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Temperature dependent measurements on two decoupled graphene monolayers
ARTICLE IN PRESS Physica E 42 (2010) 699–702
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Temperature dependent measurements on two decoupled graphene monolayers ¨ H. Schmidt , T. Ludtke, P. Barthold, R.J. Haug Institut f¨ ur Festk¨ orperphysik, Leibniz Universit¨ at Hannover, Appelstr. 2, 30167 Hannover, Germany
a r t i c l e in fo
abstract
Article history: Received 3 July 2009 Received in revised form 14 October 2009 Accepted 11 November 2009 Available online 5 December 2009
Single and coupled bilayer graphene exhibit outstanding electronic properties, making them candidates for new electronic devices. Apart from these, bilayer systems consisting of two decoupled monolayers can be produced. We investigate the electronic properties of such a decoupled double layer system formed by the folding of a single one. Electronic transport through these two very closely spaced monolayers is analyzed, varying carrier concentration, magnetic field and temperature from 1.5 up to 50 K. Using the temperature dependence of the Shubnikov–de Haas oscillations observed in the longitudinal resistance, the cyclotron masses for electrons and holes in both layers are obtained for different carrier concentrations. These values are higher than those expected from a single graphene monolayer, confirming a reduced Fermi velocity in such twisted samples. & 2009 Elsevier B.V. All rights reserved.
Keywords: Graphene Decoupled monolayers Shubnikov–de Haas effect
1. Introduction Since the evidence of the existence of freestanding atomically thin layers of carbon in 2004 [1], this so-called graphene monoand bilayers have drawn a lot of attention and its outstanding electronic and chemical properties are subject to numerous publications [2–5]. Monolayer graphene is a gapless semiconductor with a Dirac-type spectrum of charge carriers, which are considered to be massless Dirac fermions. While monolayer graphene shows an unconventional half integer quantum Hall effect with a Berry phase of p [2], a peculiar double step is observed in conventional bilayer graphene. These Bernal stacked bilayers, referred to as single crystal (SC) bilayers, have been intensively studied, also showing outstanding properties [6,7]. In contrast to these SC bilayers, samples with twisted layers are also found and investigated [8–10]. Folding a monolayer forms a system with two layers which are rotated with respect to Bernal stacking and are therefore decoupled. Measurements on such systems show independent electronic transport through the two monolayers, despite their proximity [8].
2. Sample preparation Micromechanical cleavage is used to rip off thin flakes of natural graphite [11] and to place them on a silicon wafer. This Corresponding author.
E-mail address: [email protected] (H. Schmidt). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.144
wafer is covered with 330 nm of silicon dioxide, acting as a dielectric between sample and silicon. The flakes are identified with an optical microscope using optical bandwidth filter to increase the contrast. After locating a suitable flake, the sample is processed into a Hall bar via e-beam lithography and oxygen plasma etching. In the next step, contacts are defined using PMMA resist which is exposed to e-beam, and 8 nm of chromium and 40 nm of gold is evaporated. Fig. 1 shows a sketch of such two layers, processed into a Hall bar and contacted with six leads.
3. Measurements and results 3.1. Magnetotransport measurements For a first characterization, the sample is cooled down to 1.5 K and transport measurements with an additional perpendicular magnetic field up to 13 T are performed. Tuning the backgate voltage VBG to positive (negative) values, electrons (holes) are induced into the two layers as majority charge carriers. Note that an offset due to unintentional doping has to be taken into account, shifting the transition from holes to electrons, the neutrality point, to approximately þ 12 V in the bottom layer and to þ 17 V in the upper layer. Away from these neutrality points, the carrier concentration in the bottom layer, n1 , is significantly larger than the concentration n2 in the top layer due to screening effects [8]. The inset in Fig. 2 shows the longitudinal resistance while changing the gate voltage, exhibiting the typical field effect peak known from mono- and SC bilayer systems. The gray lines mark the backgate voltages at which further temperature dependent
ARTICLE IN PRESS 700
H. Schmidt et al. / Physica E 42 (2010) 699–702
B
300
ΔR (Ω)
0
VBG
-300
Fig. 1. Sketch of the investigated sample. Two graphene layers are lying on top of each other rotated with respect to Bernal stacking. These layers have been processed into a Hall bar and contacted with Cr/Au contacts. The sample is capacitively coupled with the silicon backgate and exposed to a perpendicular magnetic field B.
0.2
0.4 1/B (1/T)
15
R (kΩ)
5
-6 0
0
60
26
22
ΔR (Ω)
4
R (kΩ)
500 10
18
0
0
VBG (V)
-500 0.2
10
2 0
2
6
4
8
12
B (T) Fig. 2. Longitudinal resistance at 1:5 K and VBG ¼ 60 V, showing the superposition of two different Shubnikov–de Haas oscillations. Filling factors are marked for both oscillations, higher (lower) numbers correspond to the bottom (top) layer. Inset: Electrical field effect at B ¼ 0 T showing a peak at 12 V. The gray lines mark the voltages at which further measurements have been performed.
measurements have been carried out, 60; 40 and 20 V in the hole region, and þ 40; þ 50 and þ 60 V in the electron region. To distinguish this system from a monolayer or a SC bilayer, magnetotransport measurements are performed. Fig. 2 shows for example a measurement at a backgate voltage of 60 V. Sweeping the perpendicular magnetic field, two superimposed Shubnikov– de Haas (SdH) oscillations are observed in the longitudinal resistance. The most important fact is that these two oscillations both exhibit a Berrys phase of p. This phase, typical for graphene monolayers, indicates charge transport through two decoupled monolayers in our system. Due to the different carrier concentrations n1 and n2 , the oscillations belonging to the upper and bottom layers are quite different and can be separated well. The one belonging to n2 yields minima with lower filling factors (2, 6, 10) while the one belonging to n1 yields larger ones (18, 22, 26) in the same magnetic field region. From such filling factors the carrier densities are obtained for both layers, being n1 ¼ 4:9 1012 =cm2 and n2 ¼ 0:7 1012 =cm2 in this case. 3.2. Temperature dependent measurements To learn more about the charge carriers, the temperature dependence of the oscillations’ amplitudes is examined. Measurements at different temperatures T ¼ 1:5, 10, 15, 25, 35 and 50 K are performed, sweeping the magnetic field at fixed gate voltages.
0.4 0.6 1/B (1/T)
0.8
Fig. 3. Shubnikov–de Haas oscillations at 60 V (a) and þ 60 V (b), plotted over the inverse magnetic field at different temperatures: 1.5, 10, 15, 25 and 35 K. A background at 50 K has been subtracted. The arrows indicate the maxima, at which the damping is shown in Fig. 4.
Fig. 3 shows the longitudinal resistance at 760 V at different temperatures over 1=B, exhibiting two sets of equidistant minima with amplitudes increasing with decreasing temperature for both oscillations. DR ¼ RðTÞRð50 KÞ is plotted to pronounce the two oscillations. This temperature dependence of the oscillations in the longitudinal resistance is described by the amplitude factor RT [12], being RT ¼
w
ð1Þ
sinhðwÞ
with w ¼ 2p2 kB T=‘ oc , the cyclotron frequency oc ¼ eB=mc , the cyclotron mass mc and the fundamental constants kB ; ‘ and e. Analyzing only the part of the oscillations amplitude which is affected by the temperature [2], geometry factors can be ignored and, with the parameter P ¼ 2p2 kB mc =‘ e, the amplitude can be approximated as Ap
w sinhðwÞ
p
ðT=BÞ : sinhðP ðT=BÞÞ
ð2Þ
From the measured SdH oscillations, the amplitude is taken at a fixed backgate voltage, different temperatures and a fixed magnetic field, corresponding to a minimum or maximum of one of the two oscillations. A curve following Eq. (2) is fitted to these data, obtaining the parameter including the cyclotron mass. Fig. 4 shows the temperature damping for the two maxima marked in Fig. 3b, at 60 V backgate voltage and B ¼ 8:14 T and 2:44 T, belonging to maxima of the oscillations of layers 1 and 2, respectively. The measured amplitude for both cases is plotted over B=T as squares and circles, while the corresponding fits are shown as solid lines. Since the mass is the only non-constant parameter in the formula, it becomes immediately clear from this
ARTICLE IN PRESS H. Schmidt et al. / Physica E 42 (2010) 699–702
701
0.8
40 ΔR (Ω)
Amplitude (normalized)
60
20
0.4
0 0.0 0
8
-300
16 T/B (K/T)
Fig. 4. Normalized amplitudes at two fixed magnetic fields of the SdH oscillations at a gate voltage of 60 V, plotted over T=B for better comparison. Squares correspond to the n1 oscillation at B ¼ 8:14 T while the circles belong to the other one at 2:44 T. Fits through both data are shown as gray lines.
0.075
mc/me
0.050
0.025
0.000 -4
-2
0
0
300
B (mT)
2
4
n (1012/cm2) Fig. 5. Cyclotron masses of charge carriers in units of me in the two layer system, obtained from the temperature dependence of the SdH oscillations. The experimental obtained masses for the upper and bottom layer are shown as squares and dots, respectively. Negative charge carrier concentration indicates holes, positive electrons. The expected value of vF ¼ 1 106 m=s is represented by the black dashed line. Fits through the first layers’ masses are shown as dotted gray (symmetric fit) and the solid black line (asymmetric fit).
graph, that the masses of the charge carriers in layer 1 and 2 are significantly different. This difference of the masses in the two layers can be explained by the different carrier densities. As shown in Ref. [13], the cyclotron mass in monolayer graphene can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ mc ¼ h2 n=4pv2F with the Fermi velocity vF , assumed to be 1 106 m=s in a single layer. Using the same fitting procedure as shown in Fig. 4, the cyclotron masses for both layers at various backgate voltages are obtained and plotted in Fig. 5 over the according carrier concentration. Masses of charge carriers in the upper layer are drawn as squares, while those for carriers in the bottom layer are drawn as dots. The black dotted line shows the values one would expect using Eq. (3) with a Fermi velocity of vF ¼ 1 106 m=s, measured for single monolayer systems [13]. While the values for the second layer are in good accordance with the monolayer
Fig. 6. Longitudinal resistance at VBG ¼ 60 V at different temperatures (top down: 1.5, 10, 15, 25, 35 K) as a function of magnetic field. The curves have been shifted for better visibility.
values, the ones for the bottom layer are significantly larger and also seem to yield different values for holes and electrons. This asymmetry could be due to the doping effects. The dotted gray line is a fit of Eq. (3), containing all n1 values, implying same Fermi velocity for electrons and holes. This fit gives a velocity of vF ¼ 0:71 106 m=s, reduced with respect to the monolayer value. The black line shows the fit assuming different values for both carrier types, giving vF ¼ 0:66 106 m=s for holes and 0:81 106 m=s for electrons. In both the symmetric and the asymmetric fit, the Fermi velocity is significantly smaller in the bottom layer of the decoupled system than the expected value for monolayers. Such a reduction has been predicted theoretically [10] especially for very small angles of rotation. Experiments on folded flakes via Raman spectroscopy [9] have shown reductions down to approximately six percent, much smaller than our results. This difference might be due to different rotation angles between the two layers with respect to Bernal stacking. Larger reductions comparable to our results have been observed in epitaxial graphene [14], yielding Fermi velocities down to 0:7 106 m=s. 3.3. Low field region In addition to the high magnetic field region, data close to B ¼ 0 T are shown in Fig. 6, shifted for better visibility. Two different effects are observed. Around zero magnetic field a sharp peak in resistance can be seen, showing the weak localization as predicted and observed in monolayer devices [15,16]. With higher temperatures this effect becomes smaller and is hardly visible at 25 and 35 K. Apart from the weak localization, conductance fluctuations being especially strong at B 7 250 mT are visible, also vanishing with increasing temperature. Both effects are similar to those in monolayer devices, but most probably containing contributions from both layers which cannot be clearly separated.
4. Conclusion Measurements on decoupled graphene monolayers have been performed, changing carrier concentration, magnetic field and temperature. Close to zero magnetic field a weak localization peak is observed while at higher magnetic fields Shubnikov–de Haas oscillations become visible. From the temperature dependent
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damping of these oscillations of both layers, cyclotron masses are pffiffiffi obtained. These masses are proportional to n1=2 as expected, but yield a reduced Fermi velocity for the bottom layer of 66–80 percent in comparison to the value for a monolayer of vF ¼ 1 106 m=s. Acknowledgments The authors like to thank V.I. Fal’ko and E. McCann for useful discussions. This work has been financially supported by the excellence cluster QUEST within the German Excellence Initiative. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] Y. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature 438 (2005) 201.
[3] A.K. Geim, K.S. Novoselov, Nature Mater. 6 (2007) 183. [4] A.K. Geim, Science 324 (2009) 1530. [5] S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, D.C. Elias, J.A. Jaszczak, A.K. Geim, Phys. Rev. Lett. 100 (2008) 016602. [6] K.S. Novoselov, E. McCann, S.V. Morozov, V.I. Fal’ko, M.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A.K. Geim, Nature Phys. 2 (2006) 177. [7] E. McCann, V.I. Fal’ko, Phys. Rev. Lett. 96 (2006) 086805. ¨ [8] H. Schmidt, T. Ludtke, P. Barthold, E. McCann, V.I. Fal’ko, R.J. Haug, Appl. Phys. Lett. 93 (2008) 172108. [9] Z. Ni, Y. Wang, T. Yu, Y. You, Z. Shen, Phys. Rev. B 77 (2008) 235403. [10] J.M.B. Lopes dos Santos, N.M.R. Peres, A.H. Castro Neto, Phys. Rev. Lett. 99 (2007) 256802. [11] NGS Naturgraphit GmbH. [12] V.P. Gusynin, S.G. Sharapov, Phys. Rev. B 71 (2005) 125124. [13] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [14] W.A. de Heer, et al., Solid State Comm. 143 (2007) 92. [15] E. McCann, K. Kechedzhi, V.I. Fal’ko, H. Suzuura, T. Ando, B.L. Altshuler, Phys. Rev. Lett. 97 (2006) 146805. [16] F.V. Tikhonenko, D.W. Horsell, R.V. Gorbachev, A.K. Savchenko, Phys. Rev. Lett. 100 (2008) 056802.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Temperature dependent measurements on two decoupled graphene monolayers ¨ H. Schmidt , T. Ludtke, P. Barthold, R.J. Haug Institut f¨ ur Festk¨ orperphysik, Leibniz Universit¨ at Hannover, Appelstr. 2, 30167 Hannover, Germany
a r t i c l e in fo
abstract
Article history: Received 3 July 2009 Received in revised form 14 October 2009 Accepted 11 November 2009 Available online 5 December 2009
Single and coupled bilayer graphene exhibit outstanding electronic properties, making them candidates for new electronic devices. Apart from these, bilayer systems consisting of two decoupled monolayers can be produced. We investigate the electronic properties of such a decoupled double layer system formed by the folding of a single one. Electronic transport through these two very closely spaced monolayers is analyzed, varying carrier concentration, magnetic field and temperature from 1.5 up to 50 K. Using the temperature dependence of the Shubnikov–de Haas oscillations observed in the longitudinal resistance, the cyclotron masses for electrons and holes in both layers are obtained for different carrier concentrations. These values are higher than those expected from a single graphene monolayer, confirming a reduced Fermi velocity in such twisted samples. & 2009 Elsevier B.V. All rights reserved.
Keywords: Graphene Decoupled monolayers Shubnikov–de Haas effect
1. Introduction Since the evidence of the existence of freestanding atomically thin layers of carbon in 2004 [1], this so-called graphene monoand bilayers have drawn a lot of attention and its outstanding electronic and chemical properties are subject to numerous publications [2–5]. Monolayer graphene is a gapless semiconductor with a Dirac-type spectrum of charge carriers, which are considered to be massless Dirac fermions. While monolayer graphene shows an unconventional half integer quantum Hall effect with a Berry phase of p [2], a peculiar double step is observed in conventional bilayer graphene. These Bernal stacked bilayers, referred to as single crystal (SC) bilayers, have been intensively studied, also showing outstanding properties [6,7]. In contrast to these SC bilayers, samples with twisted layers are also found and investigated [8–10]. Folding a monolayer forms a system with two layers which are rotated with respect to Bernal stacking and are therefore decoupled. Measurements on such systems show independent electronic transport through the two monolayers, despite their proximity [8].
2. Sample preparation Micromechanical cleavage is used to rip off thin flakes of natural graphite [11] and to place them on a silicon wafer. This Corresponding author.
E-mail address: [email protected] (H. Schmidt). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.144
wafer is covered with 330 nm of silicon dioxide, acting as a dielectric between sample and silicon. The flakes are identified with an optical microscope using optical bandwidth filter to increase the contrast. After locating a suitable flake, the sample is processed into a Hall bar via e-beam lithography and oxygen plasma etching. In the next step, contacts are defined using PMMA resist which is exposed to e-beam, and 8 nm of chromium and 40 nm of gold is evaporated. Fig. 1 shows a sketch of such two layers, processed into a Hall bar and contacted with six leads.
3. Measurements and results 3.1. Magnetotransport measurements For a first characterization, the sample is cooled down to 1.5 K and transport measurements with an additional perpendicular magnetic field up to 13 T are performed. Tuning the backgate voltage VBG to positive (negative) values, electrons (holes) are induced into the two layers as majority charge carriers. Note that an offset due to unintentional doping has to be taken into account, shifting the transition from holes to electrons, the neutrality point, to approximately þ 12 V in the bottom layer and to þ 17 V in the upper layer. Away from these neutrality points, the carrier concentration in the bottom layer, n1 , is significantly larger than the concentration n2 in the top layer due to screening effects [8]. The inset in Fig. 2 shows the longitudinal resistance while changing the gate voltage, exhibiting the typical field effect peak known from mono- and SC bilayer systems. The gray lines mark the backgate voltages at which further temperature dependent
ARTICLE IN PRESS 700
H. Schmidt et al. / Physica E 42 (2010) 699–702
B
300
ΔR (Ω)
0
VBG
-300
Fig. 1. Sketch of the investigated sample. Two graphene layers are lying on top of each other rotated with respect to Bernal stacking. These layers have been processed into a Hall bar and contacted with Cr/Au contacts. The sample is capacitively coupled with the silicon backgate and exposed to a perpendicular magnetic field B.
0.2
0.4 1/B (1/T)
15
R (kΩ)
5
-6 0
0
60
26
22
ΔR (Ω)
4
R (kΩ)
500 10
18
0
0
VBG (V)
-500 0.2
10
2 0
2
6
4
8
12
B (T) Fig. 2. Longitudinal resistance at 1:5 K and VBG ¼ 60 V, showing the superposition of two different Shubnikov–de Haas oscillations. Filling factors are marked for both oscillations, higher (lower) numbers correspond to the bottom (top) layer. Inset: Electrical field effect at B ¼ 0 T showing a peak at 12 V. The gray lines mark the voltages at which further measurements have been performed.
measurements have been carried out, 60; 40 and 20 V in the hole region, and þ 40; þ 50 and þ 60 V in the electron region. To distinguish this system from a monolayer or a SC bilayer, magnetotransport measurements are performed. Fig. 2 shows for example a measurement at a backgate voltage of 60 V. Sweeping the perpendicular magnetic field, two superimposed Shubnikov– de Haas (SdH) oscillations are observed in the longitudinal resistance. The most important fact is that these two oscillations both exhibit a Berrys phase of p. This phase, typical for graphene monolayers, indicates charge transport through two decoupled monolayers in our system. Due to the different carrier concentrations n1 and n2 , the oscillations belonging to the upper and bottom layers are quite different and can be separated well. The one belonging to n2 yields minima with lower filling factors (2, 6, 10) while the one belonging to n1 yields larger ones (18, 22, 26) in the same magnetic field region. From such filling factors the carrier densities are obtained for both layers, being n1 ¼ 4:9 1012 =cm2 and n2 ¼ 0:7 1012 =cm2 in this case. 3.2. Temperature dependent measurements To learn more about the charge carriers, the temperature dependence of the oscillations’ amplitudes is examined. Measurements at different temperatures T ¼ 1:5, 10, 15, 25, 35 and 50 K are performed, sweeping the magnetic field at fixed gate voltages.
0.4 0.6 1/B (1/T)
0.8
Fig. 3. Shubnikov–de Haas oscillations at 60 V (a) and þ 60 V (b), plotted over the inverse magnetic field at different temperatures: 1.5, 10, 15, 25 and 35 K. A background at 50 K has been subtracted. The arrows indicate the maxima, at which the damping is shown in Fig. 4.
Fig. 3 shows the longitudinal resistance at 760 V at different temperatures over 1=B, exhibiting two sets of equidistant minima with amplitudes increasing with decreasing temperature for both oscillations. DR ¼ RðTÞRð50 KÞ is plotted to pronounce the two oscillations. This temperature dependence of the oscillations in the longitudinal resistance is described by the amplitude factor RT [12], being RT ¼
w
ð1Þ
sinhðwÞ
with w ¼ 2p2 kB T=‘ oc , the cyclotron frequency oc ¼ eB=mc , the cyclotron mass mc and the fundamental constants kB ; ‘ and e. Analyzing only the part of the oscillations amplitude which is affected by the temperature [2], geometry factors can be ignored and, with the parameter P ¼ 2p2 kB mc =‘ e, the amplitude can be approximated as Ap
w sinhðwÞ
p
ðT=BÞ : sinhðP ðT=BÞÞ
ð2Þ
From the measured SdH oscillations, the amplitude is taken at a fixed backgate voltage, different temperatures and a fixed magnetic field, corresponding to a minimum or maximum of one of the two oscillations. A curve following Eq. (2) is fitted to these data, obtaining the parameter including the cyclotron mass. Fig. 4 shows the temperature damping for the two maxima marked in Fig. 3b, at 60 V backgate voltage and B ¼ 8:14 T and 2:44 T, belonging to maxima of the oscillations of layers 1 and 2, respectively. The measured amplitude for both cases is plotted over B=T as squares and circles, while the corresponding fits are shown as solid lines. Since the mass is the only non-constant parameter in the formula, it becomes immediately clear from this
ARTICLE IN PRESS H. Schmidt et al. / Physica E 42 (2010) 699–702
701
0.8
40 ΔR (Ω)
Amplitude (normalized)
60
20
0.4
0 0.0 0
8
-300
16 T/B (K/T)
Fig. 4. Normalized amplitudes at two fixed magnetic fields of the SdH oscillations at a gate voltage of 60 V, plotted over T=B for better comparison. Squares correspond to the n1 oscillation at B ¼ 8:14 T while the circles belong to the other one at 2:44 T. Fits through both data are shown as gray lines.
0.075
mc/me
0.050
0.025
0.000 -4
-2
0
0
300
B (mT)
2
4
n (1012/cm2) Fig. 5. Cyclotron masses of charge carriers in units of me in the two layer system, obtained from the temperature dependence of the SdH oscillations. The experimental obtained masses for the upper and bottom layer are shown as squares and dots, respectively. Negative charge carrier concentration indicates holes, positive electrons. The expected value of vF ¼ 1 106 m=s is represented by the black dashed line. Fits through the first layers’ masses are shown as dotted gray (symmetric fit) and the solid black line (asymmetric fit).
graph, that the masses of the charge carriers in layer 1 and 2 are significantly different. This difference of the masses in the two layers can be explained by the different carrier densities. As shown in Ref. [13], the cyclotron mass in monolayer graphene can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ mc ¼ h2 n=4pv2F with the Fermi velocity vF , assumed to be 1 106 m=s in a single layer. Using the same fitting procedure as shown in Fig. 4, the cyclotron masses for both layers at various backgate voltages are obtained and plotted in Fig. 5 over the according carrier concentration. Masses of charge carriers in the upper layer are drawn as squares, while those for carriers in the bottom layer are drawn as dots. The black dotted line shows the values one would expect using Eq. (3) with a Fermi velocity of vF ¼ 1 106 m=s, measured for single monolayer systems [13]. While the values for the second layer are in good accordance with the monolayer
Fig. 6. Longitudinal resistance at VBG ¼ 60 V at different temperatures (top down: 1.5, 10, 15, 25, 35 K) as a function of magnetic field. The curves have been shifted for better visibility.
values, the ones for the bottom layer are significantly larger and also seem to yield different values for holes and electrons. This asymmetry could be due to the doping effects. The dotted gray line is a fit of Eq. (3), containing all n1 values, implying same Fermi velocity for electrons and holes. This fit gives a velocity of vF ¼ 0:71 106 m=s, reduced with respect to the monolayer value. The black line shows the fit assuming different values for both carrier types, giving vF ¼ 0:66 106 m=s for holes and 0:81 106 m=s for electrons. In both the symmetric and the asymmetric fit, the Fermi velocity is significantly smaller in the bottom layer of the decoupled system than the expected value for monolayers. Such a reduction has been predicted theoretically [10] especially for very small angles of rotation. Experiments on folded flakes via Raman spectroscopy [9] have shown reductions down to approximately six percent, much smaller than our results. This difference might be due to different rotation angles between the two layers with respect to Bernal stacking. Larger reductions comparable to our results have been observed in epitaxial graphene [14], yielding Fermi velocities down to 0:7 106 m=s. 3.3. Low field region In addition to the high magnetic field region, data close to B ¼ 0 T are shown in Fig. 6, shifted for better visibility. Two different effects are observed. Around zero magnetic field a sharp peak in resistance can be seen, showing the weak localization as predicted and observed in monolayer devices [15,16]. With higher temperatures this effect becomes smaller and is hardly visible at 25 and 35 K. Apart from the weak localization, conductance fluctuations being especially strong at B 7 250 mT are visible, also vanishing with increasing temperature. Both effects are similar to those in monolayer devices, but most probably containing contributions from both layers which cannot be clearly separated.
4. Conclusion Measurements on decoupled graphene monolayers have been performed, changing carrier concentration, magnetic field and temperature. Close to zero magnetic field a weak localization peak is observed while at higher magnetic fields Shubnikov–de Haas oscillations become visible. From the temperature dependent
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damping of these oscillations of both layers, cyclotron masses are pffiffiffi obtained. These masses are proportional to n1=2 as expected, but yield a reduced Fermi velocity for the bottom layer of 66–80 percent in comparison to the value for a monolayer of vF ¼ 1 106 m=s. Acknowledgments The authors like to thank V.I. Fal’ko and E. McCann for useful discussions. This work has been financially supported by the excellence cluster QUEST within the German Excellence Initiative. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] Y. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature 438 (2005) 201.
[3] A.K. Geim, K.S. Novoselov, Nature Mater. 6 (2007) 183. [4] A.K. Geim, Science 324 (2009) 1530. [5] S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, D.C. Elias, J.A. Jaszczak, A.K. Geim, Phys. Rev. Lett. 100 (2008) 016602. [6] K.S. Novoselov, E. McCann, S.V. Morozov, V.I. Fal’ko, M.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A.K. Geim, Nature Phys. 2 (2006) 177. [7] E. McCann, V.I. Fal’ko, Phys. Rev. Lett. 96 (2006) 086805. ¨ [8] H. Schmidt, T. Ludtke, P. Barthold, E. McCann, V.I. Fal’ko, R.J. Haug, Appl. Phys. Lett. 93 (2008) 172108. [9] Z. Ni, Y. Wang, T. Yu, Y. You, Z. Shen, Phys. Rev. B 77 (2008) 235403. [10] J.M.B. Lopes dos Santos, N.M.R. Peres, A.H. Castro Neto, Phys. Rev. Lett. 99 (2007) 256802. [11] NGS Naturgraphit GmbH. [12] V.P. Gusynin, S.G. Sharapov, Phys. Rev. B 71 (2005) 125124. [13] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [14] W.A. de Heer, et al., Solid State Comm. 143 (2007) 92. [15] E. McCann, K. Kechedzhi, V.I. Fal’ko, H. Suzuura, T. Ando, B.L. Altshuler, Phys. Rev. Lett. 97 (2006) 146805. [16] F.V. Tikhonenko, D.W. Horsell, R.V. Gorbachev, A.K. Savchenko, Phys. Rev. Lett. 100 (2008) 056802.