The size dependence of Young’s modulus and the critical pressure for phase transformation from the diamond structure at ambient pressure to a metallic β-Sn structure at high pressure was studied in a Si phononic crystal. We used dynamic mechanical analysis and in situ electrical characterization with an electrically conducting diamond nanoindentation tip. Experiments on several phononic periodic sizes enabled us to establish that the Young’s modulus and critical phase-transformation pressure decreased as the neck width (periodic structure) of the phononic crystal shrank. The finding enables us to understand the reduction in the thermal conductivity of the Si phononic crystal at the local neck between the nanoholes.
Recent progress in fine processing technology and analytical techniques has made it possible to fabricate nanostructured materials and analyze material properties in the nanoregion.1,2 A characteristic finding in the mechanical property analysis was the atomic deconfinement phenomenon in nanostructured materials, which differs from normal confined bulk materials.3–6 A previous study on thermal and mechanical properties in phononic crystals (PnCs) shows the relationship between a reduction in thermal conductivity and a reduction in Young’s modulus of PnCs.7 On the basis of this knowledge, we set a research target to establish the origin of Young’s modulus reduction by analyzing the mechanical and in situ electrical properties during nanoindentation dynamic tests on single-crystal Si PnCs.
The PnC structure has been fabricated mainly by the semiconductor process, which led us to select the target material as a natural Si base. Accumulated research knowledge exists on the fabrication technique and material property, especially the thermal transport properties of Si PnCs. Almost all material properties of single-crystal Si have been investigated because high-purity crystals have been mass-produced using the Czochralski method since half a century ago. We focused on the mechanical properties, especially on the phase transformation from the diamond structure at ambient pressure to a metallic β-Sn structure at high pressure,8–19 and report the dependence of critical pressure on the structural size of single-crystal Si PnCs for the first time.
The Si PnC sample was fabricated on a SOI (silicon on insulator) wafer, which was prepared by separation by implantation of oxygen with a thickness t = 100 nm in single-crystal Si (100) in the top layer and with a thickness t = 3000 nm in the SiO2 buried-oxide (BOX) layer in the middle. Phosphorus-ion implantation was performed in the SOI wafer at 10 keV and 2.3 × 1015 cm−3, and activation annealing was performed at 900 °C for 30 min. The PnC structure was fabricated on the n-type semiconductive Si (100) layer with various periodic sizes and a ratio of nanohole diameter to neck width of 4:1. The neck width w, which was the minimum length between nanoholes, ranged from 40 to 400 nm [supplementary material, Fig. S1(b)]. To minimize the contact position dependence on the sheet resistance, 5-μm-wide Au electrodes were formed at intervals of 5 µm, and the Si PnC structure was fabricated in a 5 × 5 µm2 area between the Au electrodes [Fig. S1(a)]. All Au electrodes were connected to the Cu sample holder by wire bonding, which means that every position in the fabricated n-type Si (100) PnC had an equally applied electric field [Fig. S1(a)]. The SiO2 BOX layer was retained during the nanoindentation test to yield a load–depth curve, but it was removed by using hydrofluoric acid for transmission electron microscopy (TEM) observations before/after nanoindentation tests to achieve a through-thickness direction observation without cross-sectional cutting and milling. The same fabrication process was applied to control samples without PnC structures, and the material property of the bulk single-crystal Si (100) was obtained by the same method.
The nanoindentation test was performed by using a Hysitron TI 950 TriboIndenter (Bruker Co.), which was equipped with an electrically conductive cube corner-type diamond tip with a radius of curvature R of 80 nm at the end of the tip. Boron ions were doped at ∼1021 cm−3 to achieve p-type conduction of the diamond tip.20 The indentation load P and penetration depth h were controlled to less than 50 µN and 15 nm, respectively, and basic techniques such as the calculation method for the tip radius of curvature R were based on previous studies.21–34 In situ electrical measurements during nanoindentation were performed using the nanoECR (electrical contact resistance, Bruker Co.) system [Fig. S1(c)].35 Constant-voltage measurements were performed during tip loading and unloading and voltage sweep measurements were performed during tip dwelling from −5 to +5 V [Fig. S1(c)]. TEM (JEM-ARM200F, JEOL Ltd.) observations were performed before and after the nanoindentation tests for crystal structure analysis. The SiO2 BOX layer was removed for TEM sample preparation, and the suspended top single-crystal Si layer was folded and attached to a Cu mesh grid. According to the sampling method, the through-thickness direction observation was enabled with a residual thickness t = 100 nm. Annular dark field (ADF) observations were used for atomic distance measurements, lattice image observations were conducted by inverse Fourier transform (IFT) analysis, and selected area electron diffraction (SAED) patterns at the indented surface impression were obtained at a 200 kV accelerating voltage.
The electrical resistivity was measured by a four-terminal method by using four Au electrodes on n-type Si (100) crystals shown in Fig. S1(a) before wire bonding with a Cu sample plate. The I–V result showed good Ohmic properties, and the calculated electrical resistivity was 2.8 mΩ cm [Fig. S1(d)]. The elastic modulus and electric property measurements obtained by the nanoindentation test for the control sample without PnC structures are shown in Fig. 1. The plotted black data in Fig. 1(a) show the indentation load P and penetration depth h data during loading for 5 s up to P = 50 µN, dwelling for 5 s at a constant P = 50 µN, and unloading for 5 s down to P = 0 µN. The plotted red data in Fig. 1(a) show the in situ current value I with a constant voltage V = +5 V during the same nanoindentation test sequence as mentioned above. By analyzing the data in Fig. 1(a), Young’s modulus E yielded the same value as the theoretical Si (100) E = 130 GPa, and the electrical current flow turned on at h = 10 nm. Figure 1(b) shows the voltage sweep measurement during tip dwelling from −5 to +5 V, which suggests that the diode structure formed at the contact between the p-type diamond tip and n-type Si (100) crystal by defining the current flow direction as the forward biased direction. The obtained phenomenon indicated that the Schottky barrier height was high at the initial contact stage around h = 0 nm; however, it decreased at around h = 10 nm with a sufficiently high pressure for phase transformation from the diamond structure at ambient pressure to a metallic β-Sn structure.8–19
The dependence of phase-transformation critical pressure τmax from a diamond to a β-Sn structure for a neck width w in the phononic crystal is shown in Fig. 2. Note that the indentations for these measurements have been carried out at then middle of the neck. Figure 2(a) shows the same sample and data as in Fig. 1 which was from a control sample without PnC structures. The maximum shear stress τmax during indentation occurred at z = 0.48a, which was located immediately beneath the diamond tip, with a value that was calculated as follows: 23,36
where P is the indentation load and a is the contact radius. According to Eq. (1), the maximum shear stress τmax for phase transformation was calculated as τmax = 8.4 GPa. The maximum shear stress indicates the transformation point during uniaxial pressure such as the indentation test; however, the dominant component for phase transformation is hydrostatic pressure, which is conventionally obtained by the diamond-anvil cell test. In order to compare the difference between shear stress and hydrostatic pressure, the mean contact pressure pm under the indentation tip was also calculated as 14.4 GPa according to Meyer’s definition; pm = P/A(hc), where P is the load and A(hc) is the contact area.37 These values are in close agreement with the literature value of 11–12 GPa, and this suggests that phase transformation has taken place under the indenter when a load P of 35 µN is reached at a penetration depth h of 10 nm.8–11
The same analyses were performed for three samples with the parameter of neck width w = 200, 100, and 40 nm to observe the dependence of phase-transformation critical pressure τmax on neck width w. The neck width w is defined as the minimum length in the periodic structure, and we performed all mechanical and in situ electrical measurements at the neck point. Figures 2(b)–2(d) show the results from w = 200, 100, and 40 nm, respectively. All results show the current I = 0 µA at around P = 0 µN, and the current I, which was turned on after the indentation load P was applied, increased. Figure 2 shows that the critical indentation load P that is required to turn on the current I depends on the neck width w, and the reduction in w reduces τmax and pm. τmax and pm were calculated as τmax = 6.9 GPa and pm = 15.7 GPa for w = 200 nm, τmax = 5.1 GPa and pm = 12.9 GPa for w = 100 nm, and τmax = 1.4 GPa and pm = 3.20 GPa for w = 40 nm from Eq. (1) and Meyer’s definition. Young’s moduli E were E = 124 GPa for w = 200 nm, E = 90.6 GPa for w = 100 nm, and E = 40.0 GPa for w = 40 nm from P–h curves in Figs. 2(b)–2(d). Although changes in the critical pressure for phase transformation have been previously observed in diamond anvil studies,38–40 this is the first time they have been observed in nanostructured Si under nanoindentation testing.
To promote a better understanding of the phenomenon of the turning on of current flow from the region of penetration depth h < 1 nm [Fig. 2(d)], we measured the current I and voltage V (shown in the supplementary material, Fig. S2) by the V sweep mode during the dwelling period in the measurement of Fig. 2(d). Figure S2 shows a rectification diode behavior, and based on these results an appropriate set up for the nanoindentation measurement was determined. According to Figs. 2(d) and S2, the Schottky barrier height was higher than 5 V at h = 0 nm; however, the current I was turned on immediately after the load P increased. There are two possibilities for explaining the phenomenon. First is that the phase transformation to generate the metallic β-Sn phase reduced the barrier height and stimulated current flow during a shallow indentation. Second is that electrical breakdown occurred when the volume of material between the tip and the Au contact. Additionally, the electrical current density seems to be very high at the contacting indentation tip with a very small contact area. It is also possible that the local Joule heating with small contact volume stimulates the phase transformation in the surrounding nanostructured Si, which in turn induces a continuous increase in current.
The significant size dependency on the mechanical properties of single-crystal Si, as shown in Fig. 2, led to TEM observations to analyze the atomic distance variation in Si PnCs. ADF images were obtained at every seven points, as indicated in Fig. 3(a) from 1 to 7 (in red), and the atomic distance d was calculated statistically from the signal intensity in the seven lattice images [Figs. 3(b) and 3(c)]. Figures 3(b) and 3(c) are plotted for the same data without an offset [Fig. 3(b)] and with an offset on the intensity axis [Fig. 3(c)]. Figures 3(b) and 3(c) show the same values for d = 0.1920 ± 0.0001 nm from point Nos. 2 and 4, which are defined as the “neck width part,” and from point No. 7, which is defined as the “bulk part.” Therefore, we cannot obtain the d variation data from the averaged position information of ∼520 atoms that were oriented in the through-thickness direction with a thickness t = 100 nm.
The SAED pattern analysis (Fig. 4) and IFT analysis (Fig. 5) were performed at the indented surface impression on Si PnC to detect the residual evidence of phase transformation or the specific plastic deformation mechanism of the nanostructures. Figure 4(a) shows the dark-field image of PnC with a neck width w = 60 nm after the nanoindentation test, and the indented surface impression is indicated by a green square. Figure 4(b) shows the SAED pattern analysis at the indented surface impression as indicated by the framed square in Fig. 4(a) (in green). The SAED pattern shows the existence of a diamond (Si-I) structure; however, it does not show another crystal structure. Figure 5 shows the IFT analysis from lattice images on the surface impression, which is the same point as in Fig. 4(b), and it shows only diamond (Si-I) structures again. One explanation for the electrical behavior is that phase transformation to metallic β-Sn Si (Si-II) occurs on loading, and although they are not detected in the TEM results shown above, bc8 and r8 Si form on unloading. As the contact radius decreases on unloading, the current decreases [Fig. 1(a)], consistent with previous studies in Si.18,19,41–45 To confirm that phase transformation has occurred, cross-sectional TEM would need to be carried out46 on residual indent impressions just following the increase in current. However, it was not possible in the current indentation arrangement to stop the indent immediately after the sudden current rise.
In summary, this study has indicated that a phase transformation from the diamond (Si-I) structure at ambient pressure to the metallic β-Sn (Si-II) structure at high pressure was most likely observed in nanostructured Si PnCs. The critical pressure for the phase transformation (maximum shear stress τmax) decreased with the periodic structure decrease (neck width w) of Si PnCs (Fig. 6). This trend was consistent with Young’s modulus E, where E decreased with a decrease in w (Fig. 6). These facts imply that the PnC nanostructures induce a change in the material mechanical behavior. For example, they stimulated the probability of a decrease in the occurrence of high-pressure phase transformation events and decreased the elastic modulus. This explanation is consistent with the experimental behavior, despite phase transformation not being confirmed by TEM. The obtained TEM information in this study was from macroscopic averaged data of ∼520 atoms in the through-thickness direction for t = 100 nm, so further TEM observation in the microscopic region is required. The relationship between the Young’s modulus and phase-transformation pressure in the nanofabricated phononic crystals (Fig. 6) appears to correlate with the previous study of local reduction7 in thermal conductivity in phononic crystals.
See the supplementary material for more details on nano-mechanical measurements, Young’s modulus calculation, and nano-electrical measurements for Si phononic crystals.
The authors thank Specially Appointed Professor Katsuaki Suganuma for providing permission to use the nanoindentation equipment. The authors would also like to thank Dr. Hiroyuki Tanaka, Dr. Atsushi Himeno, and Mr. Kunihiko Nakamura for fruitful discussions and thank Dr. Laura Kuhar from Edanz for editing a draft of this manuscript.
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are available from the corresponding author upon reasonable request.