That will appear in the main menu of the game is the first episode of the solar systems planets will be out against. To view and change the outer appearance of direct competition Gulfstream want to do before you start. To do this, click on hangar from the bottom of the game screen. You can add new weapons and equipment in the Gulfstream HANGAR race, color, change, or you can buy a new jet. Computational fluid dynamics (CFD) has become a premier tool for resolving the flow in. Re two orders of magnitude higher and lower than the relevant Re ∼ 1 for. Jackson, B.; Evangelisa, D.J.; Ray, D.D.; Hedrick, T.L. 3D for the people:.
Published online 2013 Apr 11. doi: 10.1186/1556-276X-8-167
PMID: 23578253
This article has been cited by other articles in PMC.
Abstract
Recently, superconductivity was found on semiconductor surface reconstructions induced by metal adatoms, promising a new field of research where superconductors can be studied from the atomic level. Here we measure the electron transport properties of the Si(111)-()-In surface near the resistive phase transition and analyze the data in terms of theories of two-dimensional (2D) superconductors. In the normal state, the sheet resistances (2D resistivities) R□ of the samples decrease significantly between 20 and 5 K, suggesting the importance of the electron-electron scattering in electron transport phenomena. The decrease in R□ is progressively accelerated just above the transition temperature (Tc) due to the direct (Aslamazov-Larkin term) and the indirect (Maki-Thompson term) superconducting fluctuation effects. A minute but finite resistance tail is found below Tc down to the lowest temperature of 1.8 K, which may be ascribed to a dissipation due to free vortex flow. The present study lays the ground for a future research aiming to find new superconductors in this class of materials.
Semiconductor surface reconstructions induced by metal adatoms constitute a class of two-dimensional (2D) materials with an immense variety [1,2]. They are considered one form of atomic layer materials which can possess novel electronic properties and device applications [,]. Recently, superconductivity was measured by scanning tunneling microscopy (STM) for atomically thin Pb films [,] and three kinds of Si(111) surface reconstructions: SIC-Pb, ()-Pb, and ()-In [7]. This discovery was followed by a demonstration of macroscopic superconducting currents on Si(111)-()-In by direct electron transport measurements []. These findings are important because they enable us to create superconductors from the atomic level using state-of-the-art nanotechnology. In addition, the space inversion symmetry breaking due to the presence of surface naturally leads to the Rashba spin splitting [,] and may consequently help realize exotic superconductors [11].
In reference[], we have unambiguously clarified the presence of Si(111)-()-In (referred to as ()-In here) superconductivity. However, systematic analysis on electron transport properties above and below the transition temperature (Tc) is still lacking. For example, 2D superconductors are known to exhibit the precursor of phase transition due to the thermal fluctuation effects just above Tc[12-14]. Superconductivity is established below Tc, but vortices can be thermally excited in a 2D system. Their possible motions can cause the phase fluctuation and limit the ideal superconducting property of perfect zero resistance [15]. These fundamental properties should be revealed before one proceeds to search for new superconductors in this class of 2D materials.
In this paper, the resistive phase transition of the ()-In surface is studied in detail for a series of samples. In the normal state, the sheet resistances (2D resistivities) R□ of the samples decrease significantly between 20 and 5 K, which amounts to 5% to 15% of the residual resistivity Rn,res. Their characteristic temperature dependence suggests the importance of electron-electron scattering in electron transport phenomena, which are generally marginal for conventional metal thin films. Tc is determined to be 2.64 to 2.99 K and is found to poorly correlate with Rn,res. The decrease in R□ is progressively accelerated just above Tc due to the superconducting fluctuation effects. Quantitative analysis indicates the parallel contributions of fluctuating Cooper pairs due to the direct (Aslamazov-Larkin term) and the indirect (Maki-Thompson term) effects. A minute but finite resistance tail is found below Tc down to the lowest temperature of 1.8 K, which may be ascribed to a dissipation due to free vortex flow.
Methods
The experimental method basically follows the procedure described in reference [] but includes some modifications. The whole procedure from the sample preparation through the transport measurement was performed in a home-built ultrahigh vacuum (UHV) apparatus without breaking vacuum (see Figure Figure1a)1a) [16,17]. First, the ()-In surface was prepared by thermal evaporation of In onto a clean Si(111) substrate, followed by annealing at around 300°C for approximately 10 s in UHV [18-], and was subsequently confirmed by low-energy electron diffraction and STM. The sample was then patterned by Ar + sputtering through a shadow mask to define the current path for four-terminal resistance measurements. Typical STM images before and after sputtering are displayed in Figure Figure1b,c,1b,c, respectively. The former shows a clear periodic structure corresponding to the unit cell, while the latter shows a disordered bare silicon surface.
Instrumentation and sample preparation. The whole procedure from the sample preparation through the transport measurement was performed in a home-built UHV apparatus without breaking vacuum (a). Typical STM images of a ()-In sample before (b) (Vsample = −0.015 V) and after (c) (Vsample=2.0 V) are displayed. (d) The design of sample patterning in the black area shows the Ar +-sputtered region. The color indicates the degree of calculated current density (green, high; purple, low). (e) Optical microscope image of a patterned sample.
We note that, although the nominal coverage of the evaporated In is more than several monolayers (ML), post annealing removes surplus In layers and establishes the ()-In surface. The In coverage of this surface reconstruction was originally proposed to be 1 ML for the ‘hexagonal’ phase (()-In-hex) and 1.2 ML for the ‘rectangular’ phase (()-In-rect) [18], where 1 ML corresponds to the areal density of the top-layer Si atoms of the ideal Si(111) surface. However, recent theoretical studies point to the coverages of 1.2 ML for the ()-In-hex and of 2.4 ML for the ()-In-rect [,22]. For our experiments, the dominant phase is likely to be the ()-In-hex judging from the resemblance of the obtained STM images (Figure (Figure1b)1b) to the simulated image of the ()-In-hex (Figure two, panel b in [22]). The relation between the surface structure and the superconducting properties is intriguing and will be the subject of future work.
In the previous study, van der Pauw’s measurement was adopted to check the anisotropy of electron conduction and to exclude the possibility of spurious supercurrents. In this setup, however, transport characteristics should be analyzed with care because the spatial distribution of bias current is not uniform. To circumvent this problem, in the present study, we adopted a configuration with a linear current path between the voltage terminals (Figure (Figure1d).1d). The black regions represent the area sputtered by Ar + ions through the shadow mask. The figure also shows the current density distribution calculated by the finite element method in color scale, which confirms that it is homogeneous between the voltage probes. This allows us to determine the sheet resistance R□ of the sample in a more straightforward way: R□=(V/I)×(W/L), where V is the measured voltage, I is the bias current, W=0.3 mm is the width of the current path, and L=1.2 mm is the distance between the voltage probes. Figure Figure1e1e shows the optical microscope image of a sample, confirming the clear boundary between the shadow-masked and sputtered regions. Although the sputtering was very light, the resulting atomic-scale surface roughening was enough to make an optical contrast between the two regions.
Following the sample preparation, four Au-coated spring probes were brought into contact with the current/voltage terminal patterns in a UHV-compatible cryostat. Four-terminal zero bias sheet resistance R□ was measured with a DC bias current I=1 µA, and the offset voltage was removed by inverting the bias polarity. To access the electron conduction only through the ()-In surface at low temperatures, Si(111) substrates without intentional doping (resistivity R>1,000 Ω cm) were used. Leak currents through the substrate and the Ar +-sputtered surface region were undetectably small below 20 K, which allowed precise measurements in this temperature region.
Results and discussion
Electron transport properties above Tc
In the present study, we investigated seven samples referred to as S1, S2,... and S7. They were prepared through the identical procedure as described above, but due to subtle variations in the condition, they exhibit slightly different electron transport properties. As representative data, the temperature dependences of sheet resistance R□ for S1 and S2 are displayed in Figure Figure22 (red dots, S1; blue dots, S2). R□ drops to zero at Tc≈2.6 K for S1 and at Tc≈3.0 K for S2, consistent with the previous study on the superconducting phase transition []. The rest of the samples show the same qualitative behaviors. As shown below, S1 and S2 exhibit the lowest and the highest Tc, respectively, among all the samples. Here we note two distinctive features: (i) For the high-temperature region of 5 K<T<20 K, R□ decreases with decreasing T, i.e., dR□/dT>0. The temperature dependence of R□ is slightly nonlinear with a concave curvature, i.e., d2R□/dT2>0. (ii) The decrease in R□ is progressively accelerated as T approaches Tc.
Electron transport properties above Tc. The red and blue dots represent the temperature dependences of sheet resistance R□ for sample S1 and S2, respectively, while the yellow and green lines are the results of fitting analysis using Equations 1 to 3. ΔR□ is defined as the decrease in R□ between 20 and 5 K. The inset shows Tc as a function of Rn,res, revealing no clear correlation between them.
The data were analyzed to deduce characteristic parameters as follows. Feature (i) can be phenomenologically expressed by the 2D normal state conductivity G□,n of the following form:
(1)
where Rn,res is the residual resistance in the normal state, C is the prefactor, and a is the exponent of the power-law temperature dependence. Feature (ii) is naturally attributed to the superconducting fluctuation effects [14]. Just above Tc, parallel conduction due to thermally excited Cooper pairs adds to the normal electron conduction (Aslamazov-Larkin (AL) term), and this effect is enhanced in a 2D systems [12]. The 2D conductivity due to the Cooper pair fluctuation G□,sf takes the following form:
(2)
where R0 is a temperature-independent constant. In addition to this direct effect, another indirect contribution may be important near Tc, which originates from the inertia of Cooper pairs after decaying into pairs of quasiparticles (Maki-Thompson (MT) term) [13]. Since its temperature dependence is similar to Equation 2 but involves more material-dependent parameters, we combine these two effects and adopt Equation 2. Importantly, for the pure AL term, R0 = 16ℏ/e2 = 65.8 kΩ regardless of the thickness. Then the total sheet resistance above Tc is given by the following equation:
(3)
The experimental data were fitted excellently using Equations 1 to 3 with Rn,res, C, a, R0, and Tc being fitting parameters, as shown in Figure Figure22 (yellow line, S1; green line, S2). Since Equation 2 is only valid for T>Tc, the data of the normal state region (defined as R□>50 Ω) were used for the fitting. All parameters thus determined are listed in Table Table11 for the seven samples. We note that the obtained values for R0 are all smaller by a factor of 2.4 to 5.4 than R0=65.8 kΩ for the AL term. This indicates that the observed fluctuation-enhanced conductivities originate from both AL and MT terms. We also tried to fit the data by explicitly including the theoretical form for the MT term [13], but this resulted in poor fitting convergence.
Table 1
Summary of the fitting analysis on the resistive transition of the ()-In surface
Sample
R0 (kΩ)
Rn,res (Ω)
Tc(K)
b
ΔR□/Rn,res(%)
S1
12.1
293
2.64
1.80
8.0
S2
20.0
171
2.99
1.54
10.8
S3
15.6
146
2.81
1.78
12.6
S4
17.6
108
2.76
1.67
15.3
S5
27.7
394
2.76
1.86
5.0
S6
14.3
160
2.67
1.69
11.5
S7
20.9
124
2.88
1.48
13.7
The determined Tc ranges from 2.64 to 2.99 K. This is in reasonable agreement with the previously determined value of Tc=2.8 K, but there are noticeable variations among the samples. The normal residual resistance Rn,res also shows significant variations, ranging from 108 to 394 Ω. These two quantities, Tc and Rn,res, could be correlated because a strong impurity electron scattering might cause interference-driven electron localization and suppress Tc[23]. However, they are poorly correlated, as shown in the inset of Figure Figure2.2. This is ascribed to possible different impurity scattering mechanisms determining Rn,res and Tc as explained in the following. Electron scattering should be strong at the atomic steps because the surface layer of ()-In is severed there. Therefore, they contribute to most of the observed resistance [,]. However, the interference between scatterings at the atomic steps can be negligibly weak if the average separation between the atomic steps dav is much larger than the phase relaxation length Lϕ. This is likely to be the case because dav≈400 nm for our samples, and Lϕ is several tens of nanometer for typical surfaces [25]. In this case, electron localization and resultant suppression of Tc are dominated by other weaker scattering sources within the size of Lϕ, not by the atomic steps that determine Rn,res.
The exponent a was determined to be 1.48 to 1.85 in accordance with feature (i). This might be seen as a typical metallic behavior due to the electron-phonon scattering. However, this mechanism would lead to Re-ph∝T for T>ΛD and Re-ph∝T5 for T≪ΛD[26], neither of which is consistent with the observed temperature dependence. (Here Re-ph is the resistance due to the electron-phonon scattering, and ΛD is the Debye temperature.) Considering the exponent a to be slightly smaller than 2, we attribute its origin to the electron-electron scattering. In a 2D Fermi liquid, it leads to a resistivity Re−e with the following form [27],
(4)
where C′ is a proportional constant, εF is the Fermi energy, and kB is the Boltzmann constant. The log term in Equation 4 results in a weaker temperature dependence than that in a 3D Fermi liquid (∝T2). Fitting the data with Equation 4 instead of the CTa term in Equation 1 gives εF≈0.1 eV, although the uncertainty is quite large.
We note that a decrease in resistance in a conventional metal film is usually very small in this temperature range. For example, it is less than 1% within the range of 2<T<20 K for 2-nm-thick single-crystal Nb films, although R□=122 Ω of the film is comparable to the observed Rn,res in the present experiment []. For a metal thin film with a large resistance, R□ even increases slightly with decreasing T as a consequence of the electron localization [29]. In clear contrast, a decrease in R□ between 20 and 5 K in our samples, ΔR□, amounts to as much as 5% to 15% of Rn,res (see Figure Figure22 and Table Table1).1). In this sense, the observed temperature dependence is rather unusual. The ()-In surface studied here has an atomic-scale dimension in the normal direction and may thus have an enhanced electron-electron interaction because of insufficient electrostatic screening. In comparison, the contribution from the electron-phonon interaction can be smaller because it decreases rapidly at low temperatures as Re-ph∝T5.
Residual resistance in the superconducting phase below Tc
The superconducting fluctuation theories state that R□ becomes exactly zero at Tc, as indicated by Equation 2. However, a close inspection into the magnified plots (Figure (Figure3a)3a) reveals that R□ has a finite tail below Tc. To examine whether R□ becomes zero at sufficiently low temperatures, we have taken the current-voltage (I-V) characteristics of sample S1 below Tc down to the lowest temperature of 1.8 K. Figure Figure3b3b displays the data in the log-log plot form. Although the I-V characteristics exhibit strong nonlinearity at the high-bias current region, they show linear relations around the zero bias at all temperatures. The sheet resistances R□ determined from the linear region of the I-V curves are plotted in Figure Figure3c3c as red dots. R□ decreases rapidly as temperature decreases from Tc, but it becomes saturated at approximately 2×10−2Ω below 2 K.
Residual resistance in the superconducting phase below Tc. (a) Magnified view of Figure Figure22 around Tc. The broken circles indicate the presence of residual resistances below Tc. (b) Temperature dependence of the I-V characteristics of sample S1 below Tc. The data are plotted in the log-log scales. The measured temperatures are indicated in the graph. (c) Red dots show the sheet resistance determined from the low-bias linear region of the I-V characteristics of sample S1. The blue line shows the result of the fitting analysis using Equation 6 within the range of 2.25 K<T<2.61 K while Tc=2.64 K is fixed.
This residual resistance can be attributed to dissipation due to free vortex flow, which is caused by the Lorentz force between the magnetic flux and the current [15], since the stray magnetic field is not shielded in the experiment. The sheet resistance due to the free vortex flow R□,v is given by the following equation:
(5)
where ξ is the Ginzburg-Landau coherence length, R□,n is the normal sheet resistance of the sample, B is the magnetic field perpendicular to the suface plane, and Φ0=h/2e is the fluxoid quantum. A crude estimation using ξ=49 nm,R□,n=290 Ω, and B=3×10−5 T gives R□,v=6.3×10−2Ω, which is in the same order of magnitude as the observed value of approximately 2×10−2Ω. We note that ξ=49 nm was adopted from the value for the Si(111)-SI-Pb surface [7], and ξ is likely to be smaller here considering the difference in Tc for the two surfaces. The present picture of free vortex flow at the lowest temperature indicates that strong pinning centers are absent in this surface superconductor. This is in clear contrast to the 2D single-crystal Nb film [], where the zero bias sheet resistance was undetectably small at sufficiently low temperatures. In accordance with it, the presence of strong vortex pinning was concluded from the observation of vortex creep in []. This can be attributed to likely variations in local thickness of the epitaxial Nb film at the lateral scale of vortex size [30]. The absence of ‘local thickness’ variation in the present surface system may be the origin of the observed free vortex flow phenomenon.
As mentioned above, R□ rapidly decreases just below Tc. This behavior could be explained by the Kosterlitz-Thouless (KT) transition [31,32]. In a relatively high-temperature region close to Tc, thermally excited free vortices cause a finite resistance due to their flow motions. As temperature decreases, however, a vortex and an anti-vortex (with opposite flux directions) make a neutral bound-state pair, which does not move by current anymore. According to the theory, all vortices are paired at TK, and resistance becomes strictly zero for an infinitely large 2D system. The temperature dependence of R□ for TK<T<Tc is predicted as follows:
(6)
where C′′ is a prefactor, and b a material-dependent constant. For this transition to be observable, the transverse penetration depth λ⊥ for magnetic field must be larger than the sample size so that vortices can interact with each other logarithmically as a function of the mutual distance. The ultimate atomic-scale thickness of the present system leads to a very large λ⊥ in the order of millimeters [], thus making it a candidate for observing the KT transition. We fitted the experimental data of R□ using Equation 6 within the range of 2.25 K<T<2.61 K while Tc=2.64 K is fixed. The result is shown in Figure Figure3c3c as a blue line. The reasonable fitting over two orders of magnitude in R□ points to the precursor of the KT transition. The obtained value of TK=1.69 K is deviated from the relation [31]
(7)
where Rc = ℏ/e2 = 4.11 kΩ and R□,n are identified with Rn,res=293 Ω of sample S1 here. However, Equation 7 is derived from the assumption of the dirty-limit BCS superconductor, which is not applicable to the ()-In surface with high crystallinity. Unfortunately, the present experimental setup does not allow us to observe the expected temperature dependence of Equation 6 down to TK because of the presence of the stray magnetic field. Furthermore, the predicted I-V characteristics V∝Ia where the exponent a jumps from 1 to 3 at TK should be examined to conclude the observation of the KT transition [31,32]. Construction of a UHV-compatible cryostat with an effective magnetic shield and a lower achievable temperature will be indispensible for such future studies.
Conclusions
We have studied the resistive phase transition of the ()-In surface in detail for a series of samples. In the normal state, the sheet resistances R□ of the samples decrease significantly between 20 and 5 K, which amounts to 5% to 15% of the residual resistivity Rres. Their characteristic temperature dependence suggests the importance of electron-electron scattering in electron transport phenomena. The poor correlation between the variations in Tc and Rres indicate different mechanisms for determining these quantities. The decrease in R□ was progressively accelerated just above Tc due to the superconducting fluctuation effects. Quantitative analysis indicates the parallel contributions of fluctuating Cooper pairs corresponding to the AL and MT terms. A minute but finite resistance tail was found below Tc down to the lowest temperature of 1.8 K, which may be ascribed to a dissipation due to free vortex flow. The interpretation of the data based on the KT transition was proposed, but further experiments with an improved cryostat are required for the conclusion.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
TU and PM carried out the sample fabrication/characterization and the electron transport measurements. TU and TN conceived of the study. TU analyzed the data and drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was partly supported by World Premier International Research Center (WPI) Initiative on Materials Nanoarchitectonics, MEXT, Japan, and by the Grant-in-Aid for JSPS Fellows. The authors thank M. Aono at MANA, NIMS, for his stimulous discussions.
References
Lifshits VG, Saranin AA, Zotov AV. Surface Phases on Silicon: Preparation, Structures, and Properties. Chichester: Wiley; 1994. [Google Scholar]
Mönch W. Semiconductor Surfaces and Interfaces. Berlin: Springer; 2001. [Google Scholar]
Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA. Electric field effect in atomically thin carbon films. Science. 2004;8(5696):666–669. doi: 10.1126/science.1102896. [PubMed] [CrossRef] [Google Scholar]
Radisavljevic B, Radenovic A, Brivio J, Giacometti V, Kis A. Single-layer MoS2 transistors. Nature Nanotech. 2011;8(3):147–150. doi: 10.1038/nnano.2010.279. [PubMed] [CrossRef] [Google Scholar]
Qin SY, Kim J, Niu Q, Shih CK. Superconductivity at the two-dimensional limit. Science. 2009;8(5932):1314–1317. doi: 10.1126/science.1170775. [PubMed] [CrossRef] [Google Scholar]
Brun C, Hong IP, Patthey F, Sklyadneva I, Heid R, Echenique P, Bohnen K, Chulkov E, Schneider WD. Reduction of the superconducting gap of ultrathin Pb islands grown on Si(111) Phys Rev Lett. 2009;8(20):207002. [PubMed] [Google Scholar]
Zhang T, Cheng P, Li WJ, Sun YJ, Wang G, Zhu XG, He K, Wang LL, Ma XC, Chen X, Wang YY, Liu Y, Lin HQ, Jia JF, Xue QK. Superconductivity in one-atomic-layer metal films grown on Si(111) Nature Phys. 2010;8(2):104–108. doi: 10.1038/nphys1499. [CrossRef] [Google Scholar]
Uchihashi T, Mishra P, Aono M, Nakayama T. Macroscopic superconducting current through a silicon surface reconstruction with indium adatoms: Si(111)-()-In. Phys Rev Lett. 2011;8(20):207001. [PubMed] [Google Scholar]
Sakamoto K, Oda T, Kimura A, Miyamoto K, Tsujikawa M, Imai A, Ueno N, Namatame H, Taniguchi M, Eriksson PEJ, Uhrberg RIG. Abrupt rotation of the Rashba spin to the direction perpendicular to the surface. Phys Rev Lett. 2009;8(9):096805. [PubMed] [Google Scholar]
Yaji K, Ohtsubo Y, Hatta S, Okuyama H, Miyamoto K, Okuda T, Kimura A, Namatame H, Taniguchi M, Aruga T. Large Rashba spin splitting of a metallic surface-state band on a semiconductor surface. Nature Commun. 2010;8:17.[PMC free article] [PubMed] [Google Scholar]
Bauer E, Sigrist M. Non-Centrosymmetric Superconductors. Berlin: Springer; 2012. [Google Scholar]
Aslamasov LG, Larkin AI. The influence of fluctuation pairing of electrons on the conductivity of normal metal. Phys Lett. 1968;8:238–239.[Google Scholar]
Thompson RS. Microwave, flux flow, and fluctuation resistance of dirty type-II superconductors. Phys Rev B. 1970;8:327–333. doi: 10.1103/PhysRevB.1.327. [CrossRef] [Google Scholar]
Skocpol WJ, Tinkham M. Fluctuations near superconducting phase-transitions. Rep Prog Phys. 1975;8(9):1049–1097. doi: 10.1088/0034-4885/38/9/001. [CrossRef] [Google Scholar]
Bardeen J, Stephen MJ. Theory of the motion of vortices in superconductors. Phys Rev. 1965;8(4A):A1197–A1207. doi: 10.1103/PhysRev.140.A1197. [CrossRef] [Google Scholar]
Uchihashi T, Ramsperger U. Electron conduction through quasi-one-dimensional indium wires on silicon. Appl Phys Lett. 2002;8(22):4169–4171. doi: 10.1063/1.1483929. [CrossRef] [Google Scholar]
Uchihashi T, Ramsperger U, Nakayama T, Aono M. Nanostencil-fabricated electrodes for electron transport measurements of atomically thin nanowires in ultrahigh vacuum. Jpn J Appl Phys. 2008;8(3):1797–1799. doi: 10.1143/JJAP.47.1797. [CrossRef] [Google Scholar]
Kraft J, Surnev SL, Netzer FP. The structure of the indium-Si(111) () monolayer surface. Surf Sci. 1995;8(1-2):36–48. doi: 10.1016/0039-6028(95)00516-1. [CrossRef] [Google Scholar]
Rotenberg E, Koh H, Rossnagel K, Yeom H, SchÃd’fer J, Krenzer B, Rocha M, Kevan S. Indium on Si(111): a nearly free electron metal in two dimensions. Phys Rev Lett. 2003;8(24):246404. [PubMed] [Google Scholar]
Yamazaki S, Hosomura Y, Matsuda I, Hobara R, Eguchi T, Hasegawa Y, Hasegawa S. Metallic transport in a monatomic layer of in on a silicon surface. Phys Rev Lett. 2011;8(11):116802. [PubMed] [Google Scholar]
Park J, Kang M. Double-layer in structural model for the In/Si(111)-() surface. Phys Rev Lett. 2012;8(16):166102. [PubMed] [Google Scholar]
Uchida K, Oshiyama A. New identification of metallic phases of in atomic layers on Si(111) surfaces. 2012,. arXiv:1212.1261. [ http://arxiv.org/abs/1212.1261]
Goldman AM, Markovic N. Superconductor-insulator transitions in the two-dimensional limit. Phys Today. 1998;8(11):39–44. doi: 10.1063/1.882069. [CrossRef] [Google Scholar]
Matsuda I, Ueno M, Hirahara T, Hobara R, Morikawa H, Liu CH, Hasegawa S. Electrical resistance of a monatomic step on a crystal surface. Phys Rev Lett. 2004;8(23):236801. [PubMed] [Google Scholar]
Jeandupeux O, Burgi L, Hirstein A, Brune H, Kern K. Thermal damping of quantum interference patterns of surface-state electrons. Phys Rev B. 1999;8(24):15926–15934. doi: 10.1103/PhysRevB.59.15926. [CrossRef] [Google Scholar]
Ziman JM. Principles of the Theory of Solids. Cambridge: Cambridge University Press; 1972. [Google Scholar]
Hodges C, Smith H, Wilkins J. Effect of fermi surface geometry on electron-electron scattering. Phys Rev B. 1971;8(2):302–311. doi: 10.1103/PhysRevB.4.302. [CrossRef] [Google Scholar]
Hsu J, Kapitulnik A. Superconducting transition, fluctuation, and vortex motion in a two-dimensional single-crystal Nb film. Phys Rev B. 1992;8(9):4819–4835. doi: 10.1103/PhysRevB.45.4819. [PubMed] [CrossRef] [Google Scholar]
Bergmann G. Weak localization in thin films: a time-of-flight experiment with conduction electrons. Phys Rep. 1984;8:1–58. doi: 10.1016/0370-1573(84)90103-0. [CrossRef] [Google Scholar]
Özer MM, Thompson JR, Weitering HH. Hard superconductivity of a soft metal in the quantum regime. Nature Phys. 2006;8(3):173–176. doi: 10.1038/nphys244. [CrossRef] [Google Scholar]
Epstein K, Goldman A, Kadin A. Renormalization effects near the vortex-unbinding transition of two-dimensional superconductors. Phys Rev B. 1982;8(7):3950–3953. doi: 10.1103/PhysRevB.26.3950. [CrossRef] [Google Scholar]
Mooij JE. In: Percolation, Localization, and Superconductivity. Goldman AM, Wolf SA, editor. Berlin: Springer; 1984. Two-dimensional transition in superconducting films and junction array. [Google Scholar]
Articles from Nanoscale Research Letters are provided here courtesy of Springer