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Critical Behavior in Doping-Driven Metal–Insulator Transition on Single-Crystalline Organic Mott-FET Yoshiaki Sato, Yoshitaka Kawasugi, Masayuki Suda, Hiroshi M. Yamamoto, and Reizo Kato Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b03817 • Publication Date (Web): 30 Dec 2016 Downloaded from http://pubs.acs.org on December 31, 2016

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Critical Behavior in Doping-Driven Metal–Insulator Transition on Single-Crystalline Organic Mott-FET Yoshiaki Sato,*,† Yoshitaka Kawasugi, *,† Masayuki Suda,†,‡ Hiroshi M. Yamamoto,*,†,‡ and Reizo Kato†

†

‡

RIKEN, Hirosawa, Wako, Saitama 351-0198, Japan

Research Center for Integrative Molecular System (CIMoS), Institute for Molecular Science, 38 Nishigo-naka, Myodaiji, Okazaki 444-8585, Japan

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ABSTRACT

We present the carrier transport properties in the vicinity of a doping-driven Mott transition observed at a field-effect transistor (FET) channel using a single crystal of the typical two-dimensional organic Mott insulator κ-(BEDT-TTF)2CuN(CN)2Cl (κ-Cl). The FET shows a continuous metal–insulator transition (MIT) as electrostatic doping proceeds. The phase transition appears to involve two-step crossovers, one in Hall measurement and the other in conductivity measurement. The crossover in conductivity occurs around the conductance quantum /ℎ, and hence is not associated with ‘bad metal’ behavior, in stark contrast to the MIT in half-filled organic Mott insulators or that in doped inorganic Mott insulators. Through in-depth scaling analysis of the conductivity, it is found that the above carrier transport properties in the vicinity of the MIT can be described by a high-temperature Mott quantum critical crossover, which is theoretically argued to be a ubiquitous feature of various types of Mott transitions.

KEYWORDS: Mott transition, field-effect transistor, SAM patterning, metal–insulator transition, Hall effect, quantum critical scaling

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MAIN TEXT Strongly correlated electrons confined in two-dimensional materials exhibit a rich variety of anomalous phases, such as high-Tc superconductivity, Mott insulators, and quantum spin liquids. Enormous effort has been devoted to understanding the mechanisms of the phase transitions between them,1,2 which are driven by the complex interplay between many-body interactions and charge–spin fluctuations. Mott insulators are particularly remarkable in viewpoints of nanoscience as well as fundamental physics since they are not only mother materials for a diverse range of strongly correlated systems, but also serve as a molecular or atomic limit for coherently coupled arrays of single electron transistors (SETs).3,4 In Mott insulating state, strong Coulomb repulsion between electrons in half-filled band leads to carrier localization, which can be viewed as a situation that the collective Coulomb blockade confines an electron to the individual ‘subnano-sized SET’. The electronic/magnetic properties of Mott insulators are governed by quantum mechanics, and indeed various types of unconventional phase transitions occur due to imbalance in the particle–wave duality of electrons which is triggered by charge doping,5 a magnetic/electric field6,7 pressure8,9 or introduction of disorders.10,11 An important issue is the possibility of quantum phase transitions in Mott insulators,12,13 where quantum fluctuations play an essential role in the transitions with the emergence of quantum critical points at 0 K.14–18 This would hold true for a coherently coupled array of SETs, although it has not yet been recognized. 3


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Even at finite temperatures, quantum fluctuations with an energy scale overwhelming the temperature give rise to a critical crossover feature observed as a power-law singularity in physical quantities in the vicinity of the quantum critical point.13,19 It has been argued that the unconventional non-Fermi liquid behavior commonly observed near the Mott transition should pertain to such quantum criticality, although there is no general consensus on its origin. To properly investigate the doping-driven Mott transition, it is required to dope carriers in a clean manner without introducing impurities since the modification of the local electronic environment by such disorders should destroy the intrinsic many-body states. Electrostatic doping techniques using the electric double-layer transistor (EDLT) or field-effect transistor (FET) configurations are promising for such investigation. Combined with recent advances in fabrication technology for crystalline nanosheets1 and interfaces,20–22 these techniques not only enable fine control of the charge density as an independent tunable parameter, but also extend the possibility of novel device applications that utilize abrupt changes in physical properties triggered by a phase transition. The organic Mott insulator κ-(BEDT-TTF)2CuN(CN)2Cl, referred to as κ-Cl, is one of the two-dimensional materials suitable for the channel of electrostatic doping devices. The crystal structure of κ-Cl consists of alternately stacked π-conducting [(BEDT-TTF)2]+ cation layers and atomically thin insulating polyanion layers (Figure 1b), showing excellent crystallinity.23,24 An effectively half-filled 4


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band is formed by a carrier density of one hole per BEDT-TTF dimer. As the areal carrier density (∼ 10 cm ) is almost an order of magnitude lower than those of inorganic Mott insulators, a large region of the phase diagram can be investigated by electrostatic doping. Indeed, we have demonstrated that by charge doping up to a density of ∼ 10 cm using an EDLT, both electron- and hole-doping-driven Mott transitions were successfully realized on a single κ-Cl crystal.25 Nonetheless, the EDLT technique is inconvenient for fine tuning of the band filling at low T owing to freezing of the ionic liquid.26 On the other hand, a solid-gate FET is highly advantageous in terms of the applicability of various micrometer-scale interface engineering methods as well as the high controllability of the doping level in real time at low T. In this Letter, we report a doping-driven Mott MIT on a solid-gate FET using a single crystal with a highly flat surface as well as a gate dielectric covered with self-assembled monolayers (SAMs). We selected hydrophobic 1H,1H,2H,2H-perfluorodecyltriethoxysilane (PFES) and octyltriethoxysilane (OTS) as SAM reagents, which are widely used to improve device performance and are also expected to eliminate the adsorbed solvents on the surface.27,28 By micropatterning these SAMs,29,30 we successfully induced a continuous MIT on a κ-Cl single-crystal channel. Owing to the high resistivity of the bulk channel below ~50 K, the observed MIT was governed by carriers confined in two dimensions at the interface, exhibiting sheet conductivity equal to ~ /ℎ ( /ℎ = 3.874 × 10 Ω) on the metal– 5


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insulator crossover (MIC) line. Through in-depth analysis of the conductivity on the verge of the MIT, we demonstrated the quantum critical feature of the doping-driven Mott transition in the intermediate-T crossover region. This is distinct from the quantum criticality in the low-T regime for high-Tc cuprates,21,31,32 and follows the ‘high-T Mott quantum criticality’ expected to be universal to various types of Mott transition.33,34 On the basis of the scaling plot for the quantum criticality, we propose a candidate crossover line that delineates the boundary between metallic and insulating regions. The FET device was fabricated on a SiO2 (300 nm)/p++-Si substrate with a micropatterned SAM (area: 50 × 50 μm ), as schematically shown in Figure 1a. We fabricated four devices: one with gate surface modification by PFES (device F#1) and the other three with gate surface modification by OTS (devices C#1–#3), details of which are shown in Table S1. We found difference in SAM species has only minor effects on the MIT behaviors, as discussed in Supporting Information D. The patterning of the SAMs and the attachment of electrodes were performed using a conventional photolithography technique (Supporting Information A). Then thin crystals of κ-Cl (thickness: 40 nm for device F#1 and 80 nm for device C#2) were laminated on top of the substrate. Through the improvement of the electrochemical crystal growth process (Supporting Information B), we successfully obtained thin crystals of κ-Cl with a flat surface. Figure 1c shows an atomic force microscopy (AFM) image of the surface of the typical κ-Cl crystal laminated on the substrate. Over micrometer-scale areas, the roughness of the surface is below 6


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1.5 nm, or the thickness of one BEDT-TTF molecular layer (Figure 1b). For comparison, shown in Figure S3 is an AFM image of a typical thin crystal of the sister material κ-(BEDT-TTF)2CuN(CN)2Br (κ-Br), which was synthesized in accordance with a previous method reported elsewhere.35 One can recognize the steps of the BEDT-TTF molecular layers and many small vacancies for κ-Br. The availability of the single crystal with perfect surface flatness for κ-Cl is a major advantage for precisely investigating the MIT without a significant effect of disorder. Figure 1e shows the transfer curves of device F#1, which has a partly covered PFES layer and provides both κ-Cl/PFES/SiO2 and κ-Cl/SiO2 interfaces on an identical single crystal. Both on the SiO2 and PFES regions, the sheet conductivity () at low T is suppressed to below 10 /ℎ by applying a negative gate voltage ( ), whereas a positive induces a continuous increase in . This ‘n-type polarity’ stems from the natural n-type doping for κ-Cl surface (typically density of ∼ 10 − 10" cm ),25 and implies that electron doping breaks the strong carrier localization of Mott insulator state.35 This description is also supported by the carrier density estimated from Hall effect measurements (Figure 1f): # = $/ tan () , where $ and () are the magnetic field and Hall angle, respectively. The carrier density is nearly zero at negative , corresponding to the gapped incompressible state (small |d#/d,|, with , being the chemical potential) characterizing Mott insulators.12,36 Upon electron doping by as small amount as ∼ 10 cm , however, hole carriers with the density of 1.3 × 7


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10 cm (corresponding to the cross-sectional area of 75% of the first Brillouin zone) rapidly recovered, evidencing a compressible state with a large Fermi surface (large |d#/d,|). This change in the Hall carrier density indicates an abrupt increase in the mobile carrier spectral weight accompanied by closing of the Mott gap upon doping.25,37 Plus, continuous increase in after establishment of large Fermi surface state with almost a constant hole carrier density is also a unique feature of non-rigid bands of strongly correlated electrons, on which doping enhances carrier coherence through modification of the charge mass or lifetime. A stark difference between the PFES and SiO2 substrate areas manifests itself at a highly positive . On the PFES region, exceeds 3 /ℎ and its behavior becomes metallic (i.e., increasing with decreasing ). Importantly, an MIC in the conductivity occurs at = 75 V (the point of zero activation energy in Figure 2), while the crossover between an incompressible Mott insulator state and a compressible state with a large Fermi surface occurs at < 0 (the jump of the #/012 3 4 curve in Figure 1f). This is in contrast to the MIC of other two-dimensional electron systems, where continuous changes in the carrier density and metallic transport (i.e., d⁄d < 0) take place at the same time.38,39 The conductivity at the MIC is = 1.4 /ℎ, which agrees with the Mott–Ioffe–Regel (MIR) limit in two dimensions (minimum conductivity anticipated within semiclassical Boltzmann transport theory). The device mobility 678 = 31/9 4d/d (9 = 7 × 10: cm V : areal capacitance of back gate) 8


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also becomes very high, reaching 220 cm /V s at 5 K. On the SiO2 region, however, remains insulating with a value below e /h, and the maximum mobility is 70 cm /V s, much lower than that on the PFES region (Figure 1e inset). Focusing on the insulating regime, on the other hand, the conductivity is well expressed by activated-transport behavior modeled with ?@A 3, 4 = C exp3−F3 4/2 4 irrespective of SAM functionalization (Figure 2, inset), where F is the transport gap and C (≈ 1.4e /h) denotes the high temperature limit of activated-transport conductivity. In the insulating limit ( ~ − 120 V, Figure 2), F (≈ 35 meV) is comparable to that of bulk κ-Cl (Supporting Information E), which confirms that the original electronic properties of κ-Cl are retained through the device fabrication processes. On the other hand, electron doping continuously reduces F toward zero, which occurs more rapidly on the PFES-patterned area than on the bare SiO2 area (Figure 2). The difference in dF/d cannot be explained solely by the doping effect from the SAM,27,40 which suggests that the Mott transition is sensitively affected by interfacial disorder conditions such as impurities, the contamination by the solvent, and charge traps. Figures 3a and 3b display the detailed T-dependence of the conductivity near the MIC, which is measured on the SAM-patterned area of device C#2. Similar () behaviors were observed for all the devices on the SAM-patterned areas, and hereafter we concentrate on device C#2 (for devices C#1, C#3, 9


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and F#1, see Figure S8). The conductivity curve for each is continuous without jumps or kinks, and reproducible without hysteresis even after sweeping many times, indicating that no structural transition occurs. For 22 K ≲ ≲ 36 K, a well-defined MIC behavior is observed; a crossover line exists with T-independent conductivity ∗ = MNO = 1.35 ⁄ℎ at ∗ = MNO = 51 V , which separates the metallic and insulating regions in the vs plane. (Hereafter, variables with asterisks represent those on the crossover line.) In the lightly doped region ( < MNO ), all the 3 , 4 curves gradually decrease with decreasing T. On the other hand, the heavily doped region ( > MNO ) shows non-monotonic behavior with conductivity maxima, below which the transport becomes insulating. The low-T insulating transport for ≳ 0.5 /ℎ can be scaled linearly with log . Negative magnetoresistance is also observed in the highly doped regime below ~7 K, which indicates that the linear-in-log insulating behavior is explained by weak localization correction.41 Interestingly, U is a device-dependent value, while MNO is almost identical for each device (see Figures 1 and S8). This means that the conductivity in the vicinity of the MIC is not affected by the variation in the thickness of κ-Cl crystals, and thus the MIC is clearly caused by carriers confined in two dimensions at the interface. Notably, MNO corresponds to the minimum metallic conductivity, and hence one cannot identify a bad-metal (BM) feature, that is, metallic transport that violates the MIR limit over the entire T range. (Note that above 40 K, it is difficult to accurately extract the carrier transport on the interface from the 10


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observed conductivity because of the dominant contribution of the undoped bulk channel. Nevertheless, one cannot clearly observe BM behavior at high T. See Supporting Information E.) This is in contrast to Mott transitions in bulk κ-Cl at half-filling23,42 or those in inorganic Mott insulators.43–45 Recently, theoretical studies based on dynamical mean-field theory (DMFT)33,34,46 have advocated that the high-T MICs associated with Mott transitions bear a type of quantum criticality. This high-T quantum criticality is expected to be observed on any Mott systems since it is governed by high-energy-scale quantum fluctuations due to the competing electron–electron interaction and Fermi energy on Mott insulators. Although DMFT is exact in the limit of infinite dimensions, it becomes a reasonable approximation at high T even for low-dimensional system such as κ-Cl. Indeed, it has been experimentally demonstrated that various half-filled organic Mott insulators including κ-Cl show quantum critical nature in the high-T crossover region for a pressure-driven Mott transition.23 The experimental and theoretical phase diagrams around the quantum critical Mott crossovers are summarized in Figure S6. In brief, despite the complexity of the low-T material-specific phases, the half-filled organic Mott insulators show material-independent rapid crossovers at high T, with the carrier transport continuously changing from insulating to BM behavior. At these crossovers, quantum criticality is commonly observed with the crossover conductivity much smaller than e /h (i.e., BM behavior). These behaviors are completely consistent with the quantum criticality suggested by 11


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DMFT34,46 Therefore, the high-T Mott quantum critical crossover scenario is appropriate for describing the continuous MIC in the conductivity, at least for a pressure-driven Mott transition. A question arises: does this hold true for a doping-driven Mott transition? According to the theory,33 the quantum criticality of a doping-driven Mott transition should have notable features that have not been observed in a pressure-driven Mott transition: (1) A continuous crossover is sustained even in the low-T region (well below 1% of the half bandwidth V, with V ∼ 0.25 eV for W-type BEDT-TTF compounds47). (2) The crossover conductivity coincides with the MIR limit at high T, while it may deviate at low T. (3) The MIT can show two-step crossovers at finite T; one is a rapid change in charge compressibility |d#/d,| and the other is a sign change in d/d (ref. 33, Supplementary Information). These three points appear to be consistent with the present experimental results, and the third point can be observed in the discrepancy between the jump of the Hall carrier density and the MIC in the conductivity (Figures 1e and 1f). Thus, the high-T quantum critical crossover in the doping-driven Mott transition is a reasonable scenario that explains the observed MIC. According to DMFT studies,33,34,46 the MIC of a Mott transition can be well described within the framework of finite-size quantum critical scaling theory.19,48–50 This theory predicts that the conductivity is amenable to a scaling law written as51

(, X) = ∗ ℱ± 3|X|⁄ /[\ 4,

(1) 12


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where X is the distance from the crossover line and the power exponents ] and ^ are called the correlation exponent and dynamical exponent, respectively. In contrast to magnetic/charge ordering transitions and superconducting phase transitions, the MIT generally has no obvious order parameter. Yet, in the present doping-driven Mott transition, it is natural to regard as the scaling variable, i.e., X = − ∗ . ℱ_ (`) and ℱ (`) are the universal scaling functions for the metallic and insulating regimes, respectively, satisfying ℱ± (`) → 1 and having mirror symmetry ℱ_ (`) = 1/ℱ (`) for ` → 0. Importantly, the crossover line in eq. (1) should be provided by the ‘quantum Widom line (QWL)’,33,46 on which the system becomes the least stable. For scaling analysis, we attempt to propose a possible crossover line to find the QWL in the following. In analogy with the Widom line52 in the supercritical region of the classical gas–liquid-type phase transition, the location of the QWL in the T vs plane is generally dependent on temperature33. Thus, the crossover line ∗ () should have a complex form and deviate from the constant- crossover line widely adopted for conventional doping-driven MITs.53 Since it is fundamentally difficult to identify the location of the QWL from the transport properties, we instead derive a candidate crossover line through a simple optimization calculation (for details, see Supporting Information G). In short, the crossover line is taken as ∗ = MNO for > 25 K to satisfy the theoretically suggested high-T feature of the QWL, ∗ /?

/b

≃ ⁄ℎ (constant). Then it is smoothly extended to the low-T region so that the renormalized 13


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conductivity data (, )/ ∗ () converge onto the universal scaling curves (eq. (1)) with the highest scaling quality. If the resultant crossover line is appropriate for the candidate QWL, the quantum critical scaling along the crossover line should hold for sufficiently low T. Figure 3e depicts the results of the scaling analysis based on the high-T Mott quantum critical crossover scenario, which is obtained with ]^ = 1.55. It is found that the quantum critical scaling law persists over three decades of / ∗ and that the critical region extends down to ~7 K. As mapped in Figure 3c, the optimized crossover line exhibits a bow-shaped trajectory in the T vs plane and deviates from the conventional critical crossover line with = MNO at low T. The crossover conductivity ∗ falls from MNO with decreasing T (Figure 3d), which is consistent with the theory.33 In the lower-T regime of < 7 K, it turns out that the data for / ∗ are no longer scaled by the universal curves. This can be understood from two possible points of view. First, the quantum critical description of the continuous MIT is obscured because of the weak localization in the low-T region. Second, since long wavelength (low energy) fluctuations should come to play a significant role in low dimensional systems at a sufficiently low T, the high-T Mott critical crossover scenario derived from a local mean field approximation such as DMFT may be no longer appropriate. We emphasize that the quantum critical crossover with similar critical exponents down to ~10 K was also confirmed in devices C#1, C#3 and F#1 (Figure S8), even though details of the carrier transport characteristics varied 14


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from device to device. These findings manifest the essence of the present phase transition that the critical behavior is independent of specific details of the system. Furthermore, we also attempted the scaling analysis using conventional crossover lines (i.e., constant- ∗ lines), as displayed in Figures S7 and S9. Our attempt to fit eq. (1) with a constant ∗ was only successful in a limited T range, which indicates that the crossover line in Figure 3d is appropriate for describing the observed quantum criticality. Taken together, the proposed crossover line meets the requirements of the QWL, and the quantum critical crossover region spreads over a wide range in the vs plane. One might suspect the possibility of other types of MICs associated with, for example, a percolation transition,54 or a Mott MIC with classical criticality.24,55 However, the continuous MIC in the present study is inconsistent with criticalities derived from these mechanisms, indicting the uniqueness of the quantum critical scaling of the charge transport in the intermediate-T crossover region of the doping-induced Mott transition. Applicability for classical criticality is discussed in detail in Supporting Information H. Finally, we discuss the lack of BM behavior in the observed doping-driven Mott transition. In a wide variety of high-Tc superconductors as well as heavy fermion systems, metallic transport without a coherent Drude peak and with high resistivity exceeding the MIR limit is observed in the crossover region between the localized and delocalized states, without showing the saturation observed for 15


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conventional metals.12,56 This BM behavior is intuitively understood as collective carrier transport retaining the strongly correlated feature of the localized state (e.g., Mott insulator), while its origin remains to be fully understood. Early studies based on a quantum critical model57 did not appear to explain the BM behavior, promoting various theoretical considerations such as those based on marginal Fermi liquid theory58 and holographic duality.59,60 A DMFT study33 pointed out that the BM behavior is not contradictory to quantum criticality; the linear-in-T resistivity beyond the MIR limit can result from the inclined quantum critical region in the T vs doping density plane. In other words, the emergence of BM behavior depends on the relationship between the characteristics of the QWL and the material-specific band dispersion. Taking into account the remarkable similarity between the high-T criticality in the present experiment and the DMFT study, we speculate that the lack of BM behavior in the present experiment is due to the disordered environment and the two-dimensional band structure that can modify the QWL. So far, BM behavior has been believed to be an inherent property of Mott insulators. However, our results provide experimental counterevidence, which indicates that the non-Fermi-liquid feature in the crossover region should be understood in terms of the high-T Mott quantum criticality. Optical conductivity measurements may reveal a clearer picture of the quantum critical region of the doping-driven Mott transition in future.

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In summary, we successfully obtained single-crystal Mott-FETs with high performance, the device mobility exceeding ~200 cm2/Vs at low temperatures by a combination of thin κ-Cl crystals with a flat surface and interfacial modification of the device using SAMs. Fine control of the charge doping via gate voltage revealed two-step metal–insulator crossovers in the intermediate-temperature region: a rapid increase in the hole carrier density, followed by inversion of the sign of the temperature coefficient of the conductivity at ∼ /ℎ without the occurrence of BM behavior. The high device quality also provided an opportunity to accurately examine the transport properties in the vicinity of the MIC, which revealed quantum criticality in the conductivity. The quantum criticality was retained down to ~7 K, while at lower temperatures metallic transport became insulating owing to a disorder-induced interference effect (weak localization). A bow-shaped crossover line of the MIT with the high ∗ temperature limit of /?

/b ~

/ℎ was successfully proposed, which implies that the origin of the

observed criticality was the high-T Mott quantum critical crossover, a ubiquitous feature for various types of Mott transition suggested on the basis of DMFT. This work provides a key insight into the criticality in the doping-driven Mott transition, as well as its relationship with the pressure-driven Mott transition at half-filling and the interaction-driven MIT in two-dimensional electron gas systems.

ASSOCIATED CONTENT 17


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Supporting Information

Supporting Information Available: (A) Preparation of SAM-patterned substrate, (B) crystal growth and lamination process, (C) assessment of thin-layer crystal channel, (D) SAM species dependence on transport properties, (E) high-T transport, (F) phase diagrams of doping/pressure-driven Mott transitions, (G) scaling analysis for quantum criticality, and (H) classical critical scaling. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Y. S.)

*E-mail: [email protected] (Y. K.)

*E-mail: [email protected] (H. M. Y.)

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGEMENTS

We thank K. Yokota, Y. Yokota and T. Sato for useful discussions. Financial support of this work was provided by Grants-in-Aid for Scientific Research (S) (Grant No. 22224006, 16H06346), RIKEN Molecular System, JST ERATO, and Nanotechnology Platform Program (Molecule and Material Synthesis) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

REFERENCES

(1)

Cui, X.; Lee, G.-H.; Kim, Y. D.; Arefe, G.; Huang, P. Y.; Lee, C.; Chenet, D. A.; Zhang, X.; Wang, L.; Ye, F.; Pizzocchero, F.; Jessen, B. S.; Watanabe, K.; Taniguchi, T.; Muller, D. A.; Low, T.; Kim, P.; Hone, J. Nat. Nanotechnol. 2015, 10, 534–540.

(2)

Tsen, A. W.; Hunt, B.; Kim, Y. D.; Yuan, Z. J.; Jia, S.; Cava, R. J.; Hone, J.; Kim, P.; Dean, C. R.; Pasupathy, A. N. Nat. Phys. 2015, 12, 208–212.

(3)

Stafford, C. A.; Das Sarma, S. Phys. Rev. Lett. 1994, 72, 3590.

(4)

Byrnes, T.; Kim, N. Y.; Kusudo, K.; Yamamoto, Y. Phys. Rev. B 2008, 78, 75320.

(5)

Lee, P. A.; Wen, X.-G. Rev. Mod. Phys. 2006, 78, 17–85.

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Page 20 of 35

(6)

Cario, L.; Vaju, C.; Corraze, B.; Guiot, V.; Janod, E. Adv. Mater. 2010, 22, 5193–5197.

(7)

Zhu, M.; Peng, J.; Zou, T.; Prokes, K.; Mahanti, S. D.; Hong, T.; Mao, Z. Q.; Liu, G. Q.; Ke, X. Phys. Rev. Lett. 2016, 116, 216401.

(8)

Sipos, B.; Kusmartseva, A. F.; Akrap, A.; Berger, H.; Forró, L.; Tutis, E. Nat. Mater. 2008, 7, 960–965.

(9)

Powell, B. J.; McKenzie, R. H. Rep. Prog. Phys. 2011, 74, 56501.

(10)

Sasaki, T. Crystals 2012, 2, 374–392.

(11)

Lahoud, E.; Meetei, O. N.; Chaska, K. B.; Kanigel, A.; Trivedi, N. Phys. Rev. Lett. 2014, 112, 206402.

(12)

Imada, M.; Fujimori, A.; Tokura, Y. Rev. Mod. Phys. 1998, 70, 1039–1263.

(13)

Vojta, M. Rep. Prog. Phys. 2003, 66, 2069–2110.

(14)

Löhneysen, H. V.; Rosch, A.; Vojta, M.; Wölfle, P. Rev. Mod. Phys. 2007, 79, 1015–1075.

(15)

Millis, A. J. Phys. Rev. B 1993, 48, 7183.

(16)

Gegenwart, P.; Si, Q.; Steglich, F. Nat. Phys. 2008, 4, 186.

20


Page 21 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(17)

Sachdev, S. Phys. Status Solidi B 2009, 543, 6.

(18)

Sebastian, S. E.; Harrison, N.; Altarawneh, M. M.; Mielke, C. H.; Liang, R.; Bonn, D. A.; Hardy, W. N.; Lonzarich, G. G. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 6175–6179.

(19)

Sachdev, S. Quantum phase transition, 2nd Ed.; Cambridge University Press, 2011.

(20)

Caviglia, A. D.; Gariglio, S.; Reyren, N.; Jaccard, D.; Schneider, T.; Gabay, M.; Thiel, S.; Hammerl, G.; Mannhart, J.; Triscone, J.-M. Nature 2008, 456, 624–627.

(21)

Bollinger, A. T.; Dubuis, G.; Yoon, J.; Pavuna, D.; Misewich, J.; Božović, I. Nature 2011, 472, 458–460.

(22)

Hwang, H. Y.; Iwasa, Y.; Kawasaki, M.; Keimer, B.; Nagaosa, N.; Tokura, Y. Nat. Mater. 2012, 11, 103–113.

(23)

Furukawa, T.; Miyagawa, K.; Taniguchi, H.; Kato, R.; Kanoda, K. Nat. Phys. 2015, 11, 221–224.

(24)

Kagawa, F.; Miyagawa, K.; Kanoda, K. Nature 2005, 436, 534–537.

(25)

Kawasugi, Y.; Seki, K.; Edagawa, Y.; Sato, Y.; Pu, J.; Takenobu, T.; Yunoki, S.; Yamamoto, H. M.; Kato, R. Nat. Commun. 2016, 7, 12356.

(26)

Hayes, R.; Warr, G. G.; Atkin, R. Chem. Rev. 2015, 115, 6357–6426. 21


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(27)

Page 22 of 35

Kobayashi, S.; Nishikawa, T.; Takenobu, T.; Mori, S.; Shimoda, T.; Mitani, T.; Shimotani, H.; Yoshimoto, N.; Ogawa, S.; Iwasa, Y. Nat. Mater. 2004, 3, 317–322.

(28)

Wang, X.; Xu, J.-B.; Wang, C.; Du, J.; Xie, W. Adv. Mater. 2011, 23, 2464–2468.

(29)

Yokota, K.; Takai, K.; Kudo, Y.; Sato, Y.; Enoki, T. Phys. Chem. Chem. Phys. 2014, 16, 4313– 4319.

(30)

Najmaei, S.; Zou, X.; Er, D.; Li, J.; Jin, Z.; Gao, W.; Zhang, Q.; Park, S.; Ge, L.; Lei, S.; Kono, J.; Shenoy, V. B.; Yakobson, B. I.; George, A.; Ajayan, P. M.; Lou, J. Nano Lett. 2014, 14, 1354– 1361.

(31)

Leng, X.; Garcia-Barriocanal, J.; Bose, S.; Lee, Y.; Goldman, A. M. Phys. Rev. Lett. 2011, 107, 27001.

(32)

Garcia-Barriocanal, J.; Kobrinskii, A.; Leng, X.; Kinney, J.; Yang, B.; Snyder, S.; Goldman, A. M. Phys. Rev. B 2013, 87, 24509.

(33)

Vučičević, J.; Tanasković, D.; Rozenberg, M. J.; Dobrosavljević, V. Phys. Rev. Lett. 2015, 114, 246402.

22


Page 23 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(34)

Nano Letters

Terletska, H.; Vučičević, J.; Tanasković, D.; Dobrosavljević, V. Phys. Rev. Lett. 2011, 107, 26401.

(35)

Kawasugi, Y.; Yamamoto, H.; Tajima, N.; Fukunaga, T.; Tsukagoshi, K.; Kato, R. Phys. Rev. Lett. 2009, 103, 116801.

(36)

Sordi, G.; Haule, K.; Tremblay, A.-M. S. Phys. Rev. B 2011, 84, 75161.

(37)

Cooper, S. L.; Thomas, G. A.; Orenstein, J.; Rapkine, D. H.; Millis, A. J.; Cheong, S.; Cooper, A. S.; Fisk, Z. Phys. Rev. B 1990, 41, 11605–11608.

(38)

Radisavljevic, B.; Kis, A. Nat. Mater. 2013, 12, 815–820.

(39)

Knyazev, D.; Omel’yanovskii, O.; Pudalov, V.; Burmistrov, I. Phys. Rev. Lett. 2008, 100, 46405.

(40)

Suda, M.; Kato, R.; Yamamoto, H. M. Science. 2015, 347, 743–746.

(41)

Hikami, S.; Larkin, A. I.; Nagaoka, Y. Prog. Theor. Phys. 1980, 63, 707–710.

(42)

Limelette, P.; Wzietek, P.; Florens, S.; Georges, A.; Costi, T.; Pasquier, C.; Jérome, D.; Mézière, C.; Batail, P. Phys. Rev. Lett. 2003, 91, 16401.

(43)

Emery, V. J.; Kivelson, S. A. Phys. Rev. Lett. 1995, 74, 3253–3256.

23


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(44)

Page 24 of 35

Takagi, H.; Batlogg, B.; Kao, H.; Kwo, J.; Cava, R.; Krajewski, J.; Peck, W. Phys. Rev. Lett. 1992, 69, 2975–2978.

(45)

Si, Q.; Yu, R.; Abrahams, E. Nat. Rev. Mater. 2016, 1, 16017.

(46)

Vučičević, J.; Terletska, H.; Tanasković, D.; Dobrosavljević, V. Phys. Rev. B 2013, 88, 75143.

(47)

Komatsu, T.; Matsukawa, N.; Inoue, T.; Saito, G. J. Phys. Soc. Jpn. 1996, 65, 1340–1354.

(48)

Philips, P. Advancded solid state physics, 2nd Ed.; Cambridge University Press, 2012.

(49)

Gantmakher, V. F.; Dolgopolov, V. T. Physics-Uspekhi 2010, 53, 57.

(50)

Sondhi, S. L.; Girvin, S. M.; Carini, J. P.; Shahar, D. Rev. Mod. Phys. 1997, 69, 315–333.

(51)

Dobrosavljević, V.; Abrahams, E.; Miranda, E.; Chakravarty, S. Phys. Rev. Lett. 1997, 79, 455– 458.

(52)

Simeoni, G. G.; Bryk, T.; Gorelli, F. a.; Krisch, M.; Ruocco, G.; Santoro, M.; Scopigno, T. Nat. Phys. 2010, 6, 503–507.

(53)

Abrahams, E.; Sarachik, M. P. Rev. Mod. Phys. 2001, 73, 251–266.

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(54)

Nano Letters

Dobrosavljević, V. In Conductor-insulator quantum phase transition; Dobrosavljević, V., Trivedi, N., Valles, Jr., J. M., Eds.; Oxford University Press, 2012.

(55)

Abdel-Jawad, M.; Kato, R.; Watanabe, I.; Tajima, N.; Ishii, Y. Phys. Rev. Lett. 2015, 114, 1–5.

(56)

Gunnarsson, O.; Calandra, M.; Han, J. E. Rev. Mod. Phys. 2003, 75, 1085–1099.

(57)

Phillips, P.; Chamon, C. Phys. Rev. Lett. 2005, 95, 107002.

(58)

Varma, C. M.; Littlewood, P. B.; Schmitt-Rink, S.; Abrahams, E.; Ruckenstein, A. E. Phys. Rev. Lett. 1989, 63, 1996–1999.

(59)

Hartnoll, S. A. Nat. Phys. 2014, 11, 54–61.

(60)

Davison, R. A.; Schalm, K.; Zaanen, J. Phys. Rev. B 2014, 89, 245116.

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Figure 1. Schematic of the Mott-FET device and the doping-driven metal–insulator transition of κ-Cl. (a) Schematic of FET device with the micropatterned SAM interface. (b) Crystal structure of κ-Cl with alternately stacked cationic dimer [(BEDT-TTF)2]+ layers and anionic insulator layers. (c-d) AFM topographical image measured on the surface of the κ-Cl crystal. (e) Transfer curves for various T (device F#1) on a PFES-covered substrate (colored circles) and on bare SiO2 (gray squares). Inset: device mobility 678 = (1/9 )d/d (9 = 7.0 × 10 cm V ; areal capacitance of back gate) as a function of in the highly doped regime ( = 116 V). (f) Gate voltage dependence of carrier density #/012 (device F#1) for PFES device area determined from Hall coefficients (B = 8 T). In the highly 26


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Nano Letters

doped regime where the MIC occurs, #/012 is almost independent of with #/012 ~1.3 × 10 cm , corresponding to the cross-sectional area of ~75 % of the first Brillouin zone. Error bars were calculated from the standard deviation of the Hall angle. The solid line (light blue) is a guide to the eye.

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Figure 2. Characteristics of activated transport in insulating regime (device F#1). (Main panel) Gate voltage dependence of the transport gap F in the insulating regime for the PFES-covered (red) and bare SiO2 substrate (gray) areas. (Inset) Arrhenius plots of sheet conductivity for different gate voltages on the PFES-covered and bare substrate. Symbols are experimental results; dotted lines are fits using ?@A () = C exp(−F/2 ). Each step is F = 8 V.

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Figure 3. Criticality in the vicinity of the MIC point (device C#2, on OTS-covered area). (a) − plot for various (= 120, 112, ⋯ , −54, −60 V). For 21 K < < 36 K, there is a distinct MIC point at MNO = 51 V where the sheet conductivity becomes T-independent, namely 3MNO, 4 = MNO = 1.35e /h (thick line in panel b). On the metallic side ( > MNO ), () curves exhibit maxima at certain temperatures, below which () decreases in a linear-in-log manner. For < MNO , on the 29


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other hand, monotonically decreases with decreasing T. (b) Magnified view of − plot for various (in increments of 1 V) in the vicinity of the MIC. Dots represent measurement points. (c) Color-coded plot of conductivity with the trajectory of the optimized crossover line (orange solid line) under the assumption of high-T Mott quantum criticality (candidate QWL, see text for details). For comparison, = MNO is marked with a dashed line (green). (d) Conductivities on the optimized crossover line (orange) and the = MNO line (green). (e) Quantum critical scaling plot of normalized by ∗ (). For > 7 K, all of the conductivity data on the insulating and metallic sides collapse onto two different branches of the universal scaling function (eq. (1)). The critical exponent is ]^ = 1.55.

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TABLE OF CONTENTS GRAPHIC

31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

h+

h+ h+

Mott Insulator

h

h+ h+

h

+

+

h

+

high-T quantum criticality + h

e-doped

Metal

h+

h

h+ h+

h+

h

+

+

h

+

40

h+ h

+

h+

h+

Temperature [K]

half-filled

σ/σ*

Nano Letters

h+ h+


h+

h+

h+

0.01

0.1

1

5 Page 32 of 35

Insulator

Metal

30 20 10 0 νz = 1.55 1400 700

0

|Vg–Vg*|νz [Vνz ]

700

Page 33 of 35

e

4 250

3

κ-Cl

σ (e2/h)

VH

V SAM

100 50 0

10

20

T [K]

SiO2

30

31.5 K

c

-40

0

40

80

120

f

κ-Cl

1.5 nm 5

500 nm

0

-80

Vg [V]

d

500 nm

nhole [1014 cm–2]

(nm) 10

N

7.0 K 10.5 K 14.0 K 17.5 K 21.0 K 24.5 K 28.0 K 31.5 K

150

0

0 -120

C Cu

200

7.0 K

SiO2 /p++-Si

S Cl

PFES

Vg = 116 V on PFES on SiO2

1

Vg

V

2

μFE [cm2 V−1 s−1]

a

Height

1 2 3 4 5 6 7 8 9 10 11 12 13b 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Nano Letters

1.5 1 0.5 0

1.5 Plus nm ACS Paragon Environment –120 –80

x

30 K –40

0

Vg [V]

40

80

120

10

σ (e2/h)

Nano Letters

40

30

10–2

–112 V

10

SiO2

+112 V

1 10–1 10–2 10–3

20

PFES Page 34 of 35

10–1

10–3

σ (e2/h)

Δ [meV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

+112 V

1

–112 V 0

0.1

0.05

1/T [K–1]

MIC point

PFES SiO2

0

ACS Paragon -100 -50 Plus Environment 0 50

Vg [V]

100

a

b

Nano Letters1.7

Page 35 of 35

3 1.5

0

2

1

10 T [K]

σ (e2/h)

0

3.5

1.1 15

e

20

25

30

35

T [K]

40

10

40 30

1

20 10 0

Vg* (T)

−60

VMIC

0

60

120

Vg [V]

d σ (e2/h)

1.3

−60 V 20 30 40

3 45

Vg =51 V (VMIC)

σMIC

1

c T [K]

σ (e2/h)

2

σ (Vg, T ) / σ*(T)

σ (e2/h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Vg =120 V

40 K 39 K 38 K 37 K 36 K 35 K 34 K 33 K 32 K 31 K 29 K 28 K 27 K 26 K 25 K

0.1

1.5

0.01

1 0.5 0

σ*(T) σ(VMIC,T)

0

10

20

T [K]

30

0.001

40 Paragon Plus Environment ACS

0.1

24 K 23 K 22 K 21 K 20 K 19 K 18 K 17 K 16 K 15 K 14 K

1

13 K 12 K 11 K 10 K 9K 8K 7K

|Vg-Vg*| /

10

T 1/1.55

100

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