advances, it is expected that the influence of the large contact resistance becomes more apparent, resulting in deterioration in the performance of the integrated circuits using GCLs. However, we show that the contact resistivity of nanometer-sized GCL/gold (Au) interfaces is extremely smaller than that of micrometer-sized or larger GCL/Au interfaces. In conventional silicon semiconductor technology, as the integrated circuits are miniaturized, the contact resistance with devices and electrodes increases. In contrast, in the case of nanometer-sized GCL/Au interfaces, the increase in the contact resistance can be suppressed by miniaturization to nanometer scales. This feature is important for, e.g., designing high-frequency field-effect transistors using GCLs as channels. When their miniaturization and high operating speed are realized simultaneously, the high integration of circuits is promoted. The present study opens new ways of the applications of ultimate nanometer-sized GCLs in advanced nanodevices.

Introduction

Studies of miniaturized graphitic carbon layers (GCLs) have progressed to the stage of practical applications in electronic circuits and sensors.^1–8 It has been attempted to apply GCLs, exhibiting excellent electron mobility, in devices such as high-frequency field-effect transistors.^9–13 When GCLs are incorporated into integrated circuits, interfaces with metallic electrodes or wires are definitely formed, and their characteristics significantly affect the performance of integrated circuits.^14–18 In particular, the electron mobility of GCLs decreases owing to charging and scattering by impurities,^14,19 and the surface roughness of the connecting electrodes or substrates.^20–22 In addition, the current injection from metal electrodes to GCLs is proportional to the transmission probability and the density of states (DOS) at both the metals and GCLs, which are influenced by their interface structures.^23,24 Theoretical studies on the relationships between the atomic configuration of GCL/metal interfaces and electrical properties have been performed;^25–29 the variations in the Fermi energy and the DOS at GCL/metal interfaces depend on their interlayer spacings and lattice mismatches.^25,26 The DOS near the Fermi energy is also influenced by the adsorption sites of metal atoms.²⁷ In particular, the transmission probability at GCL/metal interfaces depends on the bond strength and interatomic distance between carbon (C) and metal atoms at the interfaces.^28,29 These studies have been intensively carried out for micrometer-sized or larger GCLs.^{9–12,14–22,30} At present, the miniaturization of integrated circuits has reached nanometer scales. GCLs are expected to be miniaturized on the same scales and be incorporated into them. However, as the device size decreases as 1/k, where k is a unitless scaling constant, the contact resistance increases as k² while maintaining the contact resistivity (ρ_C);³¹ the device properties change dramatically as their size extremely decreases. Thus, to realize nanometer-sized GCL devices, first it is necessary to elucidate their electrical properties, especially when they form interfaces with metallic electrodes. The local electrical properties of the submicrometer-sized interfaces between highly oriented pyrolytic graphite (HOPG) and metals were investigated by contact-type atomic force microscopy (AFM) and molecular dynamics (MD) simulation.^32–35 These studies show that the variation in contact resistance occurs due to changes in the real contact area.^32,33 In addition, the decrease in the penetration depth along the c-axis of HOPG and the increase in the spreading resistance are attributed to the miniaturization of the contact area.^34,35 However, it has been difficult to control the structures of nanometer-sized GCL/metal interfaces at the atomic level, and to investigate their electrical properties. In this study, we applied in situ transmission electron microscopy (TEM) combined with picometer-precise nanotip manipulation^36–39 to nanometer-sized GCL/gold(Au) interfaces. Furthermore, we performed first-principles calculations for the atomic configuration of GCL/Au interface models that have been constructed on the basis of the observed high-resolution TEM images to investigate the transmission probability and the conduction pathways at the interfaces. The obtained results revealed that the nanometer-sized GCL/Au interfaces exhibit peculiar electrical conduction, which is inherently different from that of micrometer-sized or larger GCL/Au interfaces.

Results

The nanometer-sized GCL/Au interfaces were formed in single nanoparticle junctions (SNPJs) consisting of two electrodes sandwiching a single carbon nanocapsule (CNC).^38,39Fig. 1(a) shows a high-resolution TEM image of a Au/LaC₂-encapsulated CNC/Au SNPJ. The LaC₂ crystal is surrounded by eight C layers. To produce stable SNPJs, we used LaC₂-encapsulated CNCs because the encapsulation increases the stability of CNCs.^40–42 The current–voltage characteristic in the SNPJs was linear (see Fig. S1, ESI†), which was similar to that in the graphene/metal interfaces implying that the SNPJs were zero-gap semiconductors.⁴³Fig. 1(b-1) and (b-2) show high-resolution TEM images of the interfaces with different widths between the outermost GCL of the CNC and the negative Au electrode in the SNPJ in Fig. 1(a). The observed interface-widths are 13 Au atoms wide (Fig. 1(b-1)) and 5 Au atoms wide (Fig. 1(b-2)), respectively. The spacings of the lattice fringes observed in the upper region of the GCL and the lower region of the Au electrode in Fig. 1(b-1) and (b-2) are 0.34 and 0.24 nm, which correspond to (0002)_C and (111)_Au, respectively. Fig. 1(c-1) and (c-2) show the enlarged images of the (0001)_C/(11̄1)_Au interface and the model of its atomic configuration superimposed on the image in Fig. 1(c-1), respectively. The analysis of the observed direction of the GCL in Fig. 1(c-1) is shown in Fig. 1(d). Fig. 1(d-1) shows an enlarged high-resolution TEM image of the region indicated by the red broken-line frame in Fig. 1(c-1), in which arrays of black dots are visible along a lattice fringe. Fig. 1(d-2) depicts the line intensity along the fringe in Fig. 1(d-1). The distances between the black dots are estimated to be 0.21 ± 0.05 nm from the line intensity, corresponding to the distance between the dumbbells of C atomic rows, which emerge when (0001)_C is projected along [1̄1̄20]_C.^40–42,44 From the analysis in Fig. 1(c-1)–(d-2), the orientational relationship at the GCL/Au interface in Fig. 1(b-1) is identified to be (0001)[1̄1̄20]_C//(11̄1)[110]_Au. Fig. 1(e-1)–(e-3) show the atomic configurations of this orientational relationship. In this study, other GCL/Au interfaces were formed and analyzed similarly. As a result, the following six GCL/Au interfaces with four kinds of orientational relationships were observed (interfaces A–F), i.e.,

(A) (0001)[1̄1̄20]_C//(11̄1)[110]_Au

(B) (0001)[1̄1̄20]_C//(001)[110]_Au

(C) (0001)[1̄1̄20]_C//(1̄1̄3)[110]_Au

(D) (0001)[1̄1̄20]_C//(1̄1̄2)[110]_Au.

Fig. 1 Structure of a GCL/Au interface. (a) High-resolution TEM image of a Au/LaC₂-encapsulated CNC/Au SNPJ. The upper left and the lower right dark regions are the Au electrodes on the negative side and positive side, respectively. The bright regions around the SNPJ correspond to a vacuum. (b) Enlarged high-resolution TEM images of the (0001)_C/(11̄1)_Au interface between the outermost GCL of the CNC (upper) and Au electrode (lower) indicated by the arrow in (a) before the retraction of the Au electrode from the CNC (b-1) and after the retraction (b-2) (see Movie S1, ESI†). (c-1) Analyzed region of the atomic configuration of the interface shown in (b-1). The region surrounded by the red broken-line rectangle frame is the outermost GCL of the CNC contacted on the Au. (c-2) Model of the atomic configuration superimposed on the image contrast at the interface in (c-1). The larger open circles and the smaller filled circles represent the positions of Au and C atomic columns, respectively. (d-1) Enlarged high-resolution TEM image inside the red broken-line frame in (c-1). (d-2) Image intensity profile (brightness) along the line between the two blue arrowheads in (d-1). Two coarse red broken lines on the left side and two fine red broken lines on the right side indicate the positions of the two imaged black dots separated by 0.21 nm and the distance between neighboring C atomic rows separated by 0.07 nm, respectively, as shown (d-1) and (d-2). Models of the atomic configuration at the (0001)[1̄1̄20]_C//(11̄1)[110]_Au interface: C layer and Au plane at the interface viewed along [1̄1̄20]_C and [110]_Au (e-1) and [0001]_C and [11̄1]_Au (e-2). (e-3) Overlapping of the atomic configurations of the C layer and the Au plane projected along the direction perpendicular to the interface, i.e., [0001]_C and [11̄1]_Au.

Fig. 2(a) shows the relationships between the total resistance (R_t) and the inverse of the contact areas for interfaces A–F. For all interfaces, R_t increases as the contact area decreases. ρ_C of the interfaces can be estimated from the gradient of the linear approximation line in Fig. 2(a) (see Methods for details). The ρ_C values are shown in Table 1. ρ_C depends on the orientational relationships of the interfaces. Fig. 2(b) shows the distance of the interlayer spacing (d_i) between the outermost GCL of the CNC and the Au planes (upper) and ρ_C (lower) for interfaces A–F against the areal density of the Au planes. The areal density was estimated from the orientation of the Au planes at the interfaces. Although the areal density changed between 4.9 and 13.9 atoms per nm², d_i exhibited similar values (0.28 ± 0.05–0.30 ± 0.05 nm). In contrast, ρ_C increases as the areal density decreases. Hence, ρ_C is determined not by d_i but the areal density. Fig. 2(c) shows the relationships between the conductance per one C atom (hereafter referred to as the conductance density (G^u)) and the areal density of the Au planes for interfaces A–F. G^u increases linearly as the areal density increases.

Fig. 2 Analysis of the electrical conductivity of six GCL/Au interfaces. (a) Relationships between the total resistance of the SNPJs and the inverse of the contact areas for observed interfaces A–F. Orientational relationships are A, B: (0001)[1̄1̄20]_C//(11̄1)[110]_Au, C, D: (0001)[1̄1̄20]_C//(001)[110]_Au, E: (0001)[1̄1̄20]_C//(1̄1̄3)[110]_Au, and F: (0001)[1̄1̄20]_C//(1̄1̄2)[110]_Au. The line in (a) is a linear approximation. (b) Variation in the distance of the interlayer spacing (d_i) between the outermost GCL of the CNC and the Au planes (upper) and the contact resistivity (ρ_C) (lower) against the areal density of the Au planes for interfaces A–F. (c) Relationships between the conductance per one C atom, i.e., the conductance density (G^u), and the areal density of the Au planes for interfaces A–F. The orientational relationships of interfaces A–F in (b) and (c) correspond to the same relationships in (a). The contact area was 4.0–60 nm², and G^u was calculated using ρ_C measured when the contact area was the smallest area (4.0 nm²) and the areal density of graphene (38 atoms per nm²). The bars indicate the measurement errors. For example, these factors consist of errors in analyzing images and those in the electrical resistance measurements.

Table 1 Orientational relationships, contact resistivity, areal density, distance of the interlayer spacing, conductance density, and the misfit for each GCL/Au interface in Fig. 2(a). [uvw]_Au is the direction perpendicular to the viewing direction ([110]_Au) at the interfaces in each orientational relationship

Interface	Orientational relationship	Contact resistivity (10^−10 Ω cm^2)	Areal density (atoms per nm^2)	Interlayer spacing (nm)	Conductance density (μS)	Misfit (%)
						[1̄1̄20]C//[110]Au	[1̄100]C//[uvw]Au
A	(0001)[1̄1̄20]C//(11̄1)[110]Au	1.6 ± 0.2	13.9	0.30 ± 0.05	1.7 ± 0.3	20	[1̄1̄2]Au: 37
B		1.7 ± 0.3		0.29 ± 0.05	1.5 ± 0.3
C	(0001)[1̄1̄20]C//(001)[110]Au	2.5 ± 0.4	12.0	0.29 ± 0.05	1.0 ± 0.1		[11̄0]Au: −31
D		2.5 ± 0.1		0.28 ± 0.05	1.0 ± 0.2
E	(0001)[1̄1̄20]C//(1̄1̄3)[110]Au	3.8 ± 0.8	7.3	0.28 ± 0.05	0.7 ± 0.2		[33̄2]Au: 124
F	(0001)[1̄1̄20]C//(1̄1̄2)[110]Au	8.5 ± 1.2	4.9	0.29 ± 0.05	0.3 ± 0.1		[11̄1]Au: 68

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Fig. 3 shows the results of the first-principles calculations for the relationships between the transmission probability (T_ij) and the C_i–Au_j interatomic distances at the GCL/Au interfaces, where i and j are the ordinal numbers of the C atom (C_i atom) on the GCL and the Au atom (Au_j atom). The models of the atomic configurations of the interfaces for the calculation were constructed on the basis of the d_i and the orientational relationships estimated from the high-resolution TEM images. In each model, the relative positions of the GCL and Au are set so that the positions of a C atom and a Au atom match at the center of the interface. In the model used for the calculation, we selected two C layers, because the electrical conduction at the interface is determined by the number of channels and T_ij at the interface between the outermost C layer and metal.²⁵ Therefore, it is expected that the electrical conduction of the interfaces with the eight C layers observed in this study can be compared with that of the interfaces with two C layers. When the area of the GCL was selected to be 1.4 nm² in all orientational relationships, the features of T_ij at each interface emerged. Therefore, the calculation was carried out using this contact area for all models. The distribution of T_ij is 10⁻³–10⁻¹ at C_i–Au_j interatomic distances of 0.4 nm or less (the right frames in Fig. 3(a)–(d)), regardless of the position of the C atom or Au atom; conduction pathways with relatively large T_ij exist in the entire area of the interfaces. T_ij decreases as the C_i–Au_j interatomic distance increases for all orientational relationships (Fig. 3(e)). Note that the distribution of T_ij in each orientational relationship is similar. For the models of the atomic configuration of the interfaces shown in Fig. 3(a)–(d), we investigated the total number and T_ij of the channels of the interfaces between the Au and the C layer. The sum of T_ij of all the channels was 0.49% (that for pure ballistic conduction is defined as 100%).

Fig. 3 Relationships between the transmission probability (T_ij) and C_i–Au_j interatomic distance at the GCL/Au interfaces obtained from the first-principles calculations. The left frames in (a)–(d) the models of the atomic configurations of the GCL/Au interfaces for the first-principles calculations, viewed along the direction parallel and perpendicular to the interface. The orientational relationships are (a) (0001)[1̄1̄20]_C//(11̄1)[110]_Au, (b) (0001)[1̄1̄20]_C//(001)[110]_Au, (c) (0001)[1̄1̄20]_C//(1̄1̄3)[110]_Au, and (d) (0001)[1̄1̄20]_C//(1̄1̄2)[110]_Au. The smaller gray circles and the larger yellow circles represent the positions of the atomic columns of C and Au atoms, respectively. The distances of the interlayer spacing are (a), (b), (d) 0.29 nm and (c) 0.28 nm. The right frames in (a)–(d) T_ij between the i-th C atom (C_i atom) on the GCL and the j-th Au atom (Au_j atom) on the Au planes against the C_i–Au_j interatomic distance, obtained from the first-principles calculations. Each colored and shaped plot indicates different adjacency of a Au atom to C atoms. Blue circles: the case for the nearest neighbor atoms – the 3rd adjacent atoms. Red squares: the case for the 4–6th adjacent atoms. Orange triangles: the case of the 7–9th adjacent atoms. Green diamonds: the case of atoms farther than the 10th adjacent atoms. (e) T_ij for all orientational relationships; the diagram obtained by overlapping those in (a)–(d).

For the (0001)[1̄1̄20]_C//(11̄1)[110]_Au interface, the measured value was 1.59 μS, which was 48% of the simulated value (3.34 μS) (see the ESI†). Matsuda et al. demonstrated by first-principles calculations that G^u of the interfaces between the close-packed surface of Au and graphene, having a contact area of 2 × 4 graphene unit cells, was 0.8 μS per one C atom,⁴⁵ corresponding to half of the value measured in the present study.

ρ _C of the nanometer-sized GCL/Au interfaces decreases as the areal density of the Au planes increases (Fig. 2(b)). G^u increases linearly as the areal density increases (Fig. 2(c) and Table 1). The results of our first-principles calculations show that T_ij decreases as the C_i–Au_j interatomic distance increases for all orientational relationships (Fig. 3(e)). At the interfaces where the areal density of the Au planes was the largest, the number of C_i–Au_j interatomic bonds with the shortest interatomic distance and with high T_ij was increased; the conduction pathways of large current were increased. Therefore, G^u increases as the areal density increases, resulting in the decrease in ρ_C.

Discussion

The conduction pathways in the CNCs of the SNPJs and the similarities to the conduction of graphene/metal junction systems

The CNCs have multi-GCL spherical shell structures consisting of basic units of six-membered C rings (Fig. 1(a)).^46,47 In HOPG/metal interfaces, when their contact radius is reduced to 60 nm, the penetration depth along the c-axis of HOPG is decreased to 1 nm corresponding to a three layer thickness.³⁴ Thus, the decrease in the penetration depth along the c-axis of HOPG is attributed to the miniaturization of the contact area. In addition, the c-axis resistivity of HOPG is exponentially decreased as the bias voltage is increased up to 200 V.³⁵ Under the measurement conditions in this study, i.e., the number of layers of the CNCs is eight on average and the bias voltage is 13 mV, the c-axis resistivity was maintained at ∼10⁻³ Ω cm. The c-axis resistivity of the HOPG layer is three orders of magnitude higher than the in-plane resistivity, resulting in the increase in carriers in-plane.^48,49 In this study, linear relationships between current and voltage were also observed, as shown in Fig. S1 (ESI†). This suggests that the c-axis conduction hardly influences the total conduction of the SNPJs. Therefore, most conduction electrons pass through only the outermost C layer of the CNCs.

The encapsulating LaC₂ crystals were located at the center of the eight spherical C shell-layers, and LaC₂/GCL interfaces were formed, as shown in Fig. 1(a). The influence of the LaC₂ crystals on the electronic states of the C layers was investigated by first-principles calculations (see the ESI†). It was found that the LaC₂ crystals are unlikely to contribute to the partial density of states of the C layers adjacent to the Au electrode (Fig. S5, ESI†). Thus, it is inferred that the CNC contains a LaC₂ crystal exhibiting a resistivity of 5.5 × 10⁻⁵ Ω cm in the center;⁵⁰ it hardly contributes to the conduction of the entire SNPJ.

It is also noted that although the GCLs of CNCs are curved, the observed interfaces between the outermost GCL and the Au electrode surface are planar (Fig. 1(b)), which are similar to the contact interfaces between planar graphene and metal electrodes. Therefore, on the basis of these similarities, the electrical conduction of the SNPJs in this study is discussed with reference to previous studies concerning graphene in the following sections.

The decrease in contact resistivity and contact resistance due to the miniaturization of the GCL/Au interfaces to several nanometers

ρ _C of the nanometer-sized GCL/Au interfaces was 1.6 ± 0.2–8.5 ± 1.2 × 10⁻¹⁰ Ω cm² (Table 1). In contrast, ρ_C of micrometer-sized or larger GCL/Au interfaces is 3.3 × 10⁻⁶–1 × 10⁻⁵ Ω cm²,^15,43,51,52 which were estimated from the transmission line model (TLM).^53,54 Thus, ρ_C of the nanometer-sized GCL/Au interfaces was at least 10⁴ times smaller than the minimum ρ_C of micrometer-sized or larger interfaces.

In micrometer-sized or larger graphene/metal interfaces, ρ_C of the graphene edge contacts was considerably decreased compared to the on-plane contacts owing to edge effects.^{18,45,55–60} Such edge effects have been applied in techniques to decrease ρ_C; numerous holes in graphene at interfaces were introduced to increase the ratio of the edge region to the interface.⁵⁹ A decrease in ρ_C has also been demonstrated by certain treatments of interfaces, e.g., the deposition of poly(4-vinylphenol)/poly(melamine-co-formaldehyde) molecules on graphene surfaces,¹⁸ the introduction of hydrogen molecules between graphene bilayers,¹⁷ and the application of a gate voltage of −30 V to graphene.¹⁸ In such previous studies, ρ_C was normalized by the area of the current concentration region and was calculated to be 1.2 × 10⁻⁸–2.5 × 10⁻⁵ Ω cm².^{17,18,32,45,55–60} In this study, the C–Au contacts were on-plane contacts, as shown in Fig. 1(b). In addition, the GCLs were not subjected to any special treatments, and no high gate voltages were applied to them. Nevertheless, the estimated ρ_C was 10²–10⁴ times smaller than the reported values in previous studies. Thus, owing to the miniaturization of the GCL/Au interfaces to several nanometers, ρ_C is decreased compared to the cases with treatments or under the application of high gate voltages in micrometer-sized or larger graphene/metal interfaces.

In submicrometer-sized HOPG/metal interfaces, the contact resistance was investigated by contact-type AFM and MD simulations; the contact resistance is estimated from R_t of the measurement system.³³ The contact areas were 3.5 and 11.1 × 10⁻⁴ nm², which were at least 10² times larger than that in this study. ρ_C normalized by the contact area was 2.5 × 10⁻⁵–1.5 × 10⁻⁴ Ω cm², which were 10⁵–10⁴ times larger than that in this study.

For Fig. 5, we assumed that ρ_C is proportional to the contact radius, and extrapolated the approximation line to the contact radius above 100 nm. As a result, in this extrapolated region, the approximated ρ_C is at least 10² times smaller than that of previous studies for large contact radii exceeding 100 nm.³² Thus, the observation of extremely small ρ_C in the present SNPJs is not simply attributed to a size effect but there is another mechanism, as demonstrated by the first-principles calculations in this study. There is a transition point of ρ_C between the contact radius of 1.0–4.4 nm and 100 nm.

The peculiar conduction pathways at the nanometer-sized GCL/Au interfaces

The results of our first-principles calculations show that conduction pathways exist in the entire area of the interfaces (Fig. 3). In contrast, at micrometer-sized or larger GCL/Au interfaces, the current flows in the region consisting of the channel width and transfer length (d_t) near the edges of the interfaces rather than in the center regions of the interfaces, which can be explained by TLM.^15,43,51,52 The radius of the contact interface in this study was 10⁻²–10⁻³ times smaller than the transfer length in a previous study on micrometer or larger graphene/metal interfaces.⁵¹ Thus, the nanometer-sized GCL/Au interfaces in this study were significantly smaller than micrometer-sized or larger GCL/Au where TLM is applicable. Fig. 4(b) and (c) illustrate such conduction pathways at small and large GCL/Au interfaces. Thus, the conduction pathways at the nanometer-sized GCL/Au interfaces are different from those at micrometer-sized or larger GCL/Au interfaces.

Fig. 4 Illustration of a Au/CNC/Au SNPJ and conduction pathways at small and large GCL/Au interfaces. (a) Illustration of the SNPJ shown in Fig. 1(a). The contact interface between the outermost GCL of the CNC and the negative Au electrode (nanotip) is a flat plane. As shown in Fig. 1(b), the observed contact interface consists of the small Au electrode and the small contact area (the bold arrow). The other contact area does not change during the experiments because the positive Au electrode and the contact area on the positive side are sufficiently large. The positive and negative Au electrodes are connected to an ammeter and a DC power supply. A DC voltage is applied between both Au electrodes. (b) and (c) Enlarged illustration of the interface between the outermost GCL and the negative Au electrode ((b) the case for d_r < d_t and (c) for d_r > d_t). The upper and lower sides in (b) and (c) correspond to the view from the direction parallel to the interface and the perspective view of the interface, respectively. The arrows represent the current pathways. The annular red regions in (b and c) are current crowding regions. In (b), the current flows through the entire area of the interfaces.

If we assume that TLM can be applied to the nanometer-sized GCL/Au interfaces in this experiment, d_t is calculated using the following formula:^53,54

where ρ_C is measured for each interface and R_s is the sheet resistance of the channel. In this calculation, because a major portion of the current through the SNPJs flowed along the outermost GCL of the CNC,³⁹ an R_s of 1.8 Ω sq^−1 61 corresponding to that of GCLs was used. Fig. 5 shows the relationships between the radii of the interfaces (d_r) and ρ_C for interfaces A–F. d_r was shorter than d_t calculated for all interfaces with different ρ_C. Therefore, the result of this calculation shows that the conduction at the nanometer-sized GCL/Au interfaces can not be explained on the basis of TLM; TLM can not be applied to the nanometer-sized GCL/Au interfaces.

Fig. 5 The radii of the interfaces (d_r) against the contact resistivity (ρ_C) for interfaces A–F. The triangles indicate the transfer lengths (d_t) estimated for the ρ_C of each interface. The dashed line in (C) indicates (see the text for details).

Franklin et al. showed that the miniaturization of graphene/Ti/Pd/Au interfaces, in which the contact length is 50–200 nm and the contact area of the interfaces is 5 × 10⁴–5 × 10⁵ nm², hardly affects the edge-dominated conduction; the contact resistance normalized by the edge length is maintained at a uniform value regardless of the contact area.⁶² In contrast, the contact area in this study was 1.9–53 nm², and tens to hundreds of atoms were included. The contact radius and contact area were 10–50 times and 10⁴–10⁵ times smaller than the contact length and contact area in previous studies, respectively. Thus, the results of this study revealed that the conduction pathways exist in the entire area of the interfaces due to size reduction of the interfaces to nanometer scales, which is an inherent conduction mechanism in the nanometer-sized GCL/Au interfaces.

The relationship between the contact resistance and the interface plane on the metal side

In graphene of graphene/Cu and graphene/Al laminated composites, the in-plane electrical conductivity is enhanced up to 118% of that of Cu with high carrier density and high mobility.⁶³ When the areal atom density of the Cu or Al side in the interfaces is highest, i.e., the lattice plane is (111), the in-plane conductivity becomes the maximum, and both the carrier density and mobility become high.^63,64 In this study, ρ_C became the lowest when the surface of the Au electrode was (111) (Table 1). From both results, it is considered that the resistivity can be reduced due to the selection of the lowest index surface of fcc metals.

The application of the SNPJ structure to nanodevices

The structure of the SNPJs in this study is similar to that of single molecule devices formed by sandwiching a single molecule such as C₆₀ and a C nanotube between two electrodes.^65,66 The fabrication and application of such devices have been developed; various nanocarbon materials can be incorporated into integrated circuits and substrates.^5,67,68 It is expected that such molecule-operation technologies can be applied to the incorporation of the present SNPJs into integrated circuits and substrates so that the SNPJs are operated as nanodevices.

The degree of ballistic conduction at the nanometer-sized GCL/Au interfaces

The first-principles calculations showed that the sum of T_ij of all the channels was 0.49%. This ratio seems to be low in comparison with perfect ballistic conduction, i.e., a T_ij of 100%. Nevertheless, the conduction at the nanometer-sized GCL/Au interfaces is still high in comparison with that at larger interfaces, as discussed above; ρ_C for 1.9 nm² was at least 10⁴ times smaller than that for 1 μm².

The influence of contact pressure on the contact resistivity

In this study, the Au electrode (nanotip) was pressed against the CNC to fabricate their contact interface. The resultant interfaces became atomically flat due to the rearrangement of the atomic configuration of the Au surfaces, implying that the contact interface was a single real contact interface (Fig. 1(b)). After contact, the position of the Au electrode was fixed, and the interface structure did not plastically deform (Movie S1 and Section S2 in the ESI†). This shows that the contact pressure applied to the interface was lower than the yield stress of Au. Although the contact resistance of HOPG/Pt contact interfaces is decreased as the contact pressure is increased, ρ_C is maintained because the real contact area is increased; the variation in the contact resistance occurs due to the change in the real contact area, rather than the contact pressure.³³ In this study, the resistance of the SNPJs was measured while maintaining two interface structures. The acting contact pressure contributed to the decrease in the resistance until the interface structure was stabilized. However, after fixation, the pressure hardly influenced the resistance. Thus, the observed significant decrease in the resistance of the nanometer-sized GCL/Au interfaces was not attributed to the pressure but the miniaturization of the interfaces.

The distribution of the injected electrons in the in-plane outermost C layer of the CNCs

The spreading resistance between a PtIr probe and HOPG was measured by injecting electrons from the HOPG into the probe. When the contact radius is reduced to 60 nm, the size of the effective resistance region, i.e., the distribution of the injected electrons, is increased beyond the contact area to several hundreds of micrometers.³⁴ Based on the previous result, it is considered that the electrons injected into the Au electrode exist widely in the in-plane outermost C layer since the contact radii of this study, i.e., 1.0–4.4 nm, are extremely smaller than those of the previous study.

Conclusions

Nanometer-sized GCL/Au interfaces with different orientational relationships and contact areas in Au/CNC/Au SNPJs were fabricated. The atomic configurations of the GCL/Au interfaces were observed by using in situ TEM at an atomic spatial resolution, and simultaneously ρ_C of the interfaces was evaluated. Models of the atomic configurations of the GCL/Au interfaces were constructed on the basis of the observed high-resolution TEM images, and first-principles calculations were performed to investigate T_ij and the conduction pathways.

When the orientational relationship of the interface was (0001)[1̄1̄20]_C//(11̄1)[110]_Au, ρ_C was a minimum, i.e., 1.6 ± 0.2 × 10⁻¹⁰ Ω cm². In such an orientational relationship, G^u was estimated to be 1.59 μS. The first-principles calculations using the models showed that G^u was 3.34 μS, similar to the observed value. ρ_C of the nanometer-sized GCL/Au interfaces was at least 10⁴ times smaller than that of micrometer-sized or larger GCL/Au interfaces.

In the first-principles calculations, the conduction pathways between C_i and Au_j exist in the entire area of the interfaces. Thus, at the nanometer-sized GCL/Au interfaces where d_r was smaller than the d_t estimated from TLM, it was found that current flows through not only near the edges of the interfaces but also the center region of the interfaces. The present results showed that the peculiar electrical conduction mechanism at the nanometer-sized GCL/Au interfaces emerges due to size reduction of the interfaces to nanometer scales, which is inherently different from the electrical conduction mechanism at micrometer-sized or larger GCL/Au interfaces.

In conventional silicon semiconductor technology, as the miniaturization of the integrated circuits advances, the contact resistance between devices and electrodes increases.⁶⁹ However, in the present nanometer-sized GCL/Au interfaces, the increase in the contact resistance via miniaturization to nanometer scales can be suppressed. This feature is important for designing high-frequency field-effect transistors using GCLs in which the parasitic resistance (contact resistance) imposes a limitation on their operating speed.^9–11 The present study opens new application ways of ultimate nanometer-sized GCLs in advanced integrated circuits.

Methods

In situ TEM combined with piezomanipulation of nanotips and electrical conductance measurements

Two Au nanotips (electrodes) were prepared by mechanical and ion beam milling. LaC₂-encapsulated CNCs were synthesized via a gas evaporation method.^46,47,70 The CNCs used in this study consisted of eight C layers on average. The CNCs were dispersed on the edge surfaces of one of the Au nanotips. The dispersed Au nanotip and a bare Au nanotip were attached to the first and second sample holders for a picometer-precision manipulation system for TEM, respectively. The two sample holders were inserted into a transmission electron microscope. Inside the microscope, first, the bare Au nanotip was brought into contact with the outermost GCL of a CNC on the dispersed Au nanotip by piezomanipulation so that nanometer-sized GCL/Au interfaces were fabricated. The contact area and orientational relationships of the GCL/Au interfaces were changed and controlled by Au nanotip manipulation using the sample piezodriving system. This process was observed in situ by high-resolution TEM. Simultaneously, the electrical resistivity of the interfaces was measured while applying a bias voltage of 13 mV. A series of observations and measurements were performed at room temperature at a vacuum of 1 × 10⁻⁵ Pa.

Estimation of the contact resistivity of the GCL/Au interfaces

The resistance of a homogeneous material is determined by its minimum cross-sectional area. In contrast, the resistance of the SNPJ system is expressed as the sum of the resistance of the CNC and the two contact interfaces. The resistance of the two electrodes (nanotips) and that of the connection cables, i.e., the connection resistance, are added to the measured R_t. The contact resistance was 2–17 kΩ, while the connection resistance was 0.5–1 Ω. Thus, this resistance of the connection system is negligibly small compared with R_t. Fig. 4(a) shows an illustration of a Au/CNC/Au SNPJ corresponding to the junction shown in Fig. 1(a). In this study, to estimate ρ_C of a GCL/Au interface, whereas the contact region between the GCL and the positive Au electrode was fixed, that between the GCL and the negative Au electrode was changed; R_t was measured as only the negative side interface structure was changed. Thus, R_t of the junction systems was measured as a function of the negative side contact area (Fig. S2, S3 and Movie S2, ESI†). We assumed the shapes of the contact area to be circular; the area was calculated from the diameter measured by the interface-width. As a result, R_t is estimated using the following formula:³⁹

R_t = R_Au+ + R_C + R_Au− = ρ_C′/A₊ + R_C + ρ_C/A₋, (2)

where R_Au+ and R_Au− are the contact resistance between the GCL and the Au electrodes on the positive and the negative side, R_C is the resistance of the CNC, ρ_C′ and ρ_C are the contact resistivity between the GCL and the Au electrodes on the positive and the negative side, and A₊ and A₋ are the contact area between the GCL and the Au electrodes on the positive and the negative side, respectively. ρ_C is expressed using a unit of Ω cm².⁷¹ The contact area on the positive side is fixed when R_t is measured; R_Au+ + R_C = C. According to eqn (2) and R_Au+ + R_C = C, R_t is

R_t = C + R_Au− = C + ρ_C/A₋. (3)

From eqn (3), ρ_C can be estimated from the gradient in a linear approximation line of some plots of the measured R_t with different 1/A₋, as shown in Fig. 2(a).

Transmission probability and conduction pathways simulated by using a theoretical study

First-principles calculations based on density functional theory and a non-equilibrium Green's function^72,73 were performed for the GCL/Au interfaces. For the calculations, we used Atomistix Toolkit for electron transport calculations and Virtual NanoLab as a graphical user interface.⁷⁴ We applied the generalized gradient approximation for the exchange–correlation potential, and a double-zeta polarized basis set for the basis functions.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by Grant-in-Aid for JSPS Fellows Number JP18J14221. The authors (M. T. and T. K.) thank the members of their laboratory for help with a part of the experiment.

References

1. J. Bai, X. Zhong, S. Jiang, Y. Huang and X. Duan, Nat. Nanotechnol., 2010, 5, 190–194.
2. M. Zhao, Y. Ye, Y. Han, Y. Xia, H. Zhu, S. Wang, Y. Wang, D. A. Muller and X. Zhang, Nat. Nanotechnol., 2016, 11, 954–959.
3. S. Wachter, D. K. Polyushkin, O. Bethge and T. Mueller, Nat. Commun., 2017, 8, 14948.
4. B. M. Blaschke, N. Tort-Colet, A. Guimerà-Brunet, J. Weinert, L. Rousseau, A. Heimann, S. Drieschner, O. Kempski, R. Villa, M. V. Sanchez-Vives and J. A. Garrido, 2D Mater., 2017, 4, 025040.
5. R. Bian, L. Meng, C. Guo, Z. Tang and H. Liu, Adv. Mater., 2019, 31, 1900534.
6. M. Samadi, N. Sarikhani, M. Zirak, H. Zhang, H.-L. Zhang and A. Z. Moshfegh, Nanoscale Horiz., 2018, 3, 90–204.
7. X. Yang, G. Liu, Y. Shi, W. Huang, J. Shao and X. Dong, Nanotechnology, 2018, 29, 222001.
8. F. Liu, W. T. Navaraj, N. Yogeswaran, D. H. Gregory and R. Dahiya, ACS Nano, 2019, 13, 3257–3268.
9. D. Berdebes, T. Low, Y. Sui, J. Appenzeller and M. S. Lundstrom, IEEE Trans. Electron Devices, 2011, 58, 3925–3932.
10. A. Zak, M. A. Andersson, M. Bauer, J. Matukas, A. Lisauskas, H. G. Roskos and J. Stake, Nano Lett., 2014, 14, 5834–5838.
11. F. Bianco, D. Perenzoni, D. Convertino, S. L. D. Bonis, D. Spirito, M. Perenzoni, C. Coletti, M. S. Vitiello and A. Tredicucci, Appl. Phys. Lett., 2015, 107, 131104.
12. P. Li, R. Z. Zeng, Y. B. Liao, Q. W. Zhang and J. H. Zhou, Sci. Rep., 2019, 9, 3642.
13. F. H. L. Koppens, T. Mueller, P. Avouris, A. C. Ferrari, M. S. Vitiello and M. Polini, Nat. Nanotechnol., 2014, 9, 780–793.
14. E. H. Hwang, S. Adam and S. D. Sarma, Phys. Rev. Lett., 2007, 98, 186806.
15. A. Venugopal, L. Colombo and E. M. Vogel, Appl. Phys. Lett., 2010, 96, 013512.
16. K. Nagashio and A. Toriumi, Jpn. J. Appl. Phys., 2011, 50, 070108.
17. J. S. Moon, M. Antcliffe, H. C. Seo, D. Curtis, S. Lin, A. Schmitz, I. Milosavljevic, A. A. Kiselev, R. S. Ross, D. K. Gaskill, P. M. Campbell, R. C. Fitch, K. M. Lee and P. Asbeck, Appl. Phys. Lett., 2012, 100, 203512.
18. H.-Y. Park, W.-S. Jung, D.-H. Kang, J. Jeon, G. Yoo, Y. Park, J. Lee, Y. H. Jang, J. Lee, S. Park, H.-Y. Yu, B. Shin, S. Lee and J.-H. Park, Adv. Mater., 2016, 28, 864–870.
19. J. H. Chen, C. Jang, S. Xiao, M. Ishigami and M. S. Fuhrer, Nat. Nanotechnol., 2008, 3, 206.
20. M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer and E. D. Williams, Nano Lett., 2007, 7, 1643–1648.
21. M. I. Katsnelson and A. K. Geim, Philos. Trans. R. Soc., A, 2008, 366, 195–204.
22. S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak and A. K. Geim, Phys. Rev. Lett., 2008, 100, 016602.
23. E. L. Wolf, Principles of Electron Tunneling Spectroscopy, Oxford Univ. Press, New York, 2012.
24. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, John Wiley & Sons, Inc., New Jersey, 2007.
25. G. Giovannetti, P. A. Khomyakov, G. Brocks, V. M. Karpan, J. Van den Brink and P. J. Kelly, Phys. Rev. Lett., 2008, 101, 026803.
26. C. Gong, G. Lee, B. Shan, E. M. Vogel, R. M. Wallace and K. Cho, J. Appl. Phys., 2010, 108, 123711.
27. G. Ramos-Sanchez and P. B. Balbuena, Phys. Chem. Chem. Phys., 2013, 15, 11950–11959.
28. N. Nemec, D. Tománek and G. Cuniberti, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 125420.
29. F. Xia, V. Perebeinos, Y. Lin, Y. Wu and P. Avouris, Nat. Nanotechnol., 2011, 6, 179–184.
30. K. Ito, M. Katagiri, T. Sakai and Y. Awano, Jpn. J. Appl. Phys., 2013, 52, 06GD08.
31. R. H. Dennard, F. H. Gaensslen, V. L. Rideout, E. Bassous and A. R. LeBlanc, IEEE J. Solid-State Circuits, 1974, 9, 256–268.
32. M. R. Vazirisereshk, S. A. Sumaiya, A. Martini and M. Z. Baykara, Appl. Phys. Lett., 2019, 115, 091602.
33. Z. Ye, H. Moon, M. H. Lee and A. Martini, Tribol. Int., 2014, 71, 109–113.
34. E. Koren, A. W. Knoll, E. Lörtscher and U. Duerig, Appl. Phys. Lett., 2014, 105, 123112.
35. E. Koren, A. W. Knoll, E. Lörtscher and U. Duerig, Nat. Commun., 2014, 5, 5837.
36. T. Kizuka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 155401.
37. M. Karita, K. Asaka, H. Nakahara and Y. Saito, Surf. Interface Anal., 2012, 44, 674–677.
38. D. Matsuura and T. Kizuka, J. Nanosci. Nanotechnol., 2014, 14, 2441–2445.
39. M. Tezura and T. Kizuka, Sci. Rep., 2016, 6, 29708.
40. D. Matsuura and T. Kizuka, J. Nanomater., 2012, 2012, 843516.
41. T. Kizuka and H. Koizumi, J. Nanosci. Nanotechnol., 2014, 14, 3228–3232.
42. G. Yazaki, D. Matsuura and T. Kizuka, J. Nanosci. Nanotechnol., 2014, 14, 2482–2486.
43. S. A. Peng, Z. Jin, P. Ma, D. Y. Zhang, J.-Y. Shi, J.-B. Niu, X. Y. Wang, S. Q. Wang, M. Li, X. Y. Liu, T. C. Ye, Y. H. Zhang, Z. Y. Chen and G. H. Yu, Carbon, 2015, 82, 500–505.
44. T. Kizuka and A. Akagawa, J. Nanosci. Nanotechnol., 2014, 14, 3176–3180.
45. Y. Matsuda, W. Q. Deng and W. A. Goddard, J. Phys. Chem. C, 2010, 114, 17845–17850.
46. R. S. Ruoff, D. C. Lorents, B. Chan, R. Malhotra and S. Subramoney, Science, 1993, 259, 346.
47. M. Tomita, Y. Saito and T. Hayashi, Jpn. J. Appl. Phys., 1993, 32, L280.
48. D. Z. Tsang and M. S. Dresselhaus, Carbon, 1976, 14, 43–46.
49. Y. Hishiyama and Y. Kaburagi, Carbon, 1992, 30, 483–486.
50. J. Rumble, CRC Handbook of Chemistry and Physics, CRC Press, Florida, 100th edn, 2019.
51. K. Nagashio, T. Nishimura, K. Kita and A. Toriumi, Appl. Phys. Lett., 2010, 97, 143514.
52. M. König, G. Ruhl, A. Gahoi, S. Wittmann, T. Preis, J. Batke, I. Costina and M. C. Lemme, IEEE J. Electron Devices Soc., 2019, 7, 219–226.
53. G. K. Reeves and H. B. Harrison, IEEE Electron Device Lett., 1982, 3, 111–113.
54. D. K. Schroder, Semiconductor Material and Device Characterization, John Wiley & Sons, Inc., New Jersey, 2006.
55. Y. Wu, Y. Wang, J. Wang, M. Zhou, A. Zhang, C. Zhang, Y. Yang, Y. Hua and B. Xu, AIP Adv., 2012, 2, 012132.
56. J. T. Smith, A. D. Franklin, D. B. Farmer and C. D. Dimitrakopoulos, ACS Nano, 2013, 7, 3661–3667.
57. S. M. Song, T. Y. Kim, O. J. Sul, W. C. Shin and B. J. Cho, Appl. Phys. Lett., 2014, 104, 183506.
58. T. Cusati, G. Fiori, A. Gahoi, V. Passi, M. C. Lemme, A. Fortunelli and G. Iannaccone, Sci. Rep., 2017, 7, 5109.
59. L. Anzi, A. Mansouri, P. Pedrinazzi, E. Guerriero, M. Fiocco, A. Pesquera, A. Centeno, A. Zurutuza, A. Behnam, E. A. Carrion, E. Pop and R. Sordan, 2D Mater., 2018, 5, 025014.
60. V. Passi, A. Gahoi, E. G. Marin, T. Cusati, A. Fortunelli, G. Iannaccone, G. Fiori and M. C. Lemme, Adv. Mater. Interfaces, 2019, 6, 1801285.
61. X. Wang, L. Zhi and K. Müllen, Nano Lett., 2008, 8, 323–327.
62. A. D. Franklin, S. Han, A. A. Bol and W. Haensch, IEEE Electron Device Lett., 2011, 32, 1035–1037.
63. M. Cao, D.-B. Xiong, L. Yang, S. Li, Y. Xie, Q. Guo, Z. Li, H. Adams, J. Gu, T. Fan, X. Zhang and D. Zhang, Adv. Funct. Mater., 2019, 29, 1806792.
64. M. Cao, Y. Luo, Y. Xie, Z. Tan, G. Fan, Q. Guo, Y. Su, Z. Li and D.-B. Xiong, Adv. Mater. Interfaces, 2019, 6, 1900468.
65. Y. Kim, W. Jeong, K. Kim, W. Lee and P. Reddy, Nat. Nanotechnol., 2014, 9, 881–885.
66. S. Parui, L. Pietrobon, D. Ciudad, S. Vélez, X. Sun, F. Casanova, P. Stoliar and L. E. Hueso, Adv. Funct. Mater., 2015, 25, 2972–2979.
67. J. Si, L. Liu, F. Wang, Z. Zhang and L.-M. Peng, ACS Nano, 2016, 10, 6737–6743.
68. L. Liu, L. Ding, D. Zhong, J. Han, S. Wang, Q. Meng, C. Qiu, X. Zhang, L.-M. Peng and Z. Zhang, ACS Nano, 2019, 13, 2526–2535.
69. H. K. Henisch, J. Electrochem. Soc., 1956, 103, 637–643.
70. Y. Saito, M. Okuda, T. Yoshikawa, A. Kasuya and Y. Nishina, J. Phys. Chem., 1994, 98, 6696–6698.
71. H. H. Berger, Solid-State Electron., 1972, 15, 145–158.
72. H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer-Verlag, Berlin, 1996.
73. S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge Univ. Press, Cambridge, 2013.
74. QuantumATK version 12.8.2., Synopsys QuantumATK (https://www.synopsys.com/silicon/quantumatk.html).

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Footnote

† Electronic supplementary information (ESI) available: The supplementary information that accompanies this paper is attached. See DOI: 10.1039/d0nh00119h

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