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Investigation of quantum confinement within the tunneling-percolation transition for ultrathin bismuth films
Physica B 475 (2015) 117–121
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Investigation of quantum confinement within the tunneling-percolation transition for ultrathin bismuth films Declan Oller n, Gustavo E. Fernandes, Jin Ho Kim, Jimmy Xu School of Engineering, Brown University, Providence, RI 02912, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 8 June 2015 Received in revised form 12 July 2015 Accepted 21 July 2015 Available online 21 July 2015
We investigate conduction phenomena in ultrathin bismuth (Bi) films that are thermally evaporated onto flat quartz. Critical points in the conductance as a function of deposition time are identified and used to scale the data from time dependence to coverage dependence. The resulting nonlinear coverage scaling equation is verified independently via analysis done on transmission electron microscope images of the evaporated films. The scaled data yields critical exponents in very good agreement with classical percolation theory, and clearly shows the transition from the tunneling regime into percolation. Surprisingly, no noticeable signatures of size-quantization effects in the nucleation sites as a function of deposition time is observed in either regime. We discuss our findings in light of Boltzmann transport modeling of 1D conduction as an approximation to the narrow percolative paths that form at the onset of percolation. Our results suggest that lack of a preferred crystallite orientation in the nucleation process may indeed cause quantum-confinement to be too smeared out to be observable in the tunneling to percolation transition. & 2015 Elsevier B.V. All rights reserved.
Keywords: Percolation Bismuth Tunneling Thin films Film morphology Nanomaterials
1. Introduction Ultrathin Bi films (i.e. a few tens of atoms thick) offer a host of opportunities to investigate interesting quantum phenomena not commonly observed in thicker films or in bulk. These include density of states quantization [1], mean-free path suppression in the direction perpendicular to film growth [2], pronounced surface scattering of electrons and phonons which can lead, e.g., to enhanced thermoelectric figures of merit [3], quantum-plasmons arising from the interaction of the carriers with photons [4], and abnormal optical constants [5]. Bi is particularly interesting in this context because it has many unique properties not observed in other materials: it can achieve a negative refractive index in the far infrared range [6], it has a very small effective mass ( ~0.001me ) which gives rise to quantum confinement effects at sizes comparatively much larger and achievable via commonly available lithography techniques than required for most materials, and because it is a semimetal, Bi undergoes a semimetal to semiconductor transition as a function of size-confinement [7]. In addition, thin films of Bi excited by an electron beam have been proposed as a candidate system for generating terahertz radiation [8]. Many previous studies have presented evidence of quantum n
Corresponding author. E-mail address: [email protected] (D. Oller).
http://dx.doi.org/10.1016/j.physb.2015.07.023 0921-4526/& 2015 Elsevier B.V. All rights reserved.
effects in Bi. Magnetoresistance measurements have shown evidence of quantum confinement effects in thin nanopatterned antidot Bi films as well, exemplifying the interesting effect that confinement to various morphologies and “in-between” dimensions can give rise to in Bi [9]. Bi nanowires have been used extensively to demonstrate quantum effects because of their confinement in two directions [10]. Additionally, a critical transition between superconducting and insulating phases has been observed in thin Bi films [11]. In practice, the evolution from a barren substrate to a continuous Bi film begins with the formation of individual nucleation sites (“islands”) that are randomly scattered over the substrate surface. As more material is added, the islands eventually merge to form a continuous film [12]. This structural evolution triggers a host of fundamental changes in the dominant underlying transport mechanisms in the structure. Early on, when only separate islands exist, the dominant transport of electrical current is the tunneling of electrons across the dielectric (or vacuum) barriers that exist between the conductive islands. This regime is accurately described in the framework of the Simmons model [13] and/ or variable-range hopping [14]. These tunneling models predict an exponential increase in the conductance as the island-to-island distances decrease. Percolation theory is needed to effectively describe the conductivity as the Bi film evolves from individual islands to a continuous film. When the areal material coverage reaches a critical amount, several different properties related to the ‘connectedness’
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D. Oller et al. / Physica B 475 (2015) 117–121
of the material across the substrate show divergent and critical behavior that are not well captured in by the tunneling models. These divergences at the critical point have interesting implications, such as a proposed “dielectric anomaly” in the far infrared [5], as well as marked changes in the behavior of the DC conductivity as a function of added material. While percolating systems are well understood theoretically, experimental evidence rigorously corroborating the theory remains scarce. The case for Bi is particularly cumbersome because the material readily oxidizes with exposure to air, requiring measurements to be performed in situ in the vacuum deposition chamber, or the use of passivation layers which can distort results [15–17]. In this paper, we study conductance phenomena in ultrathin Bi films in order to attempt to answer the question as to whether quantum confinement can be observed at some point along the tunneling to percolation transition. Our measurements reveal a consistent transition from tunneling to percolation to bulk conductivity. In the percolation region, critical exponents are found to agree very well with those predicted by percolation theory. However, no clear signatures of a quantization transition are observed. To elucidate our results, we model the effects of quantization on the island conductivity within the percolation regime using Boltzmann transport theory and find it to generally support our experimental findings, provided that certain assumptions are made. We propose that quantization is not observed in the percolation regime because the connected islands that contribute to conduction are necessarily far reaching in the in-plane directions, causing electrons to be mostly confined in the out-of-plane direction. This constraint demands very high confinement and the right crystal orientation to change the conductivity, conditions which are generally not met in the case of an amorphous and polycrystalline film, such as ours.
2. Experiment The substrates used were quartz (Chemglass Life Sciences). The substrate dimensions were 1 mm 6 mm 13 mm. Electrical contact pads were fabricated by evaporation of ~50 nm Au atop a ~5 nm Cr adhesion layer through a shadow mask and covered the entire substrate top surface except for a ~500 μm rectangular strip of width 4 mm that separated the two contact pads where the relevant percolation effects were to be investigated (see section 2a of the Supplementary information). Electrical measurements were conducted in situ inside the thermal evaporator’s steel vacuum chamber using a semiconductor parameter analyzer (Agilent 4156C), as described in section 2b of the Supplementary information. Thermal evaporation of Bi (5N, Kurt J. Lesker) was carried out at a fixed deposition rate of 0.2 Å/s, as measured by an Inficon XTM/2 quartz crystal monitor with a rate resolution of 0.1 Å/s. This slow rate allowed for better resolution in the measurements and control of the final thickness if it was desired to stop evaporation at a given measured current or thickness. The evaporation was done with the substrate at room temperature. The vacuum typically operated in the range of 10-6 Torr. The semiconductor parameter analyzer was configured to measure current at a constant applied voltage of 1 V and at 100 ms intervals. At the applied voltage, the measured current would typically fluctuate ~1 pA around a base value of ~1 pA for a bare substrate. Samples were imaged using a transmission electron microscope (TEM, Philips CM20). TEM measurements were performed on samples that were necessarily removed from vacuum after having deposition stopped at the desired point and thus experienced rapid oxidation in the atmosphere. However, the oxidation would not affect the morphology of the deposited film. For TEM analysis the evaporation substrate was a copper TEM grid with a SiO
Fig. 1. Linear (top) and log (bottom) plots of raw data collected with horizontal axes aligned. Vertical dotted lines indicate different regimes; horizontal line in log plot indicates baseline noise level.
membrane (Ted Pella, Inc.).
3. Results and analysis Fig. 1 shows a typical conductance vs time measurement result plotted in both log and linear plots so that each regime is illustrated. Regime 1 is the region in which tunneling exists but is below the noise level of the system. Regime 2 is the brief period in which tunneling current is measurable but the percolation threshold pc has not yet been reached; though brief in time it covers 8 orders of magnitude in current. Regime 3 is the percolation regime, which will be described in more detail. Regime 4 is the bulk regime in which the substrate area is sufficiently covered and most of the additional material added contributes only to an increase in film thickness. It is easily characterized by its linear nature. We begin by fitting an exponential relation to the tunneling regime to identify the end of this regime which coincides with the beginning of the percolation regime. The fitted curves and equations can be seen in section 4 of the Supplementary information; they all exhibit unambiguous exponential behavior. Once the end of the tunneling regime is determined, the data for the percolation regime can be fitted. The raw data consists of current measured at a constant voltage with respect to time. As discussed in section 3 of the Supplementary information, to fit the data to the canonical percolation equation, the data’s time variable, t , must be transformed into surface coverage of the form p = 1 − e−Rt , where R is a constant related to the deposition rate. To first verify this relation, we deposited nominal thicknesses (i.e., as reported by the crystal thickness monitor, as if the deposited material formed a complete uniform film immediately) in the range 3–200 Å on SiO membrane TEM grids. High contrast and resolution TEM images shown in Fig. 2 of different amounts of deposited material were then collected and the surface coverage as a function of time was measured by performing image analysis, as detailed in section 7a of the Supplementary information. These coverages are plotted with respect to their deposition times, as shown in Fig. 2. The multiple points at each time correspond to the same sample, but with the image analysis performed on images of different magnifications. Once the time to surface coverage transformation has been
D. Oller et al. / Physica B 475 (2015) 117–121
Fig. 2. TEM images (top) after different deposition times, all magnifications are 58kx. Nominal deposited thicknesses are (a) 10 Å, (b) 40 Å, (c) 80 Å, (d) 130 Å. All scale bars are 400 nm long. Coverage vs time plot (bottom) calculated from TEM images like the above, and curve of best fit for the relation 1 e Rt. Multiple points at each time are for analysis done at different magnifications (34kx, 58kx, 100kx, 175kx). The points calculated from the TEM images presented are shown as red squares on the plot. The best fit curve gives a value of R¼ 0.00337 s 1.
applied, the data can be fitted to the conductivity equation σ = σ0 (p − pc )q [18] for pc ≤ p ≤ 1, where σ0 is a constant dependent on the material and several other factors and q is the critical exponent for percolation in two dimensions. Note that the standard approach for fitting power laws by finding the slope of the data on a Log/Log plot cannot immediately be used because the independent variable p is offset by pc , and so the data will not appear linear on a Log/Log plot. Thus, we subtract pc from every value of p in the data set for p ≥ pc and call it Δp = p − pc so that the conductivity equation now has the form σ = σ0 (Δp)q , where Δp has the bounds 0 ≤ Δp ≤ 1 − pc = 0.32. However (discussed in section 3 of the Supplementary information), when coverage is almost complete, most newly added material will cause the film to grow in the out-of-plane direction. Thus, it is expected that the fitting of data to Δp must be cut off significantly before Δp = 0.32 ( p = 1), because results for Δp ≈ 0.32 will not represent percolation and thus will give nonphysical results if that model is applied. Analysis of the data
(Fig. 3) reveals that the valid range for fitting the percolation power law is significantly less than 0.32. This is clearly seen from the power law fits (section 5 of the Supplementary information) to various ranges of the data. Two sources of error must be identified in the data presented. First, the percolation onset occurs at different times between runs, despite the constant evaporation rate, and secondly, in the “bulk” regime for each sample where the current is growing linearly with time, the slopes are different. These two sources of error can both be attributed to unsteadiness of the evaporation rate. The thickness monitor rate has a resolution of only 0.1 Å/s, so for a reasonable uncertainty estimate of half that, the percolation onset time could potentially vary from 20% below to 33% above the expected time based on the reported constant rate. The onset times in this set of data range from 221–352 s with a mean of 299.5 s. This gives a range of evaporation rates 17% below the mean to 26% above the mean rate, which is well within what could be expected from the rate resolution of the thickness monitor. To get an estimate of the dimensions at which quantum confinement occurs for this system, we simulate Bi nanowires of varying cross sectional dimensions in the Boltzmann transport framework [1,3]. Our leading assumption is that the resistance of a percolative path is determined by structural constrictions or bottlenecks linking the nucleation islands. At some point in the percolation regime these constrictions are quantum confined in the cross sectional dimensions and thus effectively resemble nanowires. As more material is added, the thickness and width of these ‘nanowires’ increase and they become less quantized. Fig. 4 in the Supplementary information shows the conductivity of a square nanowire of growing diameter for two temperatures, 77 K and 300 K. At 77 K, the nanowire is effectively a semiconductor below 50 nm in diameter, and reaches bulk conductivity at ~150 nm in diameter. At 300 K, even though at small diameters the Fermi level is located inside the band gap, the temperature is so high that the Fermi function smearing leads to significant population of the conduction band, resulting in conductivity that is significantly larger at the same diameter compared to the nanowire at 77 K – at 300 K the nanowire behaves as a semiconductor only for diameters below 20 nm, and is effectively bulk at ~80 nm . To investigate if measurement of the SMSC transition during the percolation regime is feasible, we next analyze the average dimensions of the paths immediately after the percolation threshold. It is assumed here that each path is essentially rectangular and can be characterized by a ‘thickness’ d in the out of plane direction and a ‘width’ w in the in-plane direction, as illustrated in Fig. 4. The thickness d is easily estimated (derivation in section 3 of Supplemental information):
d=
Fig. 3. Log–Log plot (bottom) of multiple data sets after being scaled to coverage and offset by the percolation threshold pc. Dashed lines are curves of best fit for a fitting range of 0o Δp o0.20. Note how the fits begin to diverge from the data above Δp ¼0.20. Top: fits to each data set for fitting procedure applied to different ranges. Note that the mean exponent over the data sets is close to the predicted value for the different fitting regions up until Δp ¼ 0.30.
119
(
R dep −
Log (1 −p) R
)
p
where Rdep is the actual material deposition rate (measured in Å/s ). For the values found previously, the thickness at the percolation threshold is d ≈ 100 Å , as shown in section 7b of the Supplementary information. To estimate the average width of the percolating paths, we perform image analysis on TEM images of different amounts of deposited material. This is not a trivial task because even though in the aggregate the paths may be similar to nanowires, quantitatively, an effective average width is not a welldefined concept. We use two techniques (described in section 7b of the Supplemental information) that independently agree with each other to estimate the average width of the paths at different levels of surface coverage. For each coverage, we get the average width of the ensemble of clusters, weighted by the area of each cluster. It is clear from this data (section 7b, Supplementary
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Fig. 4. Upper left: TEM image with typical cluster highlighted. Scale bar is 100 nm and magnification is 100kx. Bottom: nanowire analogous to typical cluster, with important dimensions and orientations labeled. Upper left: tables of conductivity illustrating the effect of confinement in either and both growth directions for the two major crystal orientations based on the effective mass tensor. Units are Ω 1 m 1.
information) that each average converges to the same value range around 60 nm above the percolation threshold. This analysis suggests that a reasonable estimate for the average “nanowire” dimensions at the percolation threshold are 10 nm in thickness and 60 nm in width, and that by what was concluded to be the “end” of the percolation regime (i.e., where bulk growth takes over, ~92% ), those average dimensions are 17 nm thick by 90 nm wide. To illustrate that the conductivity does not change significantly as a result of this growth, we simulate the conductivity of a nanowire with the dimensions before and after the growth. For each crystal orientation, the table in Fig. 4 shows the conductivity before and after the relevant growth amounts for both the in- and out-of-plane directions separately, and then together. The results indicate that quantum confinement only significantly affects the conductivity when the crystal is oriented such that the lightest effective mass m1 corresponds to the more confined out-of-plane direction. Our interpretation of this result is as follows. For this orientation, the width is already large enough that it is effectively almost completely unconfined, so increasing it further has very little effect. For the other crystal orientation, with the larger effective mass m2 in the out-of-plane direction, energy band splitting from confinement in the out-of-plane direction is not as significant, while for the in-plane direction (with mass m1) it is also almost unconfined at the onset of percolation. Although the increase in conductivity for the orientation with m1 in the out-of-plane direction is significant, we expect the overall orientation of the grains to constitute an ensemble of different crystal orientations such as the ones above, some of which have very high resistance and some very low. In addition, only orientations for conduction along the m3 have been considered because only this conduction direction allows for the SMSC transition. In reality, however, the ensemble of grains will also include ones that have no possibility of crossing the SMSC transition at all. Thus, the possible confinement from this one orientation of grains will not affect the overall conductivity unless they constituted the
vast majority. X-ray diffraction measurements done in [19] for thin films of Bi on glass revealed that the vast majority of the grains were oriented with their trigonal directions perpendicular to the substrate, thus supporting the lack of quantum confinement effects in the measurements.
4. Conclusions In this paper we have identified the tunneling and bulk regimes, which will be useful in future studies in which we probe the boundaries between these regimes. The “dielectric anomaly” reported to have been observed in Au and Pb ultra-thin metal films occurred in the infrared [5], so it is interesting to consider at what frequency range it could be observed for Bi, which has a plasma frequency in the terahertz range an order of magnitude below that of Au [20]. We have determined the percolation threshold and verified the correct relation needed to transform the raw data into a surface coverage that can be fitted to, which allowed us to show that the experiment done with thermally evaporated Bi on a quartz substrate follows the percolation conductivity equation for 2 dimensions. Simulations and image analysis have corroborated the experimental data to demonstrate that significant quantum confinement effects are not present in the percolation regime. However, it is an interesting question as to whether the effects could be present if a mechanism such as a wetting layer or substrate temperature control caused the evaporated island size to change significantly or islands crystallization was influenced to have a different orientation. This study could also be extended to explain the critical phenomenon observed in [11] in which the radii of Cooper pairs could be framed as the “islands” that form a percolating material.
D. Oller et al. / Physica B 475 (2015) 117–121
Acknowledgments The authors would like to thank the NSF, AFOSR, AOARD, and ARO for their generous support.
[9]
[10]
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.physb.2015.07.023.
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[1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [2] M.G. Blaber, M.D. Arnold, M.J. Ford, Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver, J. Phys. Chem. C 113 (2009) 3041–3045. [3] X. Sun, Z. Zhang, M.S. Dresselhaus, Theoretical modeling of thermoelectricity in Bi nanowires, Appl. Phys. Lett. 74 (1999) 4005–4007. [4] W.P. Halperin, Quantum size effects in metal particles, Rev. Mod. Phys. 58 (1986) 533–606. [5] J.J. Tu, C.C. Homes, M. Strongin, Optical properties of ultrathin films: Evidence for a dielectric anomaly at the insulator-to-metal transition, Physical Review Letters 90 (2003) 017402. [6] V.A. Podolskiy, L.V. Alekseyev, E.E. Narimanov, Strongly anisotropic media: the THz perspectives of left-handed materials, J. Mod. Opt. 52 (2005) 2343–2349. [7] C.A. Hoffman, J.R. Meyer, F.J. Bartoli, A. Di Venere, X.J. Yi, C.L. Hou, et al., Semimetal-to-semiconductor transition in bismuth thin films, Phys. Rev. B 48 (1993) 11431–11434. [8] J. Parashar, S. Chauhan, P.K. Purohit, V.K. Agrawal, Terahertz surface plasmon
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excitation over a bismuth thin film by an electron beam, J. Phys. Chem. Solids 74 (2013) 1751–1755. Y. Park, Y. Hirose, S. Nakao, T. Fukumura, J. Xu, T. Hasegawa, Quantum confinement effect in Bi anti-dot thin films with tailored pore wall widths and thicknesses, Appl. Phys. Lett. 104 (2014) 023106. M.R. Black, Y.M. Lin, S.B. Cronin, O. Rabin, M.S. Dresselhaus, Infrared absorption in bismuth nanowires resulting from quantum confinement, Phys. Rev. B 65 (2002) 195417. M.D. Stewart, A. Yin, J.M. Xu, J.M. Valles, Superconducting pair correlations in an amorphous insulating nanohoneycomb film, Science 318 (2007) 1273–1275. K. Oura, Surface Science: An Introduction, Springer, Berlin; New York, 2003. J.G. Simmons, Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film, J. Appl. Phys. 34 (1963) 1793–1803. N.F. Mott, Conduction in non-crystalline materials, Philos. Mag. 19 (1969) 835–852. J. Toudert, R. Serna, M. Jiménez de Castro, Exploring the optical potential of nano-bismuth: tunable surface plasmon resonances in the near ultraviolet-tonear infrared range, J. Phys. Chem. C 116 (2012) 20530–20539. A.D.F. Dunbar, J.G. Partridge, M. Schulze, S. Scott, S.A. Brown, Measurement of the conductivity exponent in random percolating networks of nanoscale bismuth clusters, in: Proceedings International Conference on MEMS, NANO and Smart Systems, 2003, pp. 350–356. N.T. Liang, Y. Shan, S.-y Wang, Electronic conductivity and percolation theory in aggregated films, Phys. Rev. Lett. 37 (1976) 526–529. J.P. Clerc, G. Giraud, J.M. Laugier, J.M. Luck, The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models, Adv. Phys. 39 (1990) 191–309. P. Mikolajczak, W. Piasek, M. Subotowicz, Thermoelectric power in bismuth thin films, Phys. status solidi (a) 25 (1974) 619–628. I. Timrov, T. Kampfrath, J. Faure, N. Vast, C.R. Ast, C. Frischkorn, et al., Thermalization of photoexcited carriers in bismuth investigated by time-resolved terahertz spectroscopy and ab initio calculations, Phys. Rev. B 85 (2012) 155139.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Investigation of quantum confinement within the tunneling-percolation transition for ultrathin bismuth films Declan Oller n, Gustavo E. Fernandes, Jin Ho Kim, Jimmy Xu School of Engineering, Brown University, Providence, RI 02912, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 8 June 2015 Received in revised form 12 July 2015 Accepted 21 July 2015 Available online 21 July 2015
We investigate conduction phenomena in ultrathin bismuth (Bi) films that are thermally evaporated onto flat quartz. Critical points in the conductance as a function of deposition time are identified and used to scale the data from time dependence to coverage dependence. The resulting nonlinear coverage scaling equation is verified independently via analysis done on transmission electron microscope images of the evaporated films. The scaled data yields critical exponents in very good agreement with classical percolation theory, and clearly shows the transition from the tunneling regime into percolation. Surprisingly, no noticeable signatures of size-quantization effects in the nucleation sites as a function of deposition time is observed in either regime. We discuss our findings in light of Boltzmann transport modeling of 1D conduction as an approximation to the narrow percolative paths that form at the onset of percolation. Our results suggest that lack of a preferred crystallite orientation in the nucleation process may indeed cause quantum-confinement to be too smeared out to be observable in the tunneling to percolation transition. & 2015 Elsevier B.V. All rights reserved.
Keywords: Percolation Bismuth Tunneling Thin films Film morphology Nanomaterials
1. Introduction Ultrathin Bi films (i.e. a few tens of atoms thick) offer a host of opportunities to investigate interesting quantum phenomena not commonly observed in thicker films or in bulk. These include density of states quantization [1], mean-free path suppression in the direction perpendicular to film growth [2], pronounced surface scattering of electrons and phonons which can lead, e.g., to enhanced thermoelectric figures of merit [3], quantum-plasmons arising from the interaction of the carriers with photons [4], and abnormal optical constants [5]. Bi is particularly interesting in this context because it has many unique properties not observed in other materials: it can achieve a negative refractive index in the far infrared range [6], it has a very small effective mass ( ~0.001me ) which gives rise to quantum confinement effects at sizes comparatively much larger and achievable via commonly available lithography techniques than required for most materials, and because it is a semimetal, Bi undergoes a semimetal to semiconductor transition as a function of size-confinement [7]. In addition, thin films of Bi excited by an electron beam have been proposed as a candidate system for generating terahertz radiation [8]. Many previous studies have presented evidence of quantum n
Corresponding author. E-mail address: [email protected] (D. Oller).
http://dx.doi.org/10.1016/j.physb.2015.07.023 0921-4526/& 2015 Elsevier B.V. All rights reserved.
effects in Bi. Magnetoresistance measurements have shown evidence of quantum confinement effects in thin nanopatterned antidot Bi films as well, exemplifying the interesting effect that confinement to various morphologies and “in-between” dimensions can give rise to in Bi [9]. Bi nanowires have been used extensively to demonstrate quantum effects because of their confinement in two directions [10]. Additionally, a critical transition between superconducting and insulating phases has been observed in thin Bi films [11]. In practice, the evolution from a barren substrate to a continuous Bi film begins with the formation of individual nucleation sites (“islands”) that are randomly scattered over the substrate surface. As more material is added, the islands eventually merge to form a continuous film [12]. This structural evolution triggers a host of fundamental changes in the dominant underlying transport mechanisms in the structure. Early on, when only separate islands exist, the dominant transport of electrical current is the tunneling of electrons across the dielectric (or vacuum) barriers that exist between the conductive islands. This regime is accurately described in the framework of the Simmons model [13] and/ or variable-range hopping [14]. These tunneling models predict an exponential increase in the conductance as the island-to-island distances decrease. Percolation theory is needed to effectively describe the conductivity as the Bi film evolves from individual islands to a continuous film. When the areal material coverage reaches a critical amount, several different properties related to the ‘connectedness’
118
D. Oller et al. / Physica B 475 (2015) 117–121
of the material across the substrate show divergent and critical behavior that are not well captured in by the tunneling models. These divergences at the critical point have interesting implications, such as a proposed “dielectric anomaly” in the far infrared [5], as well as marked changes in the behavior of the DC conductivity as a function of added material. While percolating systems are well understood theoretically, experimental evidence rigorously corroborating the theory remains scarce. The case for Bi is particularly cumbersome because the material readily oxidizes with exposure to air, requiring measurements to be performed in situ in the vacuum deposition chamber, or the use of passivation layers which can distort results [15–17]. In this paper, we study conductance phenomena in ultrathin Bi films in order to attempt to answer the question as to whether quantum confinement can be observed at some point along the tunneling to percolation transition. Our measurements reveal a consistent transition from tunneling to percolation to bulk conductivity. In the percolation region, critical exponents are found to agree very well with those predicted by percolation theory. However, no clear signatures of a quantization transition are observed. To elucidate our results, we model the effects of quantization on the island conductivity within the percolation regime using Boltzmann transport theory and find it to generally support our experimental findings, provided that certain assumptions are made. We propose that quantization is not observed in the percolation regime because the connected islands that contribute to conduction are necessarily far reaching in the in-plane directions, causing electrons to be mostly confined in the out-of-plane direction. This constraint demands very high confinement and the right crystal orientation to change the conductivity, conditions which are generally not met in the case of an amorphous and polycrystalline film, such as ours.
2. Experiment The substrates used were quartz (Chemglass Life Sciences). The substrate dimensions were 1 mm 6 mm 13 mm. Electrical contact pads were fabricated by evaporation of ~50 nm Au atop a ~5 nm Cr adhesion layer through a shadow mask and covered the entire substrate top surface except for a ~500 μm rectangular strip of width 4 mm that separated the two contact pads where the relevant percolation effects were to be investigated (see section 2a of the Supplementary information). Electrical measurements were conducted in situ inside the thermal evaporator’s steel vacuum chamber using a semiconductor parameter analyzer (Agilent 4156C), as described in section 2b of the Supplementary information. Thermal evaporation of Bi (5N, Kurt J. Lesker) was carried out at a fixed deposition rate of 0.2 Å/s, as measured by an Inficon XTM/2 quartz crystal monitor with a rate resolution of 0.1 Å/s. This slow rate allowed for better resolution in the measurements and control of the final thickness if it was desired to stop evaporation at a given measured current or thickness. The evaporation was done with the substrate at room temperature. The vacuum typically operated in the range of 10-6 Torr. The semiconductor parameter analyzer was configured to measure current at a constant applied voltage of 1 V and at 100 ms intervals. At the applied voltage, the measured current would typically fluctuate ~1 pA around a base value of ~1 pA for a bare substrate. Samples were imaged using a transmission electron microscope (TEM, Philips CM20). TEM measurements were performed on samples that were necessarily removed from vacuum after having deposition stopped at the desired point and thus experienced rapid oxidation in the atmosphere. However, the oxidation would not affect the morphology of the deposited film. For TEM analysis the evaporation substrate was a copper TEM grid with a SiO
Fig. 1. Linear (top) and log (bottom) plots of raw data collected with horizontal axes aligned. Vertical dotted lines indicate different regimes; horizontal line in log plot indicates baseline noise level.
membrane (Ted Pella, Inc.).
3. Results and analysis Fig. 1 shows a typical conductance vs time measurement result plotted in both log and linear plots so that each regime is illustrated. Regime 1 is the region in which tunneling exists but is below the noise level of the system. Regime 2 is the brief period in which tunneling current is measurable but the percolation threshold pc has not yet been reached; though brief in time it covers 8 orders of magnitude in current. Regime 3 is the percolation regime, which will be described in more detail. Regime 4 is the bulk regime in which the substrate area is sufficiently covered and most of the additional material added contributes only to an increase in film thickness. It is easily characterized by its linear nature. We begin by fitting an exponential relation to the tunneling regime to identify the end of this regime which coincides with the beginning of the percolation regime. The fitted curves and equations can be seen in section 4 of the Supplementary information; they all exhibit unambiguous exponential behavior. Once the end of the tunneling regime is determined, the data for the percolation regime can be fitted. The raw data consists of current measured at a constant voltage with respect to time. As discussed in section 3 of the Supplementary information, to fit the data to the canonical percolation equation, the data’s time variable, t , must be transformed into surface coverage of the form p = 1 − e−Rt , where R is a constant related to the deposition rate. To first verify this relation, we deposited nominal thicknesses (i.e., as reported by the crystal thickness monitor, as if the deposited material formed a complete uniform film immediately) in the range 3–200 Å on SiO membrane TEM grids. High contrast and resolution TEM images shown in Fig. 2 of different amounts of deposited material were then collected and the surface coverage as a function of time was measured by performing image analysis, as detailed in section 7a of the Supplementary information. These coverages are plotted with respect to their deposition times, as shown in Fig. 2. The multiple points at each time correspond to the same sample, but with the image analysis performed on images of different magnifications. Once the time to surface coverage transformation has been
D. Oller et al. / Physica B 475 (2015) 117–121
Fig. 2. TEM images (top) after different deposition times, all magnifications are 58kx. Nominal deposited thicknesses are (a) 10 Å, (b) 40 Å, (c) 80 Å, (d) 130 Å. All scale bars are 400 nm long. Coverage vs time plot (bottom) calculated from TEM images like the above, and curve of best fit for the relation 1 e Rt. Multiple points at each time are for analysis done at different magnifications (34kx, 58kx, 100kx, 175kx). The points calculated from the TEM images presented are shown as red squares on the plot. The best fit curve gives a value of R¼ 0.00337 s 1.
applied, the data can be fitted to the conductivity equation σ = σ0 (p − pc )q [18] for pc ≤ p ≤ 1, where σ0 is a constant dependent on the material and several other factors and q is the critical exponent for percolation in two dimensions. Note that the standard approach for fitting power laws by finding the slope of the data on a Log/Log plot cannot immediately be used because the independent variable p is offset by pc , and so the data will not appear linear on a Log/Log plot. Thus, we subtract pc from every value of p in the data set for p ≥ pc and call it Δp = p − pc so that the conductivity equation now has the form σ = σ0 (Δp)q , where Δp has the bounds 0 ≤ Δp ≤ 1 − pc = 0.32. However (discussed in section 3 of the Supplementary information), when coverage is almost complete, most newly added material will cause the film to grow in the out-of-plane direction. Thus, it is expected that the fitting of data to Δp must be cut off significantly before Δp = 0.32 ( p = 1), because results for Δp ≈ 0.32 will not represent percolation and thus will give nonphysical results if that model is applied. Analysis of the data
(Fig. 3) reveals that the valid range for fitting the percolation power law is significantly less than 0.32. This is clearly seen from the power law fits (section 5 of the Supplementary information) to various ranges of the data. Two sources of error must be identified in the data presented. First, the percolation onset occurs at different times between runs, despite the constant evaporation rate, and secondly, in the “bulk” regime for each sample where the current is growing linearly with time, the slopes are different. These two sources of error can both be attributed to unsteadiness of the evaporation rate. The thickness monitor rate has a resolution of only 0.1 Å/s, so for a reasonable uncertainty estimate of half that, the percolation onset time could potentially vary from 20% below to 33% above the expected time based on the reported constant rate. The onset times in this set of data range from 221–352 s with a mean of 299.5 s. This gives a range of evaporation rates 17% below the mean to 26% above the mean rate, which is well within what could be expected from the rate resolution of the thickness monitor. To get an estimate of the dimensions at which quantum confinement occurs for this system, we simulate Bi nanowires of varying cross sectional dimensions in the Boltzmann transport framework [1,3]. Our leading assumption is that the resistance of a percolative path is determined by structural constrictions or bottlenecks linking the nucleation islands. At some point in the percolation regime these constrictions are quantum confined in the cross sectional dimensions and thus effectively resemble nanowires. As more material is added, the thickness and width of these ‘nanowires’ increase and they become less quantized. Fig. 4 in the Supplementary information shows the conductivity of a square nanowire of growing diameter for two temperatures, 77 K and 300 K. At 77 K, the nanowire is effectively a semiconductor below 50 nm in diameter, and reaches bulk conductivity at ~150 nm in diameter. At 300 K, even though at small diameters the Fermi level is located inside the band gap, the temperature is so high that the Fermi function smearing leads to significant population of the conduction band, resulting in conductivity that is significantly larger at the same diameter compared to the nanowire at 77 K – at 300 K the nanowire behaves as a semiconductor only for diameters below 20 nm, and is effectively bulk at ~80 nm . To investigate if measurement of the SMSC transition during the percolation regime is feasible, we next analyze the average dimensions of the paths immediately after the percolation threshold. It is assumed here that each path is essentially rectangular and can be characterized by a ‘thickness’ d in the out of plane direction and a ‘width’ w in the in-plane direction, as illustrated in Fig. 4. The thickness d is easily estimated (derivation in section 3 of Supplemental information):
d=
Fig. 3. Log–Log plot (bottom) of multiple data sets after being scaled to coverage and offset by the percolation threshold pc. Dashed lines are curves of best fit for a fitting range of 0o Δp o0.20. Note how the fits begin to diverge from the data above Δp ¼0.20. Top: fits to each data set for fitting procedure applied to different ranges. Note that the mean exponent over the data sets is close to the predicted value for the different fitting regions up until Δp ¼ 0.30.
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(
R dep −
Log (1 −p) R
)
p
where Rdep is the actual material deposition rate (measured in Å/s ). For the values found previously, the thickness at the percolation threshold is d ≈ 100 Å , as shown in section 7b of the Supplementary information. To estimate the average width of the percolating paths, we perform image analysis on TEM images of different amounts of deposited material. This is not a trivial task because even though in the aggregate the paths may be similar to nanowires, quantitatively, an effective average width is not a welldefined concept. We use two techniques (described in section 7b of the Supplemental information) that independently agree with each other to estimate the average width of the paths at different levels of surface coverage. For each coverage, we get the average width of the ensemble of clusters, weighted by the area of each cluster. It is clear from this data (section 7b, Supplementary
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Fig. 4. Upper left: TEM image with typical cluster highlighted. Scale bar is 100 nm and magnification is 100kx. Bottom: nanowire analogous to typical cluster, with important dimensions and orientations labeled. Upper left: tables of conductivity illustrating the effect of confinement in either and both growth directions for the two major crystal orientations based on the effective mass tensor. Units are Ω 1 m 1.
information) that each average converges to the same value range around 60 nm above the percolation threshold. This analysis suggests that a reasonable estimate for the average “nanowire” dimensions at the percolation threshold are 10 nm in thickness and 60 nm in width, and that by what was concluded to be the “end” of the percolation regime (i.e., where bulk growth takes over, ~92% ), those average dimensions are 17 nm thick by 90 nm wide. To illustrate that the conductivity does not change significantly as a result of this growth, we simulate the conductivity of a nanowire with the dimensions before and after the growth. For each crystal orientation, the table in Fig. 4 shows the conductivity before and after the relevant growth amounts for both the in- and out-of-plane directions separately, and then together. The results indicate that quantum confinement only significantly affects the conductivity when the crystal is oriented such that the lightest effective mass m1 corresponds to the more confined out-of-plane direction. Our interpretation of this result is as follows. For this orientation, the width is already large enough that it is effectively almost completely unconfined, so increasing it further has very little effect. For the other crystal orientation, with the larger effective mass m2 in the out-of-plane direction, energy band splitting from confinement in the out-of-plane direction is not as significant, while for the in-plane direction (with mass m1) it is also almost unconfined at the onset of percolation. Although the increase in conductivity for the orientation with m1 in the out-of-plane direction is significant, we expect the overall orientation of the grains to constitute an ensemble of different crystal orientations such as the ones above, some of which have very high resistance and some very low. In addition, only orientations for conduction along the m3 have been considered because only this conduction direction allows for the SMSC transition. In reality, however, the ensemble of grains will also include ones that have no possibility of crossing the SMSC transition at all. Thus, the possible confinement from this one orientation of grains will not affect the overall conductivity unless they constituted the
vast majority. X-ray diffraction measurements done in [19] for thin films of Bi on glass revealed that the vast majority of the grains were oriented with their trigonal directions perpendicular to the substrate, thus supporting the lack of quantum confinement effects in the measurements.
4. Conclusions In this paper we have identified the tunneling and bulk regimes, which will be useful in future studies in which we probe the boundaries between these regimes. The “dielectric anomaly” reported to have been observed in Au and Pb ultra-thin metal films occurred in the infrared [5], so it is interesting to consider at what frequency range it could be observed for Bi, which has a plasma frequency in the terahertz range an order of magnitude below that of Au [20]. We have determined the percolation threshold and verified the correct relation needed to transform the raw data into a surface coverage that can be fitted to, which allowed us to show that the experiment done with thermally evaporated Bi on a quartz substrate follows the percolation conductivity equation for 2 dimensions. Simulations and image analysis have corroborated the experimental data to demonstrate that significant quantum confinement effects are not present in the percolation regime. However, it is an interesting question as to whether the effects could be present if a mechanism such as a wetting layer or substrate temperature control caused the evaporated island size to change significantly or islands crystallization was influenced to have a different orientation. This study could also be extended to explain the critical phenomenon observed in [11] in which the radii of Cooper pairs could be framed as the “islands” that form a percolating material.
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Acknowledgments The authors would like to thank the NSF, AFOSR, AOARD, and ARO for their generous support.
[9]
[10]
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.physb.2015.07.023.
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