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Localization and electron-electron interactions in few-layer epitaxial graphene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 245201 (http://iopscience.iop.org/0957-4484/25/24/245201) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.100.58.76 This content was downloaded on 02/06/2014 at 08:43
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Nanotechnology Nanotechnology 25 (2014) 245201 (7pp)
doi:10.1088/0957-4484/25/24/245201
Localization and electron-electron interactions in few-layer epitaxial graphene Shun-Tsung Lo1, Fan-Hung Liu1, Chang-Shun Hsu1, Chiashain Chuang2, Lung-I Huang2, Yasuhiro Fukuyama3, Yanfei Yang4, Randolph E Elmquist4 and Chi-Te Liang1,2 1
Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan Department of Physics, National Taiwan University, Taipei 106, Taiwan 3 National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan 4 National Institute of Standards and Technology (NIST), Gaithersburg, MD 20899, USA 2
E-mail: [email protected] Received 28 February 2014, revised 10 April 2014 Accepted for publication 15 April 2014 Published 28 May 2014 Abstract
This paper presents a study of the quantum corrections caused by electron-electron interactions and localization to the conductivity in few-layer epitaxial graphene, in which the carriers responsible for transport are massive. The results demonstrate that the diffusive model, which can generally provide good insights into the magnetotransport of two-dimensional systems in conventional semiconductor structures, is applicable to few-layer epitaxial graphene when the unique properties of graphene on the substrate, such as intervalley scattering, are taken into account. It is suggested that magnetic-field-dependent electron-electron interactions and Kondo physics are required for obtaining a thorough understanding of magnetotransport in few-layer epitaxial graphene. Keywords: graphene, interactions, localization (Some figures may appear in colour only in the online journal) 1. Introduction
should be noted that the prohibition of backscattering off potential fluctuations due to the chiral nature of massless Dirac fermions is one of the most important physical phenomena found in graphene [16]. Therefore, weak antilocalization (WAL) is predicted even in the absence of strong spin–orbit coupling [17]. In perfect graphene, spin–orbit coupling is weak because of the low mass of carbon atoms. However, when the graphene sheet is influenced by a substrate, vacancies, or by extrinsic impurities, short-range disorder leads to an increase in the rate of intervalley scattering, and such scattering can restore the WL behavior [18]. The chiral character of graphene also affects the scattering processes making the e-e interactions sensitive to the details of disorder in the system [19, 20]. Magnetotransport has already been widely investigated in a variety of disordered graphene fabricated by mechanical exfoliation [17, 18, 21], chemical vapor deposition [22], and epitaxial growth on SiC [23–25]. In most cases, either WAL or WL was observed, depending on the strength of intervalley coupling. Apart from interesting results on probing e-e interactions in graphene by
Graphene continues to attract a great deal of world-wide interest because of its extraordinary electrical [1, 2], thermal [3, 4], and optical [5, 6] properties. Although mechanically exfoliated graphene is of ultra-high quality, its small size may limit graphene’s practical applications. In contrast, epitaxial graphene grown from silicon carbide (SiC) substrates [7, 8] can be of wafer size and thus may find real-world applications in high-frequency devices [9, 10]. In the presence of supporting SiC substrates, short-range disorder is always incorporated, yielding a wide variety of rich physical phenomena in epitaxial graphene. To realize fully its great potential for application in future electronic devices, fundamental studies of carrier transport in graphene grown on a substrate are highly desirable. Electron-electron (e-e) interactions and coherent-backscattering-induced localization, or weak localization (WL), are two prominent quantum effects in disordered twodimensional (2D) systems at low temperatures T [11−15]. It 0957-4484/14/245201+07$33.00
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Nanotechnology 25 (2014) 245201
compressibility experiments [26, 27], few experimental studies focus on the broader features of e-e interactions in graphene [28]. Hence, it is interesting and useful to study the behavior of massive carriers subject to a magnetic field in few-layer graphene sheets, and to compare the results with the well-established models applicable to conventional semiconductor systems. Moreover, few-layer graphene, which is less susceptible to air exposure and surface contamination because of the top and bottom protecting layers, is useful for stable device applications as well as an interesting playground for studying fundamental phenomena such as the insulatorquantum Hall transition [29−33]. Here, we investigate the effects of localization and e-e interactions on the massive carriers in few-layer epitaxial graphene by studying its magnetotransport behavior in both the intermediate and high magnetic field regimes. We use two different methods to extract the contribution of e-e interactions. Evidence of e-e interactions and WL is clearly visible at low B. However, we show that the number of effective channels contributing to the e-e interactions may vary with the applied magnetic field in the regime where the WL phenomenon is suppressed. These results appear to be vastly different from those in conventional semiconductor-based 2D electron systems [12, 13].
Figure 1. Converted σxx(B) and σxy(B) at various temperatures T ranging from T = 1.95 K to T = 8 K. The inset shows that the Hall slope δρxy/δB decreases logarithmically with T. The solid line is a guide to the eye.
the inset of figure 1 indicates that the Hall slope (δρxy/δB) shows a lnT dependence and decreases with increasing T. As will be shown later, σxy is T-independent, suggesting that the mobility and carrier density are insensitive to a change in T. Therefore the observed lnT dependence of δρxy/δB is ascribed to e-e interactions. In a conventional semiconductor heterostructure where the 2D electron gas (2DEG) exists at the interface, the observed temperature and magnetic field dependences of the resistivity at kFl ≫ 1 (kF and l are the Fermi wave vector and the mean free path respectively) are generally determined by the quantum corrections to the conductivity caused by WL and e-e interactions. The theory of WL developed by Hikami, Larkin, and Nagaoka (HLN) predicts a correction to the Drude conductivity given by
2. Experimental details Our few-layer graphene sample was grown on an opticallypolished Si-terminated 6H-SiC(0001) surface at 1850 °C for 45 min under an Ar gas pressure slightly higher than 101.3 KPa. The number of layers (≈4 to 5) was identified by comparing the AFM and TEM images. The device was patterned into a Hall-bar configuration by standard optical lithography followed by using a reactive ion etch in O2 plasma and lift-off processing for the Au contacts. The detailed growth process can be found in [29]. Four-terminal magnetoresistivities (longitudinal and Hall resistivities, ρxx and ρxy) were measured using standard dc techniques. The magnetic field is applied perpendicular to the plane of the graphene sheet. Four samples show similar characteristics, and measurements taken on one of them are presented in this paper.
⎧ ⎪ δσxxWL = G0 ⎨ ψ ⎪ ⎩
3. Results and discussion
⎛ B ⎞⎫ ⎛1 Bϕ ⎞ ⎪ ⎛1 Btr ⎞ ⎟ + ln ⎜ tr ⎟ ⎬ , (1) ⎜ + ⎟−ψ⎜ + ⎜ ⎟⎪ ⎝2 ⎝2 B⎠ B⎠ B ⎝ ϕ ⎠⎭
where ψ is the digamma function, G0 = e2/(πh), the transport magnetic field Btr = ℏ
Figure 1 shows the longitudinal and Hall conductivities (σxx and σxy) as a function of magnetic field B obtained from ρxx(B) and ρxy(B) according to σxx(B) = ρxx(B)/(ρxx(B)2 + ρxy(B)2) and σxy(B) = ρxy(B)/(ρxx(B)2 + ρxy(B)2). At low B, σxx increases with increasing B, showing a character typical of WL. It is believed that intervalley coupling is sufficiently strong for the observation of WL due to atomically sharp defects at the interface with the SiC substrate [23−25], which leads to a significant enhancement of the rate of intervalley scattering. With increasing B, σxx decreases starting around 0.5 T. Over the whole B range, σxx increases with increasing temperature T, indicative of insulating behavior. Moreover,
( 2el ), 2
and Bϕ = ℏ
( 4el ) 2
ϕ
[34].
Here lφ stands for the dephasing length. On the other hand, the strength of e-e interactions are linked to the value of kBTτ ℏ, where τ is the transport relaxation time. It is expected that the e-e interactions contribute a correction to σxx without affecting σxy in the diffusive regime ( kBTτ ℏ ≪ 1) [13]. For gμBB < kT, where μB is the Bohr magneton, this contribution does not depend on B and has the form ⎛ k Tτ ⎞ δσxxee = Aee G0 ln ⎜ B ⎟ , ⎝ ℏ ⎠ 2
(2)
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Nanotechnology 25 (2014) 245201
⎡ ln ( 1 + F0σ ) Aee = 1 + c ⎢ 1 − ⎢⎣ F0σ
⎤ ⎥, ⎥⎦
(3)
where the constant c corresponds to the number of multiplet channels participating in e-e interactions and F0σ is the Fermiliquid interaction parameter [12, 13]. As a result, in the diffusive regime at B > Btr, where the WL effect is suppressed, the conductivity component σxx and σxy can be written as σxx =
neμ 2
1 + ( μB )
σxy =
+ δσee,
neμ2 B 2
1 + ( μB )
,
(4)
(5)
where the first term represents the classical contribution to the conductivity with transport mobility μ [12, 13]. Figure 1 clearly shows that σxy is T-independent whereas σxx depends on T in the studied field range. Since this is characteristic of the diffusive e-e interactions described above, we fit the experimental σxy(B, T = 1.95 K) to equation (5) for B < 1 T with μ as the fitting parameter and n (≈2.02 × 1013 cm−2) estimated from the high-T Hall slope. These results, together with the extrapolation of the fitting curve, are presented in figure 2(a). Given the estimated μ (≈578 cm2 Vs−1) and the known n, we can obtain the classical contribution, neμ/(1 + (μB)2), to the longitudinal conductivity. From the Drude conductivity σD = neμ = 2πkFlG0 (where the factor of 2 is due to the valley degeneracy of graphene) for conventional 2DEG, kFl ≈ 24 is approximately evaluated, indicating that our graphene device is well inside in the weakly-localized regime. Thus, although it is found that equation (2) cannot explain the data at high B, we can use equations (1)−(5) to analyze the data adequately for low B. Figure 2(b) compares the data of σxx at T = 1.95 K with neμ/ [1 + (μB)2]. Consistent with equations (2) and (4), the correction due to e-e interactions appears to reduce the conductivity. In the following, we eliminate the WL contribution from the measured σxx and study the e-e interactions that are the main findings in this paper. Figure 3(a) presents the results of further magnetotransport measurements around B = 0. In figure 3(b), we show the fits of Δσxx(B) = σxx(B) − σxx(0) at T = 0.89 K to equation (1) for −0.1 T < B < 0.1 T and for −0.5 T < B < 0.5 T, which are denoted by the red and blue curves respectively. It is believed that the WL effect is strongest for B < |Btr|. We can estimate Btr ≈ 0.36 T using kFl ≈ 24 and kF = (πn)1/2. The good agreement with the HLN theory occurs when the fitting range is limited within |Btr| ≈ 0.36 T. A good WL fitting to the data over −0.5 T < B < 0.5 T according to the formula of McCann et al [35] for graphene can be observed in figure 3(c). As reported by Wu et al [36], both the conventional HLN theory and the one developed by McCann et al can provide a reasonable explanation to the low-field positive magnetoconductivity in epitaxial graphene with the presence of significant intervalley scattering. The strong intervalley scattering causes the 2DEG in epitaxial graphene to behave as an ordinary 2DEG in the conventional semiconductor
Figure 2. Comparison of the experimentally measured (a) σxy(B) and (b) σxx(B) with the calculated neμ2B/(1 + (μB)2) and neμ/(1 + (μB)2), respectively, using the parameters obtained from the fit to equation (5), which is denoted by the red curve in (a).
heterostructure. However, at relatively high B, the chiral nature of Dirac fermions in graphene may give rise to antilocalization behavior in the sense that σxx decreases with increasing B, which was considered in the model of McCann et al. Hence, we can expect that the theory of McCann et al [35] gives a good fit over a much larger range of B compared with that of Hikami et al [34] figure 3(d) plots the T dependence of lφ obtained from the fitting of these two different models to the WL peak. Our data can be fitted to the HLN theory and the fit yields lφ ∼ T−0.29. The data can be also fitted 3
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Figure 3. (a) σxx(B) at various T around B = 0. From top to bottom: T = 30 K, 25 K, 21 K, 18 K, 15 K, 12 K, 10 K, 8 K, 6 K, 4 K, 1.98 K, 0.89 K, respectively. (b) The red and blue curves correspond to the best fits of equation (1) to Δσxx(B, T = 0.89 K) over different field ranges. (c) A fit to Δσxx(B, T = 0.89 K) with the help of the model developed by McCann et al for graphene. (d) lφ versus T on a log-log scale, which is obtained from the fitting procedures in (b) and (c). (e) σxx(B = 0) versus T on a semi-log scale. Dashed lines denote the extrapolation of the fitting curves.
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to the model of McCann et al and gives lφ ∼ T−0.24. Such close exponents suggest that both models can be used to explain our results. When e-e scattering dominates the dephasing [23, 37], we can expect that lφ ∼ T−0.5. Our results, showing a weaker T dependence of lφ, indicate that scattering mechanisms other than e-e scattering should be involved. Figure 3(e) then shows the zero B field σxx as a function of T. Logarithmic temperature dependence with the prefactor of 1.62G0 is observed. For our device considering Btr ≫ Bφ, at zero B, the WL contribution described by equation (1) can be simply approximated as δσxxWL = pGo ln ( T ) + constant, [23, 38], where p is the scaling parameter of lϕ ~ T −p 2 . Therefore we show that e-e interactions play a role in σxx(B = 0, T) and determine its contribution with Aee ≈ 1.04 in equation (2). Normally both WL and e-e interaction correction terms are small, independent of each other, and proportional to 1/(kFl). In our system, since p ≈ 0.5 (0.6), the slope in the σxx versus lnT plot due to e-e interaction correction is expected to be 1.62 − 0.60 ≈ 1. Therefore, the e-e interaction correction term is about two times the WL term. Over the temperature range between 2 K and 30 K, the measured zero-B-field conductivity is varied between 1.71 mS and 1.73 mS. Such a small change of 20 μS in the conductivity is due to both interaction and WL correction terms since the Drude part is expected to be temperature-independent. Therefore, the contribution of interaction and WL correction to this change are calculated to be 13.4 μS and 6.7 μS respectively. The contribution of e-e interactions to σxx can also be extracted by suppressing the WL effect at B ≫ Btr ≈ 0.36 T. Figure 4(a) shows σxx at high B, which increases with increasing T. The inset of the same figure demonstrates that this insulating behavior follows σxx ∼ ln(T), which is expected to be due to e-e interactions. From the slope of the linear fits to this logarithmic T dependence, the prefactor Aee in equation (1) as a function of B (⩾10Btr ≈ 3.6 T) where WL is expected to be suppressed is obtained. These results are presented in figure 4(b). However, in contrast to the prediction of diffusive e-e interactions, Aee is shown to depend on the magnetic field. We find that Aee ≈ 0.23 at B = 8 T and increases to 0.72 when B is reduced to 3.6 T, which is different from the value of Aee ≈ 1.04 deduced at low fields. When a 2D system is out of diffusive regime, i.e. kBTτ ℏ > 1 referred to as the ballistic regime, the mobility μ in equations (4) and (5) and thereby σxy will become T-dependent [39]. However, in the inset of figure 4(b), it is shown that σxy is strictly T-independent, suggesting that the observed T-dependent correction to σxx does not result from the high-T ballistic effects. Interestingly, the shape of WL peak in graphene is expected to depend strongly on the rates of both intervalley and intravalley scattering. Jobst et al [25] demonstrated that increasing T, which effectively enhances the strength of intervalley and intravalley scattering, changes the number of multiplet channels in equation (3) from c = 3 to c = 7. For the epitaxial graphene on SiC, the theoretical value of F0σ is found to be about −0.09 and thus Aee ≈ 1–0.046c [25]. Therefore, our results may correspond to a crossover from c = 17 (Aee ≈ 0.23) to c = 6 (Aee ≈ 0.72) with decreasing the magnetic
Figure 4. (a) σxx(B) at various T for B > 7.5 T. Inset: semi-log plot of σxx(T) at various B. (b) The obtained prefactor Aee in equation (2) as a function of B. Inset: σxy(B) at various T for B > 3.6 T.
field from B = 8 T to B = 3.6 T. However, the maximum number of multiplet channels due to fourfold spin and fourfold valley degeneracy is c = 15 [19, 25], which occurs when intervalley and intravalley scattering is insignificant and can only be realized in perfect graphene. In our disordered fewlayer graphene, where WL has been observed at low B, c should be less than 15. Therefore, our new experimental results suggest that varying the magnetic field may cause a 5
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variation of the relative strength of intervalley and intravalley in graphene [42], and call for further studies. According to the data shown in figure 2(a), there is an additional contribution to ρxx(B), which shifts σxy below the expected value of equation (5). We note that it is not due to a classical mechanism of magnetoresistance (MR) since in our few-layer graphene, μB (≈0.46) < 1 even at B = 8 T. It has been reported that the Kondo effect is important in graphene in the presence of defects acting as magnetic moments [40, 41], which can also give a logarithmic correction to ρxx(T) and thus may be one of the possible mechanisms leading to a change in the prefactor of the logarithmic T dependence of σxx with B, as observed in figure 4(b). Consequently, our work demonstrates that in addition to the usual MR effects, mechanisms such as valley-dependent scattering and Kondo physics should be carefully considered to understand the magnetotransport in disordered few-layer epitaxial graphene.
[3] Balandin A A, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F and Lau C N 2008 Nano Lett. 8 902 [4] Ghosh S, Calizo I, Teweldebrhan D, Pokatilov E P, Nika D L, Balandin A A, Bao W, Miao F and Lau C N 2008 Appl. Phys. Lett. 92 151911 [5] Mueller T, Xia F and Avouris P 2010 Nat. Photon. 4 297 [6] Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, Wang F and Zhang X 2011 Nature 474 64 [7] Berger C et al 2004 J. Phys. Chem. B 108 19912 [8] Berger C et al 2006 Science 312 1191 [9] Lin Y-M, Dimitrakopoulos C, Jenkins K A, Farmer D B, Chiu H-Y, Grill A and Avouris P 2010 Science 327 662 [10] Lin Y-M et al 2011 Science 332 1294 [11] Simmons M Y, Hamilton A R, Pepper M, Linfield E H, Rose P D and Ritchie D A 2000 Phys. Rev. Lett. 84 2489 [12] Minkov G M, Rut O E, Germanenko A V, Sherstobitov A A, Shashkin V I, Khrykin O I and Daniltsev V M 2001 Phys. Rev. B 64 235327 [13] Minkov G M, Rut O E, Germanenko A V, Sherstobitov A A, Shashkin V I, Khrykin O I and Zvonkov B N 2003 Phys. Rev. B 67 205306 [14] Clarke W R, Yasin C E, Hamilton A R, Micolich A P, Simmons M Y, Muraki K, Hirayama Y, Pepper M and Ritchie D A 2008 Nat. Phys. 4 55 [15] Goh K E J, Simmons M Y and Hamilton A R 2008 Phys. Rev. B 77 235410 [16] Katsnelson M I, Novoselov K S and Geim A K 2006 Nat. Phys. 2 620 [17] Morozov S V, Novoselov K S, Katsnelson M I, Schedin F, Ponomarenko L A, Jiang D and Geim A K 2006 Phys. Rev. Lett. 97 016801 [18] Tikhonenko F V, Kozikov A A, Savchenko A K and Gorbachev R V 2009 Phys. Rev. Lett. 103 226801 [19] Kozikov A A, Savchenko A K, Narozhny B N and Shytov A V 2010 Phys. Rev. B 82 075424 [20] Kotov V N, Uchoa B, Pereira V M, Guinea F and Castro Neto A H 2012 Rev. Mod. Phys. 84 1067 [21] Tikhonenko F V, Horsell D W, Gorbachev R V and Savchenko A K 2008 Phys. Rev. Lett. 100 056802 [22] Baker A M R et al 2012 Phys. Rev. B 86 235441 [23] Pan W, Ross I A J, Howell S W, Ohta T, Friedmann T A and Liang C T 2011 New J. Phys. 13 113005 [24] Jouault B, Jabakhanji B, Camara N, Desrat W, Consejo C and Camassel J 2011 Phys. Rev. B 83 195417 [25] Jobst J, Waldmann D, Gornyi I V, Mirlin A D and Weber H B 2012 Phys. Rev. Lett. 108 106601 [26] Yu G L et al 2013 PNAS 110 3282 [27] Chen X L, Wang L, Li W, Wang Y, He Y H, Wu Z F, Han Y, Zhang M W, Xiong W and Wang N 2013 Appl. Phys. Lett. 102 203103 [28] Herbut I F 2009 Physics 2 57 [29] Liu F-H, Hsu C-S, Chuang C, Woo T-P, Huang L-I, Lo S-T, Fukuyama Y, Yang Y, Elmquist R and Liang C-T 2013 Nanoscale Res. Lett. 8 360 [30] Kivelson S, Lee D-H and Zhang S-C 1992 Phys. Rev. B 46 2223 [31] Jiang H W, Johnson C E, Wang K L and Hannahs S T 1993 Phys. Rev. Lett. 71 1439 [32] Hughes R J F, Nicholls J T, Frost J E F, Linfield E H, Pepper M, Ford C J B, Ritchie D A, Jones G A C, Kogan E and Kaveh M 1994 J. Phys.: Condens. Matter 6 4763 [33] Wang T, Clark K P, Spencer G F, Mack A M and Kirk W P 1994 Phys. Rev. Lett. 72 709 [34] Hikami S, Larkin A I and Nagaoka Y 1980 Prog. Theor. Phys. 63 707 [35] McCann E, Kechedzhi K, Fal’ko V I, Suzuura H, Ando T and Altshuler B L 2006 Phys. Rev. Lett. 97 146805 [36] Wu X, Li X, Song Z, Berger C and de Heer W A 2007 Phys. Rev. Lett. 98 136801
4. Conclusion In conclusion, we have investigated localization and e-e interactions in few-layer epitaxial graphene on SiC by studying the temperature and magnetic field dependences of the conductivity tensor. At zero magnetic field, the observed logarithmic T dependence of the longitudinal conductivity can be ascribed to weak localization and e-e interaction. At high magnetic fields, where WL is suppressed, the logarithmic T dependence of the longitudinal conductivity still exits, and is presumably due to e-e interactions. However, we show that the prefactor of this logarithmic correction decreases with increasing the magnetic field, and observe that the Hall conductivity deviates from its expected value. Our results demonstrate that both weak localization and e-e interactions play an important role in the magnetotransport of massive carriers in few-layer epitaxial graphene. We note that in order to obtain a thorough understanding of transport in epitaxial graphene, magnetic-field-dependent e-e interactions as well as Kondo physics need to be considered. Such effects are vastly different from those in conventional semiconductor-based 2D systems and require further experimental and theoretical studies. Acknowledgments This work was funded by the National Science Council (NSC), Taiwan, and National Taiwan University (grant no.: 103R890932 and 103R7552-2).
References [1] Zhang Y, Tan Y-W, Stormer H L and Kim P 2005 Nature 438 201 [2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 6
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[37] Lin J J and Bird J P 2002 J. Phys.: Condens. Matter 14 R501 [38] Minkov G M, Rut O E, Germanenko A V, Sherstobitov A A, Zvonkov B N, Uskova E A and Birukov A A 2002 Phys. Rev. B 65 235322 [39] Minkov G M, Germanenko A V, Rut O E, Sherstobitov A A, Larionova V A, Bakarov A K and Zvonkov B N 2006 Phys. Rev. B 74 045314
[40] Chen J-H, Li L, Cullen W G, Williams E D and Fuhrer M S 2011 Nat. Phys. 7 535 [41] Jobst J, Kisslinger F and Weber H B 2013 Phys. Rev. B 88 155412 [42] Lu Y-F, Lo S-T, Lin J-C, Zhang W, Lu J-Y, Liu F-H, Tseng C-M, Lee Y-H, Liang C-T and Li L-J 2013 ACS Nono 7 6522
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Localization and electron-electron interactions in few-layer epitaxial graphene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 245201 (http://iopscience.iop.org/0957-4484/25/24/245201) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.100.58.76 This content was downloaded on 02/06/2014 at 08:43
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 25 (2014) 245201 (7pp)
doi:10.1088/0957-4484/25/24/245201
Localization and electron-electron interactions in few-layer epitaxial graphene Shun-Tsung Lo1, Fan-Hung Liu1, Chang-Shun Hsu1, Chiashain Chuang2, Lung-I Huang2, Yasuhiro Fukuyama3, Yanfei Yang4, Randolph E Elmquist4 and Chi-Te Liang1,2 1
Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan Department of Physics, National Taiwan University, Taipei 106, Taiwan 3 National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan 4 National Institute of Standards and Technology (NIST), Gaithersburg, MD 20899, USA 2
E-mail: [email protected] Received 28 February 2014, revised 10 April 2014 Accepted for publication 15 April 2014 Published 28 May 2014 Abstract
This paper presents a study of the quantum corrections caused by electron-electron interactions and localization to the conductivity in few-layer epitaxial graphene, in which the carriers responsible for transport are massive. The results demonstrate that the diffusive model, which can generally provide good insights into the magnetotransport of two-dimensional systems in conventional semiconductor structures, is applicable to few-layer epitaxial graphene when the unique properties of graphene on the substrate, such as intervalley scattering, are taken into account. It is suggested that magnetic-field-dependent electron-electron interactions and Kondo physics are required for obtaining a thorough understanding of magnetotransport in few-layer epitaxial graphene. Keywords: graphene, interactions, localization (Some figures may appear in colour only in the online journal) 1. Introduction
should be noted that the prohibition of backscattering off potential fluctuations due to the chiral nature of massless Dirac fermions is one of the most important physical phenomena found in graphene [16]. Therefore, weak antilocalization (WAL) is predicted even in the absence of strong spin–orbit coupling [17]. In perfect graphene, spin–orbit coupling is weak because of the low mass of carbon atoms. However, when the graphene sheet is influenced by a substrate, vacancies, or by extrinsic impurities, short-range disorder leads to an increase in the rate of intervalley scattering, and such scattering can restore the WL behavior [18]. The chiral character of graphene also affects the scattering processes making the e-e interactions sensitive to the details of disorder in the system [19, 20]. Magnetotransport has already been widely investigated in a variety of disordered graphene fabricated by mechanical exfoliation [17, 18, 21], chemical vapor deposition [22], and epitaxial growth on SiC [23–25]. In most cases, either WAL or WL was observed, depending on the strength of intervalley coupling. Apart from interesting results on probing e-e interactions in graphene by
Graphene continues to attract a great deal of world-wide interest because of its extraordinary electrical [1, 2], thermal [3, 4], and optical [5, 6] properties. Although mechanically exfoliated graphene is of ultra-high quality, its small size may limit graphene’s practical applications. In contrast, epitaxial graphene grown from silicon carbide (SiC) substrates [7, 8] can be of wafer size and thus may find real-world applications in high-frequency devices [9, 10]. In the presence of supporting SiC substrates, short-range disorder is always incorporated, yielding a wide variety of rich physical phenomena in epitaxial graphene. To realize fully its great potential for application in future electronic devices, fundamental studies of carrier transport in graphene grown on a substrate are highly desirable. Electron-electron (e-e) interactions and coherent-backscattering-induced localization, or weak localization (WL), are two prominent quantum effects in disordered twodimensional (2D) systems at low temperatures T [11−15]. It 0957-4484/14/245201+07$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
S-T Lo et al
Nanotechnology 25 (2014) 245201
compressibility experiments [26, 27], few experimental studies focus on the broader features of e-e interactions in graphene [28]. Hence, it is interesting and useful to study the behavior of massive carriers subject to a magnetic field in few-layer graphene sheets, and to compare the results with the well-established models applicable to conventional semiconductor systems. Moreover, few-layer graphene, which is less susceptible to air exposure and surface contamination because of the top and bottom protecting layers, is useful for stable device applications as well as an interesting playground for studying fundamental phenomena such as the insulatorquantum Hall transition [29−33]. Here, we investigate the effects of localization and e-e interactions on the massive carriers in few-layer epitaxial graphene by studying its magnetotransport behavior in both the intermediate and high magnetic field regimes. We use two different methods to extract the contribution of e-e interactions. Evidence of e-e interactions and WL is clearly visible at low B. However, we show that the number of effective channels contributing to the e-e interactions may vary with the applied magnetic field in the regime where the WL phenomenon is suppressed. These results appear to be vastly different from those in conventional semiconductor-based 2D electron systems [12, 13].
Figure 1. Converted σxx(B) and σxy(B) at various temperatures T ranging from T = 1.95 K to T = 8 K. The inset shows that the Hall slope δρxy/δB decreases logarithmically with T. The solid line is a guide to the eye.
the inset of figure 1 indicates that the Hall slope (δρxy/δB) shows a lnT dependence and decreases with increasing T. As will be shown later, σxy is T-independent, suggesting that the mobility and carrier density are insensitive to a change in T. Therefore the observed lnT dependence of δρxy/δB is ascribed to e-e interactions. In a conventional semiconductor heterostructure where the 2D electron gas (2DEG) exists at the interface, the observed temperature and magnetic field dependences of the resistivity at kFl ≫ 1 (kF and l are the Fermi wave vector and the mean free path respectively) are generally determined by the quantum corrections to the conductivity caused by WL and e-e interactions. The theory of WL developed by Hikami, Larkin, and Nagaoka (HLN) predicts a correction to the Drude conductivity given by
2. Experimental details Our few-layer graphene sample was grown on an opticallypolished Si-terminated 6H-SiC(0001) surface at 1850 °C for 45 min under an Ar gas pressure slightly higher than 101.3 KPa. The number of layers (≈4 to 5) was identified by comparing the AFM and TEM images. The device was patterned into a Hall-bar configuration by standard optical lithography followed by using a reactive ion etch in O2 plasma and lift-off processing for the Au contacts. The detailed growth process can be found in [29]. Four-terminal magnetoresistivities (longitudinal and Hall resistivities, ρxx and ρxy) were measured using standard dc techniques. The magnetic field is applied perpendicular to the plane of the graphene sheet. Four samples show similar characteristics, and measurements taken on one of them are presented in this paper.
⎧ ⎪ δσxxWL = G0 ⎨ ψ ⎪ ⎩
3. Results and discussion
⎛ B ⎞⎫ ⎛1 Bϕ ⎞ ⎪ ⎛1 Btr ⎞ ⎟ + ln ⎜ tr ⎟ ⎬ , (1) ⎜ + ⎟−ψ⎜ + ⎜ ⎟⎪ ⎝2 ⎝2 B⎠ B⎠ B ⎝ ϕ ⎠⎭
where ψ is the digamma function, G0 = e2/(πh), the transport magnetic field Btr = ℏ
Figure 1 shows the longitudinal and Hall conductivities (σxx and σxy) as a function of magnetic field B obtained from ρxx(B) and ρxy(B) according to σxx(B) = ρxx(B)/(ρxx(B)2 + ρxy(B)2) and σxy(B) = ρxy(B)/(ρxx(B)2 + ρxy(B)2). At low B, σxx increases with increasing B, showing a character typical of WL. It is believed that intervalley coupling is sufficiently strong for the observation of WL due to atomically sharp defects at the interface with the SiC substrate [23−25], which leads to a significant enhancement of the rate of intervalley scattering. With increasing B, σxx decreases starting around 0.5 T. Over the whole B range, σxx increases with increasing temperature T, indicative of insulating behavior. Moreover,
( 2el ), 2
and Bϕ = ℏ
( 4el ) 2
ϕ
[34].
Here lφ stands for the dephasing length. On the other hand, the strength of e-e interactions are linked to the value of kBTτ ℏ, where τ is the transport relaxation time. It is expected that the e-e interactions contribute a correction to σxx without affecting σxy in the diffusive regime ( kBTτ ℏ ≪ 1) [13]. For gμBB < kT, where μB is the Bohr magneton, this contribution does not depend on B and has the form ⎛ k Tτ ⎞ δσxxee = Aee G0 ln ⎜ B ⎟ , ⎝ ℏ ⎠ 2
(2)
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⎡ ln ( 1 + F0σ ) Aee = 1 + c ⎢ 1 − ⎢⎣ F0σ
⎤ ⎥, ⎥⎦
(3)
where the constant c corresponds to the number of multiplet channels participating in e-e interactions and F0σ is the Fermiliquid interaction parameter [12, 13]. As a result, in the diffusive regime at B > Btr, where the WL effect is suppressed, the conductivity component σxx and σxy can be written as σxx =
neμ 2
1 + ( μB )
σxy =
+ δσee,
neμ2 B 2
1 + ( μB )
,
(4)
(5)
where the first term represents the classical contribution to the conductivity with transport mobility μ [12, 13]. Figure 1 clearly shows that σxy is T-independent whereas σxx depends on T in the studied field range. Since this is characteristic of the diffusive e-e interactions described above, we fit the experimental σxy(B, T = 1.95 K) to equation (5) for B < 1 T with μ as the fitting parameter and n (≈2.02 × 1013 cm−2) estimated from the high-T Hall slope. These results, together with the extrapolation of the fitting curve, are presented in figure 2(a). Given the estimated μ (≈578 cm2 Vs−1) and the known n, we can obtain the classical contribution, neμ/(1 + (μB)2), to the longitudinal conductivity. From the Drude conductivity σD = neμ = 2πkFlG0 (where the factor of 2 is due to the valley degeneracy of graphene) for conventional 2DEG, kFl ≈ 24 is approximately evaluated, indicating that our graphene device is well inside in the weakly-localized regime. Thus, although it is found that equation (2) cannot explain the data at high B, we can use equations (1)−(5) to analyze the data adequately for low B. Figure 2(b) compares the data of σxx at T = 1.95 K with neμ/ [1 + (μB)2]. Consistent with equations (2) and (4), the correction due to e-e interactions appears to reduce the conductivity. In the following, we eliminate the WL contribution from the measured σxx and study the e-e interactions that are the main findings in this paper. Figure 3(a) presents the results of further magnetotransport measurements around B = 0. In figure 3(b), we show the fits of Δσxx(B) = σxx(B) − σxx(0) at T = 0.89 K to equation (1) for −0.1 T < B < 0.1 T and for −0.5 T < B < 0.5 T, which are denoted by the red and blue curves respectively. It is believed that the WL effect is strongest for B < |Btr|. We can estimate Btr ≈ 0.36 T using kFl ≈ 24 and kF = (πn)1/2. The good agreement with the HLN theory occurs when the fitting range is limited within |Btr| ≈ 0.36 T. A good WL fitting to the data over −0.5 T < B < 0.5 T according to the formula of McCann et al [35] for graphene can be observed in figure 3(c). As reported by Wu et al [36], both the conventional HLN theory and the one developed by McCann et al can provide a reasonable explanation to the low-field positive magnetoconductivity in epitaxial graphene with the presence of significant intervalley scattering. The strong intervalley scattering causes the 2DEG in epitaxial graphene to behave as an ordinary 2DEG in the conventional semiconductor
Figure 2. Comparison of the experimentally measured (a) σxy(B) and (b) σxx(B) with the calculated neμ2B/(1 + (μB)2) and neμ/(1 + (μB)2), respectively, using the parameters obtained from the fit to equation (5), which is denoted by the red curve in (a).
heterostructure. However, at relatively high B, the chiral nature of Dirac fermions in graphene may give rise to antilocalization behavior in the sense that σxx decreases with increasing B, which was considered in the model of McCann et al. Hence, we can expect that the theory of McCann et al [35] gives a good fit over a much larger range of B compared with that of Hikami et al [34] figure 3(d) plots the T dependence of lφ obtained from the fitting of these two different models to the WL peak. Our data can be fitted to the HLN theory and the fit yields lφ ∼ T−0.29. The data can be also fitted 3
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Figure 3. (a) σxx(B) at various T around B = 0. From top to bottom: T = 30 K, 25 K, 21 K, 18 K, 15 K, 12 K, 10 K, 8 K, 6 K, 4 K, 1.98 K, 0.89 K, respectively. (b) The red and blue curves correspond to the best fits of equation (1) to Δσxx(B, T = 0.89 K) over different field ranges. (c) A fit to Δσxx(B, T = 0.89 K) with the help of the model developed by McCann et al for graphene. (d) lφ versus T on a log-log scale, which is obtained from the fitting procedures in (b) and (c). (e) σxx(B = 0) versus T on a semi-log scale. Dashed lines denote the extrapolation of the fitting curves.
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to the model of McCann et al and gives lφ ∼ T−0.24. Such close exponents suggest that both models can be used to explain our results. When e-e scattering dominates the dephasing [23, 37], we can expect that lφ ∼ T−0.5. Our results, showing a weaker T dependence of lφ, indicate that scattering mechanisms other than e-e scattering should be involved. Figure 3(e) then shows the zero B field σxx as a function of T. Logarithmic temperature dependence with the prefactor of 1.62G0 is observed. For our device considering Btr ≫ Bφ, at zero B, the WL contribution described by equation (1) can be simply approximated as δσxxWL = pGo ln ( T ) + constant, [23, 38], where p is the scaling parameter of lϕ ~ T −p 2 . Therefore we show that e-e interactions play a role in σxx(B = 0, T) and determine its contribution with Aee ≈ 1.04 in equation (2). Normally both WL and e-e interaction correction terms are small, independent of each other, and proportional to 1/(kFl). In our system, since p ≈ 0.5 (0.6), the slope in the σxx versus lnT plot due to e-e interaction correction is expected to be 1.62 − 0.60 ≈ 1. Therefore, the e-e interaction correction term is about two times the WL term. Over the temperature range between 2 K and 30 K, the measured zero-B-field conductivity is varied between 1.71 mS and 1.73 mS. Such a small change of 20 μS in the conductivity is due to both interaction and WL correction terms since the Drude part is expected to be temperature-independent. Therefore, the contribution of interaction and WL correction to this change are calculated to be 13.4 μS and 6.7 μS respectively. The contribution of e-e interactions to σxx can also be extracted by suppressing the WL effect at B ≫ Btr ≈ 0.36 T. Figure 4(a) shows σxx at high B, which increases with increasing T. The inset of the same figure demonstrates that this insulating behavior follows σxx ∼ ln(T), which is expected to be due to e-e interactions. From the slope of the linear fits to this logarithmic T dependence, the prefactor Aee in equation (1) as a function of B (⩾10Btr ≈ 3.6 T) where WL is expected to be suppressed is obtained. These results are presented in figure 4(b). However, in contrast to the prediction of diffusive e-e interactions, Aee is shown to depend on the magnetic field. We find that Aee ≈ 0.23 at B = 8 T and increases to 0.72 when B is reduced to 3.6 T, which is different from the value of Aee ≈ 1.04 deduced at low fields. When a 2D system is out of diffusive regime, i.e. kBTτ ℏ > 1 referred to as the ballistic regime, the mobility μ in equations (4) and (5) and thereby σxy will become T-dependent [39]. However, in the inset of figure 4(b), it is shown that σxy is strictly T-independent, suggesting that the observed T-dependent correction to σxx does not result from the high-T ballistic effects. Interestingly, the shape of WL peak in graphene is expected to depend strongly on the rates of both intervalley and intravalley scattering. Jobst et al [25] demonstrated that increasing T, which effectively enhances the strength of intervalley and intravalley scattering, changes the number of multiplet channels in equation (3) from c = 3 to c = 7. For the epitaxial graphene on SiC, the theoretical value of F0σ is found to be about −0.09 and thus Aee ≈ 1–0.046c [25]. Therefore, our results may correspond to a crossover from c = 17 (Aee ≈ 0.23) to c = 6 (Aee ≈ 0.72) with decreasing the magnetic
Figure 4. (a) σxx(B) at various T for B > 7.5 T. Inset: semi-log plot of σxx(T) at various B. (b) The obtained prefactor Aee in equation (2) as a function of B. Inset: σxy(B) at various T for B > 3.6 T.
field from B = 8 T to B = 3.6 T. However, the maximum number of multiplet channels due to fourfold spin and fourfold valley degeneracy is c = 15 [19, 25], which occurs when intervalley and intravalley scattering is insignificant and can only be realized in perfect graphene. In our disordered fewlayer graphene, where WL has been observed at low B, c should be less than 15. Therefore, our new experimental results suggest that varying the magnetic field may cause a 5
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variation of the relative strength of intervalley and intravalley in graphene [42], and call for further studies. According to the data shown in figure 2(a), there is an additional contribution to ρxx(B), which shifts σxy below the expected value of equation (5). We note that it is not due to a classical mechanism of magnetoresistance (MR) since in our few-layer graphene, μB (≈0.46) < 1 even at B = 8 T. It has been reported that the Kondo effect is important in graphene in the presence of defects acting as magnetic moments [40, 41], which can also give a logarithmic correction to ρxx(T) and thus may be one of the possible mechanisms leading to a change in the prefactor of the logarithmic T dependence of σxx with B, as observed in figure 4(b). Consequently, our work demonstrates that in addition to the usual MR effects, mechanisms such as valley-dependent scattering and Kondo physics should be carefully considered to understand the magnetotransport in disordered few-layer epitaxial graphene.
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4. Conclusion In conclusion, we have investigated localization and e-e interactions in few-layer epitaxial graphene on SiC by studying the temperature and magnetic field dependences of the conductivity tensor. At zero magnetic field, the observed logarithmic T dependence of the longitudinal conductivity can be ascribed to weak localization and e-e interaction. At high magnetic fields, where WL is suppressed, the logarithmic T dependence of the longitudinal conductivity still exits, and is presumably due to e-e interactions. However, we show that the prefactor of this logarithmic correction decreases with increasing the magnetic field, and observe that the Hall conductivity deviates from its expected value. Our results demonstrate that both weak localization and e-e interactions play an important role in the magnetotransport of massive carriers in few-layer epitaxial graphene. We note that in order to obtain a thorough understanding of transport in epitaxial graphene, magnetic-field-dependent e-e interactions as well as Kondo physics need to be considered. Such effects are vastly different from those in conventional semiconductor-based 2D systems and require further experimental and theoretical studies. Acknowledgments This work was funded by the National Science Council (NSC), Taiwan, and National Taiwan University (grant no.: 103R890932 and 103R7552-2).
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