- Email: [email protected]
Measuring the height-to-height correlation function of corrugation in suspended graphene
Ultramicroscopy 165 (2016) 1–7
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Measuring the height-to-height correlation function of corrugation in suspended graphene D.A. Kirilenko a,b,n, P.N. Brunkov a,c a
Ioffe Institute, Politekhnicheskaya ul. 26, 194021 St-Petersburg, Russia EMAT, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium c ITMO University, Kronverksky pr. 49, 197101 St. Petersburg, Russia b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 November 2015 Received in revised form 11 March 2016 Accepted 23 March 2016 Available online 28 March 2016
Nanocorrugation of 2D crystals is an important phenomenon since it affects their electronic and mechanical properties. The corrugation may have various sources; one of them is flexural phonons that, in particular, are responsible for the thermal conductivity of graphene. A study of corrugation of just the suspended graphene can reveal much of valuable information on the physics of this complicated phenomenon. At the same time, the suspended crystal nanorelief can hardly be measured directly because of high flexibility of the 2D crystal. Moreover, the relief portion related to rapid out-of-plane oscillations (flexural phonons) is also inaccessible by such measurements. Here we present a technique for measuring the Fourier components of the height–height correlation function H(q) of suspended graphene which includes the effect of flexural phonons. The technique is based on the analysis of electron diffraction patterns. The H(q) is measured in the range of wavevectors qE 0.4–4.5 nm 1. At the upper limit of this range H(q) does follow the T/κq4 law. So, we measured the value of suspended graphene bending rigidity κ ¼ 1.27 0.4 eV at ambient temperature T E300 K. At intermediate wave vectors, H(q) follows a slightly weaker exponent than theoretically predicted q 3.15 but is closer to the results of the molecular dynamics simulation. At low wave vectors, the dependence becomes even weaker, which may be a sign of influence of charge carriers on the dynamics of undulations longer than 10 nm. The technique presented can be used for studying physics of flexural phonons in other 2D materials. & 2016 Elsevier B.V. All rights reserved.
Keywords: Graphene corrugation Flexural phonons Transmission electron microscopy Electron diffraction
1. Introduction Graphene sheets are always corrugated to some extent. This is important because the corrugation affects electronic properties of graphene [1–6]. The corrugation may arise from the underlying substrate relief [7,8] or be an intrinsic graphene property [9] which is most pronounced in the case of suspended graphene [10]. Intrinsic corrugation of suspended graphene has attracted much interest because of the complexity of physics driving the phenomenon. The intrinsic corrugation may have various sources: ripples originating from any strain applied, lattice distortions caused by ad-atoms on the crystal surface, and out-of-plane thermal oscillations (or flexural phonons) which can reach rather high amplitudes in low-dimensional crystals. The flexural phonons play an important role in the graphene physics. They define the graphene thermal conductivity [11–14]; they are partially n Corresponding author at: Ioffe Institute, Politekhnicheskaya ul. 26, 194021 St-Petersburg, Russia. E-mail address: [email protected] (D.A. Kirilenko).
http://dx.doi.org/10.1016/j.ultramic.2016.03.010 0304-3991/& 2016 Elsevier B.V. All rights reserved.
responsible for the unusual temperature dependence of its conductivity [15–18] and for electronic properties of suspended-graphene-based field-emission transistors [19]. However, it is very difficult to directly measure the suspended graphene nanocorrugation with probe microscopies (atomic-force or scanning tunneling microscopy) because of high flexibility of the ultrathin crystal. There is just a couple of works [20,21] that have reported the evidence of such a corrugation in a suspended graphene sheet. Besides that, direct measurement of a graphene surface profile can reveal only a static corrugation, while the dynamic corrugation related to the flexural phonons remains beyond the capabilities of the probe microscopy techniques. At the same time, the flexural phonons affect to a considerable extent most of the graphene properties, and measuring of their impact on the corrugation is of great interest. Here we present a technique for measuring the Fourier components of the height–height correlation function H (q) of the suspended graphene nanocorrugation, where q is a wavevector. The H(q) is just a spectrum of flexural phonons in case of absence of any static corrugation. The technique is based on the analysis of
2
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
electron diffraction tilt series obtained with a transmission electron microscope (TEM). The electron diffraction can provide very important information on the dynamic lattice distortions caused by flexural phonons due to short interaction time of high-energy electrons with a crystal. In this case, an electron diffraction pattern contains data on h (q) 2 averaged over the time of exposure (ca.
valid as long as h2w 2 < 1 or, as follows from (2), as the rod density deviates only slightly from its maximum value observed at w ¼0:
1 s), where h(q) means the Fourier components of the corrugation. This is in contrast to direct imaging techniques for determining h (q) t and thus prevents adequate measurement of the flexural phonon part of the corrugation.
form of h(x,y) and can be referred to as the spectrum of corrugation. The term containing the Dirac δ-function describes the rod density, while the second term shows that the rod is surrounded by the image of the corrugation spectrum whose intensity grows
t
2. Method description An electron diffraction pattern (in the case of TEM operating at 80–300 kV) represents almost a plane cut of the crystal reciprocal lattice. Note that in the case of graphene that scatters only a minute portion of incident electrons the so-called kinematical approach is applicable. This means that the diffraction image gives squared reciprocal-space density modulus |F(u,v,w)|2 that can be used to extract the crystal structure characteristics. The density distribution of an ideally flat 2D crystal exhibits a set of rods that are perpendicular to the crystal plane and arranged according to the crystal symmetry. If the 2D crystal becomes corrugated, then the density distribution in the reciprocal space changes. Assume that the crystal atoms have been shifted in vertical direction, and their shifts are defined by displacement field h(x,y). Then the density distribution around a rod may be described by the following expression:
F (u, v, w ) =
∬
e2πiwh (x, y) e2πi (ux + vy) dxdy (1)
XY
where the integral is taken over the crystal area, and w is the applicate in the reciprocal space. Hereinafter we consider only outof-plane displacements, because they are much greater than the graphene in-plane displacements which thus can be neglected without dramatic loss of precision. They can also be neglected because our analysis is based on variation of the measured intensities with w, that is along a rod corresponding to one diffraction spot. This variation is only coupled with the out-of-plane displacements field h(x,y), as can be seen from (1). Whereas the information on the in-plane displacements can be extracted from the dependence of the intensities on u and v, that is by measuring differences between various diffraction spots [22,23]. Expression (1) means that the rods become shorter as corrugation amplitude increases. In the case of uncorrelated ripples, the rod density in the vertical direction may be defined as exp (−(2π )2 h2w 2), where h2 is the mean square atomic displacement. In general, one can use the following rod density profile approximation which is obtained by the second-order expansion of the integrand and valid for an arbitrary corrugation as long as h2w 2 < 1: 2
F (0, 0, w ) 2 =
∬
e2πiwh (x, y) dxdy
∝ 1 − (2π )2 h2w 2 (2)
XY
Thus, the rod density decreases proportionally to h2with increasing w. It is important that, in the case of a 2D crystal, the density integral over any plane parallel to the crystal plane (w ¼const) is a constant independent of w as can be derived from (1). This means that, as the rod density decreases, the density around the rod increases in the same manner, namely, pro2
2
portionally to h and w . In the diffraction images it looks like fading of the bright diffraction spot accompanied by intensification of the diffuse spot around it as crystal tilt increases [24]. This can be explained by the following approximation which is
F (u, v, w ) ∝ δ (u, v)⋅(1 − 2π 2 h2w 2) + i (2πw ) hq (u, v) where hq (u, v ) =
(3)
∬ h (x, y) e2πi (ux + vy) dxdy is the Fourier transXY
with increasing w. At relatively high w assumption ( h2w 2 < 1) is no longer valid, and the high-order autocorrelations of the hq (u, v ) come into play. This results in a diffused spot whose width grows with w [24,25]. Eq. (3) implies the possibility of direct measuring the corrugation spectrum by making an appropriate cut of the crystal reciprocal space at an intermediate w by obtaining an electron diffraction image for a slightly tilted graphene sheet. However, instrumentation imperfections shall be taken into account. Their main effect is that the Dirac δfunction turns into the point-spread function S(u,v) which is basically governed by electron beam coherency. In addition, S(u,v) contains a term corresponding to in-plane distortions of the graphene sheet which create a slightly diffused halation of the spot even at zero tilt. Note that the impact of the in-plane shifts is described by the second factor under the integral in (1). This corresponds to convolution of the spot image with a function independent on w and causes changes similar to those induced by S(u,v). Thus, the measurements of intensity versus w provide information only on the out-of-plane shifts. Additionally, magnitudes of the in-plane shifts are much lower than those of the out-of-plane shifts [23,26], and their impact on the diffraction spot intensity distribution is much weaker. In the case of a TEM equipped with a field emission gun, the S (u,v) FWHM is quite small, ca. 0.2 nm 1. Thus, it does not diffuse significantly the image of the corrugation spectrum hq (u, v ). However, S(u,v) has low but relatively broad tails interfering with the measurements of hq (u, v ) when S(u,v) stands for the δ (u, v ). So, what one sees in the electron diffraction image can be simply described to some approximation by the following equation:
F (u, v, w ) 2 = S (u, v)⋅(1 − (2π )2 h2w 2) + (2πw )2 hq (u, v)
2
(4)
Function S(u,v) is what the diffraction spot represents at w¼ 0, that is when its intensity reaches the maximum. The spot intensity profile normalized by its integral intensity can be then used as point-spread function S(u,v). Parameter w is not seen directly in diffraction images but can be determined through the interspot distance, which changes with the electron beam incidence angle during tilting the crystal. Fig. 1 shows the aforementioned features of the corrugated 2D crystal reciprocal lattice and illustrates their manifestations in the electron diffraction images; the latter are shown in Fig. 2. So, distance g between the diffraction spot and the center grows with w because g 2 = g02 + w 2; here g0 is the minimal distance defined by inverse of the corresponding lattice spacing of graphene. To get the sought values, one can just calculate the slope of the curve presenting intensities versus w2 measured at various crystal tilt angles. Moreover, it is convenient to calculate the slope versus directly measured g2; mathematically this is the same, but simplifies the experimental data processing.
∂I (u, v)
( )
∂ g
2
(
= (2π )2 hq (u, v)
2
− h2S (u, v)
)
(5)
Here I(u,v) is the intensity at a certain point of the diffraction image. To reduce the impact of noise on the measurements, it is
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
3
∂I (q) = H (q) − h2S (q) ∂(2πg )2
(6)
Here I(q) is the mean intensity, the averaging being performed over a circle of radius q around the center of the diffraction spot. The H 2
Fig. 1. Vertical cross-section of the model reciprocal lattice of a corrugated 2D crystal. The dashed lines present the cross-sections of the reciprocal space obtained by electron diffraction. The reciprocal-space rods produce the bright diffraction spots whose intensities decrease with increasing tilt. The diffused pattern around a rod looks like a diffuse spot (with a bright spot in the center). The diffuse spot intensity increases with increasing tilt. Value w can be determined from g 2 = g02 + w 2 , where g can be directly measured as half of the distance between two opposite bright spots in the diffraction pattern.
beneficial to average the measured intensities over a circle of a certain radius thus assuming the absence of strong anisotropy of the graphene corrugation. This averaging also serves another important goal. First, the electron diffraction image represents not the reciprocal space horizontal cross-section but the tilted crosssection. This means that the different points neighboring to the diffraction spot are characterized by different values of w. The averaging eliminates this difficulty and provides a value corresponding to w at which Ewald's sphere crosses the rod; this is just that w we measure as the distance between bright spots. In the experiments, the tilt angles did not exceed 10° that is defined by the corrugation amplitude of suspended graphene in the sense of fulfilling the h2w 2 ≪ 1 condition. Therefore, the next-order corrections are negligible at such small angles. The averaging deprives us of information on the two-dimensional spectrum and provides data on the dependence on the wave-vector modulus q. Thus, we suggest measuring the following function:
(q) stands for hq (u, v ) , where q is the modulus of the wave-vector with coordinates (u,v), and is the Fourier transform of the height– height correlation function of the corrugation. Note that in the analysis we assumed the spot integral intensity to be 1; this means that all the experimentally measured intensities, including S(q), must be normalized by the actual integral intensity of the spot. Experimental profile HM (q) = H (q) − h2S (q) is to be processed so as to extract the H(q) values. The expression defining HM(q) entering into (6) contains the height-to-height correlation function of the corrugation and the point-spread-function. If we could assume that there is no strong divergence of H(q) around q¼0 than the second term in (6) could be easily eliminated by subtracting. In this case, S (q).is measured experimentally while h2is determined as the slope of the intensity vs g2 curve measured at the spot center. However, the theory of flexible membranes suggests that it might be not the case, and H(q) may possess large values at small q-s that, in turn, may strongly affect measurements of h2. Unfortunately, expression (6) represents a linearly dependent system. If its values are obtained at all wave vectors so that
∫ H (q) d2q (which equals h2) can be calculated directly, the resulting equation does not allow independent determination of h2. Thus, the only way to get rid of the interference between H(q) and S(q) in a relatively wide range of wave vectors is to deconvolute the obtained profile HM(q) so as to shrink the S(q) domain and to correctly measure the spectrum everywhere besides the range of relatively small wave-vectors. The efficiency of this procedure will define the lower limit of our measurements. At the same time, the upper limit of our measurements is defined by the combination of statistical and background noise at large wave-vectors; in our experiments, it reaches 4 nm 1, where the measurement error is about 50%. This upper limit is mainly defined by the overall time of experiment whose reasonable duration is confined by considerations of longterm microscope stability or specimen sustainability under an electron beam in the medium of the microscope vacuum column. We propose to use two-dimensional convolution of the measured HM(q) with function 2δ(q) S(q); we have found it to be very
Fig. 2. Experimental diffraction patterns of suspended graphene obtained at different crystal tilts: (a) below 2°; (b) ca. 10°. The tilt axis is horizontal in this case. The farther away is the spots from the tilt axis, the more prominent is the effect.
4
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
comparison of these two reconstructions may be used to estimate information loss due to experimental imperfections and validity of final conclusions (as shown below). Finally, we propose the following technique for measuring the height–height correlation function of suspended 2D crystal sheets. First, it is necessary to obtain an electron diffraction tilt series. Then, a set of intensities averaged over circles of various diameters is to be measured around the diffraction spot of each image. The diameters correspond to the points of the measured spectrum. The slope of the curve presenting measured intensities versus g2 provides values of
Fig. 3. Experimental S(q) and its transform 2S(q) S*S(q). The latter decays much more rapidly than S(q). This is more pronounced in the inset presenting both plots in enlarged scale.
HM (q) = H (q) − h2S (q). The point-spread function S(q) can be determined from the spot image at the lowest value of g2. The obtained values must be normalized by the integral spot intensity which is invariant of the tilt and, in addition, equals the integral of S(q). Quantity g can be measured as the distance from the diffraction spot to the central spot of the diffraction image or as the distance between two opposite diffraction spots. Then, reconstructions HR (q) = 2HM (q) − HM (q)*S (q) and HR0 (q) = HM (q) − HM (0) /S (0) × S (q) can be obtained in order to extend the q-range where the sought height–height correlation function is determined to a fair precision towards small q values.
3. Experimental efficient in finding H(q). The main advantage of this transform is that it is integral in nature and, thus, does not lead to significant amplification of noise as it takes place in the case of deconvolution techniques. This is quite crucial for recovering the right-hand part of H(q) where its amplitude is very small. The recovered HR(q) profile will be as follows:
HR (q) = HM (q)*(2δ (q) − S (q)) = (2H (q) − H (q)*S (q)) − h2 (2S (q) − S*S (q))
(7)
So, HR(q) is slightly distorted H(q) minus function 2S(q) S*S(q) (to which S(q) has been transformed). The latter has much more rapidly decaying tails than S(q) itself (see Fig. 3); thus, the obtained HR(q) represents much less deviated values of H(q) in the mid-range of wave vectors. The effect of the suggested transform on the shape of S(q) can be explained as follows. We can see from the Fourier transform of the resulting function F[2S(q) S*S(q)] ¼2Σ |Σ|2 that its n-th derivative at q ¼0 is zero if all preceding derivatives of Σ are zero (when Σ(0) ¼1 which means that S(q) is normalized). Differentiability of the function Fourier transform is determined by convergence of ∫ qnS (q) d2q , namely, by the rate of S(q) decay at q1. Hence, 2S(q) S*S(q) decays at least by factor 1/q2 faster than S (q). This makes values of HR(q) much closer to H(q) than those of HM(q) at q exceeding HWHM of S(q). There is also another simple but rough way to recover H(q) to a fair precision. The second term in (6) can be merely subtracted; however, the value of h2may be indefinite for the reasons discussed above. At the same time, the values of height-to-height correlation function must not be negative. So, one may add S(q) with a coefficient making the resulting function equal to zero at q ¼0 where HM(q) has its minimum negative value. Then, the coefficient equals HM(0)/S(0), and the obtained approximate reconstruction HR0(q) will be:
HR0 (q) = HM (q) −
HM (0) S (q) S (0)
(8)
This subtraction is mathematically very close to as if we use as the coefficient h2measured as the peak intensity decay with g2; here h2can differ from the real amplitude because of the finite width of the point-spread function. The procedure of obtaining HR0 (q) is quite different in nature than that of HR(q). Hence, the
In our experiments, we used graphene sheets obtained by the exfoliation procedure described in [27]. The procedure provides graphene sheets with a crystalline structure whose high preservation is confirmed by the sharpness of diffraction spots. This means that the long-range order scale of the material exceeds the achievable limit of our measurements which is defined by the width of the point-spread function S(q). The material still contains a minor amount of species (e.g. oxygen groups) which might affect the sheet bending rigidity; however, our measurements have shown that the effect is unlikely to be crucial, and theoretical simulations [28] show that ad-atoms only slightly affect the graphene corrugation unless they fully cover its surface. The graphene sheets were deposited onto a conventional Lacey carbon film thus providing suspended areas 1–5 mm in size. We performed the electron diffraction experiments using accelerating voltages of 80 kV and 200 kV. Remarkably, even 200 kV showed itself as an appropriate choice for the experiments. Noticeable changes in the measured values were not observed even after 10 min of irradiating the graphene by an electron beam. This is due to widely spread and, thus, low-intense beam used for obtaining a well-focused diffraction pattern. The most important is that operation at the higher voltage provides a beam of higher coherency and, thus, extends the measurement range closer to the zero wave vector.
4. Results and discussion The technique described was applied to studying suspended graphene. Fig. 4 presents the measurement scheme and obtained intensities versus g2. The presented intensities were measured for a series of radii as shown in Fig. 4(a). The radius step were chosen equal to one pixel of the diffraction images acquired by the CCD camera, and 50 points of the resulting H(q) were obtained so that q ranged from zero to 4 nm 1. Variations in g were observed through a series of diffraction images obtained at various crystal tilts. The intensity in the diffraction spot central area decreases with the increasing crystal tilt, while the intensity at a certain distance from the center increases, which is in accordance with (6). The integral intensity of the diffraction spot remains constant as it must be in the case of a 2D crystal and is used for intensity normalization.
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
5
Fig. 4. (a) The scheme of measuring g and set of I(q) that is the intensity integrated over circles of various diameters. (b) The obtained dependencies of I(q) on g2. Each set of points corresponds to a certain value of q.
Fig. 5(a) represents the results of the measurements. Slope of each set of points presented in Fig. 4(b) corresponds to the measured HM(q) at a certain q. At small q the obtained values are large and negative due the term containing S(q) in (6); they are not shown for the purpose of clarity. Reconstructed profiles HR(q) and HR0(q) are also shown. Fig. 5(b) represents the data in the log-log scale which clearly show varying exponents of the found Fourier components of the height–height correlation function. The H(q) measurements are not significantly disturbed by S(q) at q 41.5 nm 1 where S(q) becomes negligibly small. At the same time, in the 0.4 nm 1 oq o1.5 nm 1 range reconstruction provides corrected values bearing additional information. The most remarkable feature of the measurements is that the obtained H(q) profile follows the ∝ q−4 law on the right-hand side. This is in agreement with the basic concepts of physics of flexural phonon whose spectrum is expected to be describable by T /κq 4 at high q. Here T is the crystal temperature, and κ is the graphene bending rigidity. The value of κ can be derived from the obtained H (q) profile. It is found to be 0.8–1.6 eV, which is in good agreement with the theory [26,29]. The strong dependence becomes weaker at q of ca. 3 nm 1, which is also in qualitative agreement with the theory and experiment [23], where longest phonon wavelength in the frames of linear elastic
theory was found to be about 2.5 nm. In the range q¼1–3 nm 1, it is about q 2.4 that is a bit weaker than q 3.15 predicted by the general theory of flexible membranes [30,31] and by simplified models [32– 34]. Nevertheless, accurate calculations by means of molecular dynamics simulation (MDS) [32,33] show the H(q) dependence to be very close to q 2.4 at intermediate wave vectors. The observed dependence which is weaker than q 3.15 was usually explained by that the models used were improper when the wavelength of fluctuations begins to be comparable with the size of a crystal used in simulations. However, deviations from the ∝ q−3.15dependence can be observed already at wavelengths which are an order of magnitude shorter than the crystal size. Thus, the experimental findings allowed us to suggest that the ∝ q−2.4 dependence represents the real property of graphene rather than an artifact of simulations. What is important is that we have found this dependence in the range q¼ 1–3 nm 1 where error caused by the background noise is not very high, and the influence of S (q) on the measured HM(q) is quite weak. Moreover, our measurements demonstrate at qo1 nm 1 even a weaker dependence that is close to ∝ q−1.3. This finding may be related to the fact that dynamics of large-scale the ripples (starting from several nanometers) is strongly influenced by interactions with the charge carriers [35–37], and simulation of the flexural phonon spectrum in this range needs taking into account this phenomenon.
Fig. 5. (a) Blue squares represent HM(q) derived from the slopes of plots of Fig. 3(b), orange circles are HR(q), and green triangles are HR0(q). (b) Log–log plots of the data obtained. The H(q) right-hand part follows the harmonic approximation law, namely, T /κq4 , whereas, the dependence is much weaker on the left-hand side. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
Fig. 6. Models of experimentally measured HM(q) and corresponding recoveries for various supposed profiles of H(q): (a) predicted by the elastic 2D-membrane theory; (b) derived from our measurements; (c) similar to (b) but assuming that the q 1.3 part is fully an artifact of our measurements.
To check the validity of the presented reconstruction techniques and obtained results, we have mathematically simulated the effect of the point-spread-function and proposed transform taking various H(q) dependencies as input data. Fig. 6 shows the results of modeling the measured HM(q) and its reconstructions (HR(q) and HR0(q)) for three different H(q) profiles: H1(q) that is the theoretically predicted dependence (q 3.15 switching to q 4 as in [29,30]), H2(q) that is the dependence obtained from our measurements, H3(q) that is similar to H2(q) but with the q 2.4 part extended to q¼0 in order to additionally check the validity of the found q 1.3 exponent of H(q). All the H(q) values were normalized by their integrals in order to enable presenting these results in appropriate scales. It can be seen that HR(q) gives a good indication of the H(q) exponent, especially in the case given in Fig. 6(b) where the difference between HM(q) and HR(q) (as well as H(q)) is most pronounced. However, the most crucial for our analysis is the behavior of HR0(q) which significantly deviates from both HM(q) and HR(q) in Fig. 6 (a) though approaches them in Fig. 6(b). This is a consequence of strong divergence of H1(q) in the vicinity of zero. For this reason, the most part of the H1(q) integral is concentrated inside the S(q) FWHM domain, which in turn, negatively affects all the reconstruction results. The opposite situation is presented in Fig. 6(b) that demonstrates successive coincidence of HR0(q) with HR(q) and, further, with HM(q), which is in complete accordance with the experimental results presented in Fig. 5. Moreover, if we assume that the revealed dependence ∝q−1.3is absent in the real H(q), we again get noticeable discrepancy between HR0(q) and HR(q) (Fig. 6(c)). One can see that both functions clearly follow the exponent of 2.4. Thus, only model H2(q) demonstrates results matching the experimental data in essential details and, in general, in the shape of the obtained profiles. Note that throughout our measurements we have not succeeded in revealing an evidence of nonmonotonic dependence H (q) and existence of a local maximum suggested by several simulations [38,39]. However, we believe that experiments involving in-situ mechanical compression or stretching of graphene might lead to such findings.
5. Conclusion A technique for measuring the height-to-height correlation function H(q) of the corrugation of suspended graphene has been proposed. The measured H(q) follows the T /κq 4 law at the upper limit of the q variation range, which is in accordance with theoretical predictions. This allowed us to experimentally measure bending rigidity κ of suspended graphene. This value was found to be 0.8–1.6 eV, which is in good agreement with the theory. At qo3 nm 1 H(q) follows the power-law ∝ q−2.4 which is weaker than that predicted by the general theory of flexible membranes but is in
better accordance with the results of MDS. Even weaker power-law was observed at qo1 nm 1, namely, ca. ∝ q−1.3, whose validity was confirmed by modeling. These results may be regarded as a sign of a stronger influence of the charge carriers on the dynamics of the relatively long undulations of graphene surface.
Acknowledgments D.K. thanks the RFBR (Grant no. 16-32-60165) for the partial support of this research. The work was carried out in part at the Joint Research Center “Material Science and Characterization in Advanced Technologies” (St-Petersburg, Russia) under the financial support from the Ministry of Education and Science of the Russian Federation (Agreement 14.621.21.0007, 04.12.2014, id RFMEFI62114X0007, the use of the Jeol JEM-2100F microscope) and at EMAT, Universiteit Antwerpen (Antwerpen, Belgium), (the use of the FEI Tecnai G2 microscope).
References [1] M.I. Katsnelson, A.K. Geim, Electron scattering on microscopic corrugations in graphene, Philos. Trans. R. Soc. A 366 (2008) 195–204, http://dx.doi.org/ 10.1098/rsta.2007.2157. [2] I.F. Herbut, V. Juričić, O. Vafek, Coulomb interaction, ripples, and the minimal conductivity of graphene, Phys. Rev. Lett. 100 (2008) 046403, http://dx.doi. org/10.1103/PhysRevLett.100.046403. [3] Yu.A. Firsov, N.E. Firsova, Surface corrugations influence monolayer graphene electromagnetic response, Phys. E: Low-dimens. Syst. Nanostruct. 62 (2014) 36–42, http://dx.doi.org/10.1016/j.physe.2014.04.015. [4] Z. Osváth, A. Deák, K. Kertész, Gy Molnár, G. Vértesy, D. Zámbó, C. Hwang, L. P. Biró, The structure and properties of graphene on gold nanoparticles, Nanoscale 7 (2015) 5503–5509, http://dx.doi.org/10.1039/C5NR00268K. [5] S.-Y. Lin, S.-L. Chang, F.-L. Shyu, J.-M. Lu, M.-F. Lin, Feature-rich electronic properties in graphene ripples, Carbon 86 (2015) 207–216, http://dx.doi.org/ 10.1016/j.carbon.2014.12.068. [6] A. Rossi, S. Piccini, V. Pellegrini, S. de Gironcoli, V. Tozzini, Nano-scale corrugations in graphene: a density functional theory study of structure, electronic properties and hydrogenation, J. Phys. Chem. C 119 (2015) 7900–7910, http: //dx.doi.org/10.1021/jp511409b. [7] C.H. Lui, L. Liu, K.F. Mak, G.W. Flynn, T.F. Heinz, Ultraflat graphene, Nature 462 (2009) 339–341, http://dx.doi.org/10.1038/nature08569. [8] M. Ishigami, J.H. Chen, W.G. Cullen, M.S. Fuhrer, E.D. Williams, Atomic structure of graphene on SiO2, Nano Lett. 7 (2007) 1643–1648, http://dx.doi.org/ 10.1021/nl070613a. [9] V. Geringer, M. Liebmann, T. Echtermeyer, S. Runte, M. Schmidt, R. Ruckamp, M.C. Lemme, M. Morgenstern, Intrinsic and extrinsic corrugation of monolayer graphene deposited on SiO2, Phys. Rev. Lett. 102 (2009) 076102, http://dx.doi. org/10.1103/PhysRevLett.102.076102. [10] J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, T.J. Booth, S. Roth, The structure of suspended graphene sheets, Nature 446 (2007) 60–63, http://dx. doi.org/10.1038/nature05545. [11] L. Lindsay, D.A. Broido, N. Mingo, Flexural phonons and thermal transport in graphene, Phys. Rev. B 82 (2010) 115427, http://dx.doi.org/10.1103/ PhysRevB.82.115427. [12] J.H. Seol, I. Jo, A.L. Moore, L. Lindsay, Z.H. Aitken, M.T. Pettes, X. Li, Z. Yao,
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
R. Huang, D. Broido, N. Mingo, R.S. Ruoff, L. Shi, Two-dimensional phonon transport in supported graphene, Science 328 (2010) 213–216, http://dx.doi. org/10.1126/science.1184014. M.D. Kamatagi, A.S. Nissimagoudar, N.S. Sankeshwar, B.G. Mulimani, Lattice thermal conductivity of graphene, in: Proceedings of the AIP Conf. Proc., vol. 1447, 2012, pp. 945–946 〈http://dx.doi.org/10.1063/1.4710315〉. X. Tang, S. Xu, J. Zhang, X. Wang, Five Orders of Magnitude Reduction in Energy Coupling across Corrugated Graphene/Substrate Interfaces, ACS Appl. Mater. Interfaces 6 (4) (2014) 2809–2818, http://dx.doi.org/10.1021/ am405388a. E. Mariani, F. von Oppen, Temperature-dependent resistivity of suspended graphene, Phys. Rev. B 82 (2010) 195403, http://dx.doi.org/10.1103/ PhysRevB.82.195403. I.V. Gornyi, V.Yu Kachorovskii, A.D. Mirlin, Conductivity of suspended graphene at the Dirac point, Phys. Rev. B 86 (2012) 165413, http://dx.doi.org/ 10.1103/PhysRevB.86.165413. S. Das Sarma, E.H. Hwang, Density-dependent electrical conductivity in suspended graphene: approaching the Dirac point in transport, Phys. Rev. B 87 (2013) 035415, http://dx.doi.org/10.1103/PhysRevB.87.035415. K.I. Bolotin, K.J. Sikes, J. Hone, H.L. Stormer, P. Kim, Temperature-dependent transport in suspended graphene, Phys. Rev. Lett. 101 (2008) 096802, http: //dx.doi.org/10.1103/PhysRevLett.101.096802. H. Ochoa, E.V. Castro, M.I. Katsnelson, F. Guinea, Scattering by flexural phonons in suspended graphene under back gate induced strain, Physica E 44 (2012) 963–966, http://dx.doi.org/10.1016/j.physe.2011.03.017. P. Xu, Y. Yang, S.D. Barber, M.L. Ackerman, J.K. Schoelz, D. Qi, I.A. Kornev, L. Dong, L. Bellaiche, S. Barraza-Lopez, P.M. Thibado, Atomic control of strain in freestanding graphene, Phys. Rev. B 85 (2012) 121406, http://dx.doi.org/ 10.1103/PhysRevB.85.121406. R. Zan, C. Muryn, U. Bangert, P. Mattocks, P. Wincott, D. Vaughan, X. Li, L. Colombo, R.S. Ruoff, B. Hamilton, K.S. Novoselov, Scanning tunnelling microscopy of suspended graphene, Nanoscale 4 (2012) 3065–3068, http://dx. doi.org/10.1039/c2nr30162h. B. Shevitski, M. Mecklenburg, W.A. Hubbard, E.R. White, B. Dawson, M. S. Lodge, M. Ishigami, B.C. Regan, Dark-field transmission electron microscopy and the Debye–Waller factor of graphene, Phys. Rev. B 87 (2013) 045417, http: //dx.doi.org/10.1103/PhysRevB.87.045417. C.S. Allen, E. Liberti, J.S. Kim, Q. Xu, Y. Fan, K. He, A.W. Robertson, H. W. Zandbergen, J.H. Warner, A.I. Kirkland, Temperature dependence of atomic vibrations in mono-layer graphene, J. Appl. Phys. 118 (2015) 074302, http://dx. doi.org/10.1063/1.4928324. D.A. Kirilenko, A.T. Dideykin, G. Van Tendeloo, Measuring the corrugation amplitude of suspended and supported graphene, Phys. Rev. B. 84 (2011) 235417, http://dx.doi.org/10.1103/PhysRevB.84.235417.
7
[25] J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, D. Obergfell, S. Roth, C. Girit, A. Zettl, On the roughness of single- and bi-layer graphene membranes, Solid State Commun. 143 (2007) 101–109, http://dx.doi.org/10.1016/j. ssc.2007.02.047. [26] A. Fasolino, J.H. Los, M.I. Katsnelson, Intrinsic ripples in graphene, Nat. Mater. 6 (2007) 858–861, http://dx.doi.org/10.1038/nmat2011. [27] A. Dideykin, A.E. Aleksenskiy, D. Kirilenko, P. Brunkov, V. Goncharov, M. Baidakova, D. Sakseev, A. Ya.Vul, Monolayer graphene from graphite oxide, Diam. Relat. Mater. 20 (2011) 105–108, http://dx.doi.org/10.1016/j. diamond.2010.10.007. [28] S.K. Singh, S. Goverapet Srinivasan, M. Neek-Amal, S. Costamagna, Adri C. T. van Duin, F.M. Peeters, Thermal properties of fluorinated graphene, Phys. Rev. B 87 (2013) 104114, http://dx.doi.org/10.1103/PhysRevB.87.104114. [29] J.H. Los, M.I. Katsnelson, O.V. Yazyev, K.V. Zakharchenko, A. Fasolino, Phys. Rev. B 80 (2009) 121405, http://dx.doi.org/10.1103/PhysRevB.80.121405. [30] D. Nelson, L. Peliti, Fluctuations in membranes with crystalline and hexatic order, J. Phys. France 48 (1987) 1085, http://dx.doi.org/10.1051/ jphys:019870048070108500. [31] P. Le Doussal, L. Radzihovsky, Self-consistent theory of polymerized membranes, Phys. Rev. Lett. 69 (1992) 1209, http://dx.doi.org/10.1103/ PhysRevLett.69.1209. [32] F.L. Braghin, N. Hasselmann, Thermal fluctuations of free-standing graphene, Phys. Rev. B 82 (2010) 035407, http://dx.doi.org/10.1103/PhysRevB.82.035407. [33] K.V. Zakharchenko, R. Roldán, A. Fasolino, M.I. Katsnelson, Self-consistent screening approximation for flexible membranes: application to graphene, Phys. Rev. B 82 (2010) 125435, http://dx.doi.org/10.1103/PhysRevB.82.125435. [34] R. Roldan, A. Fasolino, K.V. Zakharchenko, M.I. Katsnelson, Suppression of anharmonicities in crystalline membranes by external strain, Phys. Rev. B 83 (2011) 174104, http://dx.doi.org/10.1103/PhysRevB.83.174104. [35] P. San-Jose, J. Gonzalez, F. Guinea, Electron-induced rippling in graphene, Phys. Rev. Lett. 106 (2011) 045502, http://dx.doi.org/10.1103/ PhysRevLett.106.045502. [36] M. Gibertini, A. Tomadin, F. Guinea, M.I. Katsnelson, M. Polini, Electron-hole puddles in the absence of charged impurities, Phys. Rev. B 85 (2012) 201405, http://dx.doi.org/10.1103/PhysRevB.85.201405. [37] D. Gazit, Correlation between charge inhomogeneities and structure in graphene and other electronic crystalline membranes, Phys. Rev. B 80 (2009) 161406, http://dx.doi.org/10.1103/PhysRevB.80.161406. [38] A. Smolyanitsky, Molecular dynamics simulation of thermal ripples in graphene with bond-order informed harmonic constraints, Nanotechnology 25 (2014) 485701, http://dx.doi.org/10.1088/0957-4484/25/48/485701. [39] N. Abedpour, Reza Asgari, M. Reza Rahimi Tabar, Irreversibility in response to forces acting on graphene sheets, Phys. Rev. Lett. 104 (2010) 196804, http://dx. doi.org/10.1103/PhysRevLett.104.196804.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Measuring the height-to-height correlation function of corrugation in suspended graphene D.A. Kirilenko a,b,n, P.N. Brunkov a,c a
Ioffe Institute, Politekhnicheskaya ul. 26, 194021 St-Petersburg, Russia EMAT, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium c ITMO University, Kronverksky pr. 49, 197101 St. Petersburg, Russia b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 November 2015 Received in revised form 11 March 2016 Accepted 23 March 2016 Available online 28 March 2016
Nanocorrugation of 2D crystals is an important phenomenon since it affects their electronic and mechanical properties. The corrugation may have various sources; one of them is flexural phonons that, in particular, are responsible for the thermal conductivity of graphene. A study of corrugation of just the suspended graphene can reveal much of valuable information on the physics of this complicated phenomenon. At the same time, the suspended crystal nanorelief can hardly be measured directly because of high flexibility of the 2D crystal. Moreover, the relief portion related to rapid out-of-plane oscillations (flexural phonons) is also inaccessible by such measurements. Here we present a technique for measuring the Fourier components of the height–height correlation function H(q) of suspended graphene which includes the effect of flexural phonons. The technique is based on the analysis of electron diffraction patterns. The H(q) is measured in the range of wavevectors qE 0.4–4.5 nm 1. At the upper limit of this range H(q) does follow the T/κq4 law. So, we measured the value of suspended graphene bending rigidity κ ¼ 1.27 0.4 eV at ambient temperature T E300 K. At intermediate wave vectors, H(q) follows a slightly weaker exponent than theoretically predicted q 3.15 but is closer to the results of the molecular dynamics simulation. At low wave vectors, the dependence becomes even weaker, which may be a sign of influence of charge carriers on the dynamics of undulations longer than 10 nm. The technique presented can be used for studying physics of flexural phonons in other 2D materials. & 2016 Elsevier B.V. All rights reserved.
Keywords: Graphene corrugation Flexural phonons Transmission electron microscopy Electron diffraction
1. Introduction Graphene sheets are always corrugated to some extent. This is important because the corrugation affects electronic properties of graphene [1–6]. The corrugation may arise from the underlying substrate relief [7,8] or be an intrinsic graphene property [9] which is most pronounced in the case of suspended graphene [10]. Intrinsic corrugation of suspended graphene has attracted much interest because of the complexity of physics driving the phenomenon. The intrinsic corrugation may have various sources: ripples originating from any strain applied, lattice distortions caused by ad-atoms on the crystal surface, and out-of-plane thermal oscillations (or flexural phonons) which can reach rather high amplitudes in low-dimensional crystals. The flexural phonons play an important role in the graphene physics. They define the graphene thermal conductivity [11–14]; they are partially n Corresponding author at: Ioffe Institute, Politekhnicheskaya ul. 26, 194021 St-Petersburg, Russia. E-mail address: [email protected] (D.A. Kirilenko).
http://dx.doi.org/10.1016/j.ultramic.2016.03.010 0304-3991/& 2016 Elsevier B.V. All rights reserved.
responsible for the unusual temperature dependence of its conductivity [15–18] and for electronic properties of suspended-graphene-based field-emission transistors [19]. However, it is very difficult to directly measure the suspended graphene nanocorrugation with probe microscopies (atomic-force or scanning tunneling microscopy) because of high flexibility of the ultrathin crystal. There is just a couple of works [20,21] that have reported the evidence of such a corrugation in a suspended graphene sheet. Besides that, direct measurement of a graphene surface profile can reveal only a static corrugation, while the dynamic corrugation related to the flexural phonons remains beyond the capabilities of the probe microscopy techniques. At the same time, the flexural phonons affect to a considerable extent most of the graphene properties, and measuring of their impact on the corrugation is of great interest. Here we present a technique for measuring the Fourier components of the height–height correlation function H (q) of the suspended graphene nanocorrugation, where q is a wavevector. The H(q) is just a spectrum of flexural phonons in case of absence of any static corrugation. The technique is based on the analysis of
2
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
electron diffraction tilt series obtained with a transmission electron microscope (TEM). The electron diffraction can provide very important information on the dynamic lattice distortions caused by flexural phonons due to short interaction time of high-energy electrons with a crystal. In this case, an electron diffraction pattern contains data on h (q) 2 averaged over the time of exposure (ca.
valid as long as h2w 2 < 1 or, as follows from (2), as the rod density deviates only slightly from its maximum value observed at w ¼0:
1 s), where h(q) means the Fourier components of the corrugation. This is in contrast to direct imaging techniques for determining h (q) t and thus prevents adequate measurement of the flexural phonon part of the corrugation.
form of h(x,y) and can be referred to as the spectrum of corrugation. The term containing the Dirac δ-function describes the rod density, while the second term shows that the rod is surrounded by the image of the corrugation spectrum whose intensity grows
t
2. Method description An electron diffraction pattern (in the case of TEM operating at 80–300 kV) represents almost a plane cut of the crystal reciprocal lattice. Note that in the case of graphene that scatters only a minute portion of incident electrons the so-called kinematical approach is applicable. This means that the diffraction image gives squared reciprocal-space density modulus |F(u,v,w)|2 that can be used to extract the crystal structure characteristics. The density distribution of an ideally flat 2D crystal exhibits a set of rods that are perpendicular to the crystal plane and arranged according to the crystal symmetry. If the 2D crystal becomes corrugated, then the density distribution in the reciprocal space changes. Assume that the crystal atoms have been shifted in vertical direction, and their shifts are defined by displacement field h(x,y). Then the density distribution around a rod may be described by the following expression:
F (u, v, w ) =
∬
e2πiwh (x, y) e2πi (ux + vy) dxdy (1)
XY
where the integral is taken over the crystal area, and w is the applicate in the reciprocal space. Hereinafter we consider only outof-plane displacements, because they are much greater than the graphene in-plane displacements which thus can be neglected without dramatic loss of precision. They can also be neglected because our analysis is based on variation of the measured intensities with w, that is along a rod corresponding to one diffraction spot. This variation is only coupled with the out-of-plane displacements field h(x,y), as can be seen from (1). Whereas the information on the in-plane displacements can be extracted from the dependence of the intensities on u and v, that is by measuring differences between various diffraction spots [22,23]. Expression (1) means that the rods become shorter as corrugation amplitude increases. In the case of uncorrelated ripples, the rod density in the vertical direction may be defined as exp (−(2π )2 h2w 2), where h2 is the mean square atomic displacement. In general, one can use the following rod density profile approximation which is obtained by the second-order expansion of the integrand and valid for an arbitrary corrugation as long as h2w 2 < 1: 2
F (0, 0, w ) 2 =
∬
e2πiwh (x, y) dxdy
∝ 1 − (2π )2 h2w 2 (2)
XY
Thus, the rod density decreases proportionally to h2with increasing w. It is important that, in the case of a 2D crystal, the density integral over any plane parallel to the crystal plane (w ¼const) is a constant independent of w as can be derived from (1). This means that, as the rod density decreases, the density around the rod increases in the same manner, namely, pro2
2
portionally to h and w . In the diffraction images it looks like fading of the bright diffraction spot accompanied by intensification of the diffuse spot around it as crystal tilt increases [24]. This can be explained by the following approximation which is
F (u, v, w ) ∝ δ (u, v)⋅(1 − 2π 2 h2w 2) + i (2πw ) hq (u, v) where hq (u, v ) =
(3)
∬ h (x, y) e2πi (ux + vy) dxdy is the Fourier transXY
with increasing w. At relatively high w assumption ( h2w 2 < 1) is no longer valid, and the high-order autocorrelations of the hq (u, v ) come into play. This results in a diffused spot whose width grows with w [24,25]. Eq. (3) implies the possibility of direct measuring the corrugation spectrum by making an appropriate cut of the crystal reciprocal space at an intermediate w by obtaining an electron diffraction image for a slightly tilted graphene sheet. However, instrumentation imperfections shall be taken into account. Their main effect is that the Dirac δfunction turns into the point-spread function S(u,v) which is basically governed by electron beam coherency. In addition, S(u,v) contains a term corresponding to in-plane distortions of the graphene sheet which create a slightly diffused halation of the spot even at zero tilt. Note that the impact of the in-plane shifts is described by the second factor under the integral in (1). This corresponds to convolution of the spot image with a function independent on w and causes changes similar to those induced by S(u,v). Thus, the measurements of intensity versus w provide information only on the out-of-plane shifts. Additionally, magnitudes of the in-plane shifts are much lower than those of the out-of-plane shifts [23,26], and their impact on the diffraction spot intensity distribution is much weaker. In the case of a TEM equipped with a field emission gun, the S (u,v) FWHM is quite small, ca. 0.2 nm 1. Thus, it does not diffuse significantly the image of the corrugation spectrum hq (u, v ). However, S(u,v) has low but relatively broad tails interfering with the measurements of hq (u, v ) when S(u,v) stands for the δ (u, v ). So, what one sees in the electron diffraction image can be simply described to some approximation by the following equation:
F (u, v, w ) 2 = S (u, v)⋅(1 − (2π )2 h2w 2) + (2πw )2 hq (u, v)
2
(4)
Function S(u,v) is what the diffraction spot represents at w¼ 0, that is when its intensity reaches the maximum. The spot intensity profile normalized by its integral intensity can be then used as point-spread function S(u,v). Parameter w is not seen directly in diffraction images but can be determined through the interspot distance, which changes with the electron beam incidence angle during tilting the crystal. Fig. 1 shows the aforementioned features of the corrugated 2D crystal reciprocal lattice and illustrates their manifestations in the electron diffraction images; the latter are shown in Fig. 2. So, distance g between the diffraction spot and the center grows with w because g 2 = g02 + w 2; here g0 is the minimal distance defined by inverse of the corresponding lattice spacing of graphene. To get the sought values, one can just calculate the slope of the curve presenting intensities versus w2 measured at various crystal tilt angles. Moreover, it is convenient to calculate the slope versus directly measured g2; mathematically this is the same, but simplifies the experimental data processing.
∂I (u, v)
( )
∂ g
2
(
= (2π )2 hq (u, v)
2
− h2S (u, v)
)
(5)
Here I(u,v) is the intensity at a certain point of the diffraction image. To reduce the impact of noise on the measurements, it is
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
3
∂I (q) = H (q) − h2S (q) ∂(2πg )2
(6)
Here I(q) is the mean intensity, the averaging being performed over a circle of radius q around the center of the diffraction spot. The H 2
Fig. 1. Vertical cross-section of the model reciprocal lattice of a corrugated 2D crystal. The dashed lines present the cross-sections of the reciprocal space obtained by electron diffraction. The reciprocal-space rods produce the bright diffraction spots whose intensities decrease with increasing tilt. The diffused pattern around a rod looks like a diffuse spot (with a bright spot in the center). The diffuse spot intensity increases with increasing tilt. Value w can be determined from g 2 = g02 + w 2 , where g can be directly measured as half of the distance between two opposite bright spots in the diffraction pattern.
beneficial to average the measured intensities over a circle of a certain radius thus assuming the absence of strong anisotropy of the graphene corrugation. This averaging also serves another important goal. First, the electron diffraction image represents not the reciprocal space horizontal cross-section but the tilted crosssection. This means that the different points neighboring to the diffraction spot are characterized by different values of w. The averaging eliminates this difficulty and provides a value corresponding to w at which Ewald's sphere crosses the rod; this is just that w we measure as the distance between bright spots. In the experiments, the tilt angles did not exceed 10° that is defined by the corrugation amplitude of suspended graphene in the sense of fulfilling the h2w 2 ≪ 1 condition. Therefore, the next-order corrections are negligible at such small angles. The averaging deprives us of information on the two-dimensional spectrum and provides data on the dependence on the wave-vector modulus q. Thus, we suggest measuring the following function:
(q) stands for hq (u, v ) , where q is the modulus of the wave-vector with coordinates (u,v), and is the Fourier transform of the height– height correlation function of the corrugation. Note that in the analysis we assumed the spot integral intensity to be 1; this means that all the experimentally measured intensities, including S(q), must be normalized by the actual integral intensity of the spot. Experimental profile HM (q) = H (q) − h2S (q) is to be processed so as to extract the H(q) values. The expression defining HM(q) entering into (6) contains the height-to-height correlation function of the corrugation and the point-spread-function. If we could assume that there is no strong divergence of H(q) around q¼0 than the second term in (6) could be easily eliminated by subtracting. In this case, S (q).is measured experimentally while h2is determined as the slope of the intensity vs g2 curve measured at the spot center. However, the theory of flexible membranes suggests that it might be not the case, and H(q) may possess large values at small q-s that, in turn, may strongly affect measurements of h2. Unfortunately, expression (6) represents a linearly dependent system. If its values are obtained at all wave vectors so that
∫ H (q) d2q (which equals h2) can be calculated directly, the resulting equation does not allow independent determination of h2. Thus, the only way to get rid of the interference between H(q) and S(q) in a relatively wide range of wave vectors is to deconvolute the obtained profile HM(q) so as to shrink the S(q) domain and to correctly measure the spectrum everywhere besides the range of relatively small wave-vectors. The efficiency of this procedure will define the lower limit of our measurements. At the same time, the upper limit of our measurements is defined by the combination of statistical and background noise at large wave-vectors; in our experiments, it reaches 4 nm 1, where the measurement error is about 50%. This upper limit is mainly defined by the overall time of experiment whose reasonable duration is confined by considerations of longterm microscope stability or specimen sustainability under an electron beam in the medium of the microscope vacuum column. We propose to use two-dimensional convolution of the measured HM(q) with function 2δ(q) S(q); we have found it to be very
Fig. 2. Experimental diffraction patterns of suspended graphene obtained at different crystal tilts: (a) below 2°; (b) ca. 10°. The tilt axis is horizontal in this case. The farther away is the spots from the tilt axis, the more prominent is the effect.
4
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
comparison of these two reconstructions may be used to estimate information loss due to experimental imperfections and validity of final conclusions (as shown below). Finally, we propose the following technique for measuring the height–height correlation function of suspended 2D crystal sheets. First, it is necessary to obtain an electron diffraction tilt series. Then, a set of intensities averaged over circles of various diameters is to be measured around the diffraction spot of each image. The diameters correspond to the points of the measured spectrum. The slope of the curve presenting measured intensities versus g2 provides values of
Fig. 3. Experimental S(q) and its transform 2S(q) S*S(q). The latter decays much more rapidly than S(q). This is more pronounced in the inset presenting both plots in enlarged scale.
HM (q) = H (q) − h2S (q). The point-spread function S(q) can be determined from the spot image at the lowest value of g2. The obtained values must be normalized by the integral spot intensity which is invariant of the tilt and, in addition, equals the integral of S(q). Quantity g can be measured as the distance from the diffraction spot to the central spot of the diffraction image or as the distance between two opposite diffraction spots. Then, reconstructions HR (q) = 2HM (q) − HM (q)*S (q) and HR0 (q) = HM (q) − HM (0) /S (0) × S (q) can be obtained in order to extend the q-range where the sought height–height correlation function is determined to a fair precision towards small q values.
3. Experimental efficient in finding H(q). The main advantage of this transform is that it is integral in nature and, thus, does not lead to significant amplification of noise as it takes place in the case of deconvolution techniques. This is quite crucial for recovering the right-hand part of H(q) where its amplitude is very small. The recovered HR(q) profile will be as follows:
HR (q) = HM (q)*(2δ (q) − S (q)) = (2H (q) − H (q)*S (q)) − h2 (2S (q) − S*S (q))
(7)
So, HR(q) is slightly distorted H(q) minus function 2S(q) S*S(q) (to which S(q) has been transformed). The latter has much more rapidly decaying tails than S(q) itself (see Fig. 3); thus, the obtained HR(q) represents much less deviated values of H(q) in the mid-range of wave vectors. The effect of the suggested transform on the shape of S(q) can be explained as follows. We can see from the Fourier transform of the resulting function F[2S(q) S*S(q)] ¼2Σ |Σ|2 that its n-th derivative at q ¼0 is zero if all preceding derivatives of Σ are zero (when Σ(0) ¼1 which means that S(q) is normalized). Differentiability of the function Fourier transform is determined by convergence of ∫ qnS (q) d2q , namely, by the rate of S(q) decay at q1. Hence, 2S(q) S*S(q) decays at least by factor 1/q2 faster than S (q). This makes values of HR(q) much closer to H(q) than those of HM(q) at q exceeding HWHM of S(q). There is also another simple but rough way to recover H(q) to a fair precision. The second term in (6) can be merely subtracted; however, the value of h2may be indefinite for the reasons discussed above. At the same time, the values of height-to-height correlation function must not be negative. So, one may add S(q) with a coefficient making the resulting function equal to zero at q ¼0 where HM(q) has its minimum negative value. Then, the coefficient equals HM(0)/S(0), and the obtained approximate reconstruction HR0(q) will be:
HR0 (q) = HM (q) −
HM (0) S (q) S (0)
(8)
This subtraction is mathematically very close to as if we use as the coefficient h2measured as the peak intensity decay with g2; here h2can differ from the real amplitude because of the finite width of the point-spread function. The procedure of obtaining HR0 (q) is quite different in nature than that of HR(q). Hence, the
In our experiments, we used graphene sheets obtained by the exfoliation procedure described in [27]. The procedure provides graphene sheets with a crystalline structure whose high preservation is confirmed by the sharpness of diffraction spots. This means that the long-range order scale of the material exceeds the achievable limit of our measurements which is defined by the width of the point-spread function S(q). The material still contains a minor amount of species (e.g. oxygen groups) which might affect the sheet bending rigidity; however, our measurements have shown that the effect is unlikely to be crucial, and theoretical simulations [28] show that ad-atoms only slightly affect the graphene corrugation unless they fully cover its surface. The graphene sheets were deposited onto a conventional Lacey carbon film thus providing suspended areas 1–5 mm in size. We performed the electron diffraction experiments using accelerating voltages of 80 kV and 200 kV. Remarkably, even 200 kV showed itself as an appropriate choice for the experiments. Noticeable changes in the measured values were not observed even after 10 min of irradiating the graphene by an electron beam. This is due to widely spread and, thus, low-intense beam used for obtaining a well-focused diffraction pattern. The most important is that operation at the higher voltage provides a beam of higher coherency and, thus, extends the measurement range closer to the zero wave vector.
4. Results and discussion The technique described was applied to studying suspended graphene. Fig. 4 presents the measurement scheme and obtained intensities versus g2. The presented intensities were measured for a series of radii as shown in Fig. 4(a). The radius step were chosen equal to one pixel of the diffraction images acquired by the CCD camera, and 50 points of the resulting H(q) were obtained so that q ranged from zero to 4 nm 1. Variations in g were observed through a series of diffraction images obtained at various crystal tilts. The intensity in the diffraction spot central area decreases with the increasing crystal tilt, while the intensity at a certain distance from the center increases, which is in accordance with (6). The integral intensity of the diffraction spot remains constant as it must be in the case of a 2D crystal and is used for intensity normalization.
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
5
Fig. 4. (a) The scheme of measuring g and set of I(q) that is the intensity integrated over circles of various diameters. (b) The obtained dependencies of I(q) on g2. Each set of points corresponds to a certain value of q.
Fig. 5(a) represents the results of the measurements. Slope of each set of points presented in Fig. 4(b) corresponds to the measured HM(q) at a certain q. At small q the obtained values are large and negative due the term containing S(q) in (6); they are not shown for the purpose of clarity. Reconstructed profiles HR(q) and HR0(q) are also shown. Fig. 5(b) represents the data in the log-log scale which clearly show varying exponents of the found Fourier components of the height–height correlation function. The H(q) measurements are not significantly disturbed by S(q) at q 41.5 nm 1 where S(q) becomes negligibly small. At the same time, in the 0.4 nm 1 oq o1.5 nm 1 range reconstruction provides corrected values bearing additional information. The most remarkable feature of the measurements is that the obtained H(q) profile follows the ∝ q−4 law on the right-hand side. This is in agreement with the basic concepts of physics of flexural phonon whose spectrum is expected to be describable by T /κq 4 at high q. Here T is the crystal temperature, and κ is the graphene bending rigidity. The value of κ can be derived from the obtained H (q) profile. It is found to be 0.8–1.6 eV, which is in good agreement with the theory [26,29]. The strong dependence becomes weaker at q of ca. 3 nm 1, which is also in qualitative agreement with the theory and experiment [23], where longest phonon wavelength in the frames of linear elastic
theory was found to be about 2.5 nm. In the range q¼1–3 nm 1, it is about q 2.4 that is a bit weaker than q 3.15 predicted by the general theory of flexible membranes [30,31] and by simplified models [32– 34]. Nevertheless, accurate calculations by means of molecular dynamics simulation (MDS) [32,33] show the H(q) dependence to be very close to q 2.4 at intermediate wave vectors. The observed dependence which is weaker than q 3.15 was usually explained by that the models used were improper when the wavelength of fluctuations begins to be comparable with the size of a crystal used in simulations. However, deviations from the ∝ q−3.15dependence can be observed already at wavelengths which are an order of magnitude shorter than the crystal size. Thus, the experimental findings allowed us to suggest that the ∝ q−2.4 dependence represents the real property of graphene rather than an artifact of simulations. What is important is that we have found this dependence in the range q¼ 1–3 nm 1 where error caused by the background noise is not very high, and the influence of S (q) on the measured HM(q) is quite weak. Moreover, our measurements demonstrate at qo1 nm 1 even a weaker dependence that is close to ∝ q−1.3. This finding may be related to the fact that dynamics of large-scale the ripples (starting from several nanometers) is strongly influenced by interactions with the charge carriers [35–37], and simulation of the flexural phonon spectrum in this range needs taking into account this phenomenon.
Fig. 5. (a) Blue squares represent HM(q) derived from the slopes of plots of Fig. 3(b), orange circles are HR(q), and green triangles are HR0(q). (b) Log–log plots of the data obtained. The H(q) right-hand part follows the harmonic approximation law, namely, T /κq4 , whereas, the dependence is much weaker on the left-hand side. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
Fig. 6. Models of experimentally measured HM(q) and corresponding recoveries for various supposed profiles of H(q): (a) predicted by the elastic 2D-membrane theory; (b) derived from our measurements; (c) similar to (b) but assuming that the q 1.3 part is fully an artifact of our measurements.
To check the validity of the presented reconstruction techniques and obtained results, we have mathematically simulated the effect of the point-spread-function and proposed transform taking various H(q) dependencies as input data. Fig. 6 shows the results of modeling the measured HM(q) and its reconstructions (HR(q) and HR0(q)) for three different H(q) profiles: H1(q) that is the theoretically predicted dependence (q 3.15 switching to q 4 as in [29,30]), H2(q) that is the dependence obtained from our measurements, H3(q) that is similar to H2(q) but with the q 2.4 part extended to q¼0 in order to additionally check the validity of the found q 1.3 exponent of H(q). All the H(q) values were normalized by their integrals in order to enable presenting these results in appropriate scales. It can be seen that HR(q) gives a good indication of the H(q) exponent, especially in the case given in Fig. 6(b) where the difference between HM(q) and HR(q) (as well as H(q)) is most pronounced. However, the most crucial for our analysis is the behavior of HR0(q) which significantly deviates from both HM(q) and HR(q) in Fig. 6 (a) though approaches them in Fig. 6(b). This is a consequence of strong divergence of H1(q) in the vicinity of zero. For this reason, the most part of the H1(q) integral is concentrated inside the S(q) FWHM domain, which in turn, negatively affects all the reconstruction results. The opposite situation is presented in Fig. 6(b) that demonstrates successive coincidence of HR0(q) with HR(q) and, further, with HM(q), which is in complete accordance with the experimental results presented in Fig. 5. Moreover, if we assume that the revealed dependence ∝q−1.3is absent in the real H(q), we again get noticeable discrepancy between HR0(q) and HR(q) (Fig. 6(c)). One can see that both functions clearly follow the exponent of 2.4. Thus, only model H2(q) demonstrates results matching the experimental data in essential details and, in general, in the shape of the obtained profiles. Note that throughout our measurements we have not succeeded in revealing an evidence of nonmonotonic dependence H (q) and existence of a local maximum suggested by several simulations [38,39]. However, we believe that experiments involving in-situ mechanical compression or stretching of graphene might lead to such findings.
5. Conclusion A technique for measuring the height-to-height correlation function H(q) of the corrugation of suspended graphene has been proposed. The measured H(q) follows the T /κq 4 law at the upper limit of the q variation range, which is in accordance with theoretical predictions. This allowed us to experimentally measure bending rigidity κ of suspended graphene. This value was found to be 0.8–1.6 eV, which is in good agreement with the theory. At qo3 nm 1 H(q) follows the power-law ∝ q−2.4 which is weaker than that predicted by the general theory of flexible membranes but is in
better accordance with the results of MDS. Even weaker power-law was observed at qo1 nm 1, namely, ca. ∝ q−1.3, whose validity was confirmed by modeling. These results may be regarded as a sign of a stronger influence of the charge carriers on the dynamics of the relatively long undulations of graphene surface.
Acknowledgments D.K. thanks the RFBR (Grant no. 16-32-60165) for the partial support of this research. The work was carried out in part at the Joint Research Center “Material Science and Characterization in Advanced Technologies” (St-Petersburg, Russia) under the financial support from the Ministry of Education and Science of the Russian Federation (Agreement 14.621.21.0007, 04.12.2014, id RFMEFI62114X0007, the use of the Jeol JEM-2100F microscope) and at EMAT, Universiteit Antwerpen (Antwerpen, Belgium), (the use of the FEI Tecnai G2 microscope).
References [1] M.I. Katsnelson, A.K. Geim, Electron scattering on microscopic corrugations in graphene, Philos. Trans. R. Soc. A 366 (2008) 195–204, http://dx.doi.org/ 10.1098/rsta.2007.2157. [2] I.F. Herbut, V. Juričić, O. Vafek, Coulomb interaction, ripples, and the minimal conductivity of graphene, Phys. Rev. Lett. 100 (2008) 046403, http://dx.doi. org/10.1103/PhysRevLett.100.046403. [3] Yu.A. Firsov, N.E. Firsova, Surface corrugations influence monolayer graphene electromagnetic response, Phys. E: Low-dimens. Syst. Nanostruct. 62 (2014) 36–42, http://dx.doi.org/10.1016/j.physe.2014.04.015. [4] Z. Osváth, A. Deák, K. Kertész, Gy Molnár, G. Vértesy, D. Zámbó, C. Hwang, L. P. Biró, The structure and properties of graphene on gold nanoparticles, Nanoscale 7 (2015) 5503–5509, http://dx.doi.org/10.1039/C5NR00268K. [5] S.-Y. Lin, S.-L. Chang, F.-L. Shyu, J.-M. Lu, M.-F. Lin, Feature-rich electronic properties in graphene ripples, Carbon 86 (2015) 207–216, http://dx.doi.org/ 10.1016/j.carbon.2014.12.068. [6] A. Rossi, S. Piccini, V. Pellegrini, S. de Gironcoli, V. Tozzini, Nano-scale corrugations in graphene: a density functional theory study of structure, electronic properties and hydrogenation, J. Phys. Chem. C 119 (2015) 7900–7910, http: //dx.doi.org/10.1021/jp511409b. [7] C.H. Lui, L. Liu, K.F. Mak, G.W. Flynn, T.F. Heinz, Ultraflat graphene, Nature 462 (2009) 339–341, http://dx.doi.org/10.1038/nature08569. [8] M. Ishigami, J.H. Chen, W.G. Cullen, M.S. Fuhrer, E.D. Williams, Atomic structure of graphene on SiO2, Nano Lett. 7 (2007) 1643–1648, http://dx.doi.org/ 10.1021/nl070613a. [9] V. Geringer, M. Liebmann, T. Echtermeyer, S. Runte, M. Schmidt, R. Ruckamp, M.C. Lemme, M. Morgenstern, Intrinsic and extrinsic corrugation of monolayer graphene deposited on SiO2, Phys. Rev. Lett. 102 (2009) 076102, http://dx.doi. org/10.1103/PhysRevLett.102.076102. [10] J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, T.J. Booth, S. Roth, The structure of suspended graphene sheets, Nature 446 (2007) 60–63, http://dx. doi.org/10.1038/nature05545. [11] L. Lindsay, D.A. Broido, N. Mingo, Flexural phonons and thermal transport in graphene, Phys. Rev. B 82 (2010) 115427, http://dx.doi.org/10.1103/ PhysRevB.82.115427. [12] J.H. Seol, I. Jo, A.L. Moore, L. Lindsay, Z.H. Aitken, M.T. Pettes, X. Li, Z. Yao,
D.A. Kirilenko, P.N. Brunkov / Ultramicroscopy 165 (2016) 1–7
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
R. Huang, D. Broido, N. Mingo, R.S. Ruoff, L. Shi, Two-dimensional phonon transport in supported graphene, Science 328 (2010) 213–216, http://dx.doi. org/10.1126/science.1184014. M.D. Kamatagi, A.S. Nissimagoudar, N.S. Sankeshwar, B.G. Mulimani, Lattice thermal conductivity of graphene, in: Proceedings of the AIP Conf. Proc., vol. 1447, 2012, pp. 945–946 〈http://dx.doi.org/10.1063/1.4710315〉. X. Tang, S. Xu, J. Zhang, X. Wang, Five Orders of Magnitude Reduction in Energy Coupling across Corrugated Graphene/Substrate Interfaces, ACS Appl. Mater. Interfaces 6 (4) (2014) 2809–2818, http://dx.doi.org/10.1021/ am405388a. E. Mariani, F. von Oppen, Temperature-dependent resistivity of suspended graphene, Phys. Rev. B 82 (2010) 195403, http://dx.doi.org/10.1103/ PhysRevB.82.195403. I.V. Gornyi, V.Yu Kachorovskii, A.D. Mirlin, Conductivity of suspended graphene at the Dirac point, Phys. Rev. B 86 (2012) 165413, http://dx.doi.org/ 10.1103/PhysRevB.86.165413. S. Das Sarma, E.H. Hwang, Density-dependent electrical conductivity in suspended graphene: approaching the Dirac point in transport, Phys. Rev. B 87 (2013) 035415, http://dx.doi.org/10.1103/PhysRevB.87.035415. K.I. Bolotin, K.J. Sikes, J. Hone, H.L. Stormer, P. Kim, Temperature-dependent transport in suspended graphene, Phys. Rev. Lett. 101 (2008) 096802, http: //dx.doi.org/10.1103/PhysRevLett.101.096802. H. Ochoa, E.V. Castro, M.I. Katsnelson, F. Guinea, Scattering by flexural phonons in suspended graphene under back gate induced strain, Physica E 44 (2012) 963–966, http://dx.doi.org/10.1016/j.physe.2011.03.017. P. Xu, Y. Yang, S.D. Barber, M.L. Ackerman, J.K. Schoelz, D. Qi, I.A. Kornev, L. Dong, L. Bellaiche, S. Barraza-Lopez, P.M. Thibado, Atomic control of strain in freestanding graphene, Phys. Rev. B 85 (2012) 121406, http://dx.doi.org/ 10.1103/PhysRevB.85.121406. R. Zan, C. Muryn, U. Bangert, P. Mattocks, P. Wincott, D. Vaughan, X. Li, L. Colombo, R.S. Ruoff, B. Hamilton, K.S. Novoselov, Scanning tunnelling microscopy of suspended graphene, Nanoscale 4 (2012) 3065–3068, http://dx. doi.org/10.1039/c2nr30162h. B. Shevitski, M. Mecklenburg, W.A. Hubbard, E.R. White, B. Dawson, M. S. Lodge, M. Ishigami, B.C. Regan, Dark-field transmission electron microscopy and the Debye–Waller factor of graphene, Phys. Rev. B 87 (2013) 045417, http: //dx.doi.org/10.1103/PhysRevB.87.045417. C.S. Allen, E. Liberti, J.S. Kim, Q. Xu, Y. Fan, K. He, A.W. Robertson, H. W. Zandbergen, J.H. Warner, A.I. Kirkland, Temperature dependence of atomic vibrations in mono-layer graphene, J. Appl. Phys. 118 (2015) 074302, http://dx. doi.org/10.1063/1.4928324. D.A. Kirilenko, A.T. Dideykin, G. Van Tendeloo, Measuring the corrugation amplitude of suspended and supported graphene, Phys. Rev. B. 84 (2011) 235417, http://dx.doi.org/10.1103/PhysRevB.84.235417.
7
[25] J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, D. Obergfell, S. Roth, C. Girit, A. Zettl, On the roughness of single- and bi-layer graphene membranes, Solid State Commun. 143 (2007) 101–109, http://dx.doi.org/10.1016/j. ssc.2007.02.047. [26] A. Fasolino, J.H. Los, M.I. Katsnelson, Intrinsic ripples in graphene, Nat. Mater. 6 (2007) 858–861, http://dx.doi.org/10.1038/nmat2011. [27] A. Dideykin, A.E. Aleksenskiy, D. Kirilenko, P. Brunkov, V. Goncharov, M. Baidakova, D. Sakseev, A. Ya.Vul, Monolayer graphene from graphite oxide, Diam. Relat. Mater. 20 (2011) 105–108, http://dx.doi.org/10.1016/j. diamond.2010.10.007. [28] S.K. Singh, S. Goverapet Srinivasan, M. Neek-Amal, S. Costamagna, Adri C. T. van Duin, F.M. Peeters, Thermal properties of fluorinated graphene, Phys. Rev. B 87 (2013) 104114, http://dx.doi.org/10.1103/PhysRevB.87.104114. [29] J.H. Los, M.I. Katsnelson, O.V. Yazyev, K.V. Zakharchenko, A. Fasolino, Phys. Rev. B 80 (2009) 121405, http://dx.doi.org/10.1103/PhysRevB.80.121405. [30] D. Nelson, L. Peliti, Fluctuations in membranes with crystalline and hexatic order, J. Phys. France 48 (1987) 1085, http://dx.doi.org/10.1051/ jphys:019870048070108500. [31] P. Le Doussal, L. Radzihovsky, Self-consistent theory of polymerized membranes, Phys. Rev. Lett. 69 (1992) 1209, http://dx.doi.org/10.1103/ PhysRevLett.69.1209. [32] F.L. Braghin, N. Hasselmann, Thermal fluctuations of free-standing graphene, Phys. Rev. B 82 (2010) 035407, http://dx.doi.org/10.1103/PhysRevB.82.035407. [33] K.V. Zakharchenko, R. Roldán, A. Fasolino, M.I. Katsnelson, Self-consistent screening approximation for flexible membranes: application to graphene, Phys. Rev. B 82 (2010) 125435, http://dx.doi.org/10.1103/PhysRevB.82.125435. [34] R. Roldan, A. Fasolino, K.V. Zakharchenko, M.I. Katsnelson, Suppression of anharmonicities in crystalline membranes by external strain, Phys. Rev. B 83 (2011) 174104, http://dx.doi.org/10.1103/PhysRevB.83.174104. [35] P. San-Jose, J. Gonzalez, F. Guinea, Electron-induced rippling in graphene, Phys. Rev. Lett. 106 (2011) 045502, http://dx.doi.org/10.1103/ PhysRevLett.106.045502. [36] M. Gibertini, A. Tomadin, F. Guinea, M.I. Katsnelson, M. Polini, Electron-hole puddles in the absence of charged impurities, Phys. Rev. B 85 (2012) 201405, http://dx.doi.org/10.1103/PhysRevB.85.201405. [37] D. Gazit, Correlation between charge inhomogeneities and structure in graphene and other electronic crystalline membranes, Phys. Rev. B 80 (2009) 161406, http://dx.doi.org/10.1103/PhysRevB.80.161406. [38] A. Smolyanitsky, Molecular dynamics simulation of thermal ripples in graphene with bond-order informed harmonic constraints, Nanotechnology 25 (2014) 485701, http://dx.doi.org/10.1088/0957-4484/25/48/485701. [39] N. Abedpour, Reza Asgari, M. Reza Rahimi Tabar, Irreversibility in response to forces acting on graphene sheets, Phys. Rev. Lett. 104 (2010) 196804, http://dx. doi.org/10.1103/PhysRevLett.104.196804.