Random strain fluctuations as dominant disorder source for high-quality on-substrate graphene devices

We have performed systematic investigations of transport through graphene on hexagonal boron nitride (hBN) substrates, together with confocal Raman measurements and a targeted theoretical analysis, to identify the dominant source of disorder in this system. Low-temperature transport measurements on many devices reveal a clear correlation between the carrier mobility $\mu$ and the width $n^*$ of the resistance peak around charge neutrality, demonstrating that charge scattering and density inhomogeneities originate from the same microscopic mechanism. The study of weak-localization unambiguously shows that this mechanism is associated to a long-ranged disorder potential, and provides clear indications that random pseudo-magnetic fields due to strain are the dominant scattering source. Spatially resolved Raman spectroscopy measurements confirm the role of local strain fluctuations, since the line-width of the Raman 2D-peak --containing information of local strain fluctuations present in graphene-- correlates with the value of maximum observed mobility. The importance of strain is corroborated by a theoretical analysis of the relation between $\mu$ and $n^*$ that shows how local strain fluctuations reproduce the experimental data at a quantitative level, with $n^*$ being determined by the scalar deformation potential and $\mu$ by the random pseudo-magnetic field (consistently with the conclusion drawn from the analysis of weak-localization). Throughout our study, we compare the behavior of devices on hBN substrates to that of devices on SiO$_2$ and SrTiO$_3$, and find that all conclusions drawn for the case of hBN are compatible with the observations made on these other materials. These observations suggest that random strain fluctuations are the dominant source of disorder for high-quality graphene on many different substrates, and not only on hexagonal boron nitride.

By investigating low-temperature transport through many graphene devices on hBN substrates, we reveal a clear correlation between the carrier mobility µ and the width of the resistance peak around charge neutrality. The correlation -satisfied quantitatively also by devices realized in other laboratories, and on other substrates-indicates that a same, universal microscopic mechanism limits the carrier mobility and generates charge fluctuations for graphene-on-substrate. Weak-localization measurements show that the underlying random disorder potential is long-ranged, at least for devices whose mobility is between 1.000 and 80.000 cm 2 /Vs. We propose a theoretical interpretation based on the effects of strain in graphene, which reproduces all key aspects of our observations.
Hexagonal boron nitride (hBN) substrates enable the fabrication of graphene devices with strongly reduced disorder [1][2][3][4], exhibiting impressively high carrier mobility values, and leading to the the observation of interesting physical phenomena [5][6][7][8][9]. Progress in device quality, however, has not led to the identification of the nature of disorder affecting the properties of graphene-onsubstrate. Many open questions remain [10]. It is unclear, whether the dominant microscopic mechanism causing scattering of charge carriers is also responsible for the inhomogeneity in carrier density around the charge neutrality point (the so-called charge puddles). It is not yet established whether the dominant carrier scattering processes originate from short-or long-range potentials [11][12][13]. It is also unclear if and how, microscopically, disorder is related to the specific substrate material.
Here we present an experimental study of grapheneon-hBN addressing these issues. While the best grapheneon-hBN devices exhibit impressive mobility, more modest values are also commonly found and the resulting broad range of electrical characteristics allows the identification of correlations between different quantities. We find an unambiguous correlation between the carrier mobility µ and the width of the resistance peak around charge neutrality n * -with µ ∝ (n * ) −1 -extending over nearly two orders of magnitude. This correlation is also satisfied by devices fabricated in other laboratories and, more surprisingly, by devices on other substrate materials. It demonstrates that the physical mechanism limiting the mobility in graphene-on-substrate is the same one responsible causing charge inhomogeneity, and points to its universality. Through weak localization measurements we establish that -at least for devices with µ in the range 1.000-80.000 cm 2 /Vs-the mobility is limited by intravalley scattering, implying that the dominant disorder is long-ranged. Our results point to deformations caused by strain in graphene as the dominant source of disorder for graphene-on-substrate, and we discuss in detail our data in this context.
The fabrication of graphene-on-hBN devices relies on a technique described in the literature [1]. We exfoliate hBN crystals onto a heavily doped, oxidized Si wafer. Graphene flakes extracted from natural graphite are transferred onto a hBN crystal, following the procedure of Ref. [1]. Metallic contacts (Ti/Au, 10/75 nm) are defined by electron-beam lithography, evaporation and liftoff (see Fig.1(a)). We find "bubbles"(as in Ref. [2,3,14]) when transferring graphene on hBN : achieving high-µ requires etching Hall bar devices in parts of the flakes with no bubbles (regions with "bubbles" exhibit lower µ, comparable to SiO 2 devices). After an electrical characterization at 4K, we perform different low-temperature thermal annealing steps (150-250 C, in an environment of H 2 /Ar at 100/200 sccm) and check each time the lowtemperature transport characteristics. We find that the initial annealing step always results in a mobility increase (a factor of 2 in the very best cases), whereas subsequent annealing lead to a decrease in µ, eventually to values similar to those obtained on SiO 2 [15].
We analyzed approximately 15 distinct Hall-bar devices. Mobility values (at 4.2 K) between 30.000 cm 2 /Vs and 80.000 cm 2 /Vs at a carrier density of few 10 11 cm −2 were found regularly. Integer quantum Hall (QH) plateaus with σ Hall = 4(1/2 + N )e 2 /h (N integer) are fully developed starting from B = 1 T, and broken symmetry QH states with Hall conductivity σ Hall = ±1e 2 /h appear from B = 8 T. Full degeneracy lifting of the N = 0 and N = ±1 Landau levels is observed below 15 T (Fig1.(e)). In devices where the lattices of graphene and hBN were intentionally aligned, we observe the effect of a superlattice potential, with the appearance of satellite Dirac peaks in the measured R(V g ) curve ( Fig.1.(c)) [7,9]. These results indicate that our devices have quality comparable to those reported in the literature, fabricated using a similar procedure.
To evaluate the quality of our graphene-on-hBN devices we focus on the low-T mobility µ and on the width n * of the minimum in the conductivity σ-vs-V g curve. µ measures the elastic scattering time τ responsible for momentum relaxation, whereas n * quantifies the potential fluctuations present [10,16]. Since these potential fluctuations are not a priori the dominant source of elastic scattering, there is no reason to assume that µ and n * are related. Experimentally, the carrier mobility is obtained from µ = σ/ne(see Fig. 2(a)), with the density of charge carriers n obtained through the Hall resistance. To extract n * we plot log(σ) as a function of log(n), and determine at which n the constant value of log(σ) measured at low density crosses the value of log(σ) extrapolated (linearly) from high density (as shown in Fig. 2(b)). The mobility is estimated for n > n * . Fig. 2(c) shows µ as a function of n * for all devices, measured either immediately after fabrication, or after a subsequent annealing step. The presence of a correlation between µ and n * is unambiguous : devices with smaller density fluctuations have larger mobility. For hBN devices fabricated in our laboratory, this correlation extends from µ values of 5.000 cm 2 /Vs (for devices after multiple annealing steps, see below) to 80.000 cm 2 /Vs. Results reported in the literature [1,2,5] quantitatively fit the same trend, extending the range to µ = 100.000 cm 2 /Vs. Surprisingly, the correlation is fulfilled also by devices on different substrate materials,(the red and green dots in Fig. 2(c) represent data obtained from graphene on SiO 2 and SrTiO 3 [13] [17]). Plotting 1/µ-vs-n * (Fig. 3(a)) shows that the relation between these two quantities is essentially linear. To reduce the statistical The blue full circles represent the low-temperature mobility µ (plotted versus n * ) for all the 15 graphene-on-hBN devices realized in our laboratory, measured after fabrication or after annealing. The triangles represent data for graphene-on-hBN extracted from Ref. [1][5](orange triangles) and from Ref. [2](green triangle). The green diamonds and red squares are from devices realized in our laboratory on SiO2 and SrTiO3 substrates, respectively.
fluctuations we subdivide the n * axis into eight different intervals and plot the inverse averaged mobility as a function of the average charge density fluctuations (Fig. 3(b)), which makes the linear scaling of 1/µ with n * fully apparent. We conclude that the microscopic mechanism limiting µ for graphene-on-substrate is the same that causes density fluctuations around charge neutrality. We also conclude that the process is "universal", i.e., independent of the specific substrate material or of details of the fabrications process, which only determines the strength of disorder. To gain additional insight, we analyze the scattering times in graphene, extracted from weak-localization measurements [18][19][20]. Specifically, we compare the intervalley scattering time τ iv to the elastic scattering time τ determined from the carrier mobility. Either τ iv ≃ τ , implying that the mobility is determined by inter-valley scattering processes (i.e., the dominant source of disorder are short-range potentials), or τ iv ≫ τ , indicating that µ is limited by intra-valley scattering (i.e., long-range disorder potentials dominate). Surprisingly, this straightforward argument has not been used systematically in previous work to identify the dominant disorder, nor it has been suggested in theoretical work (for an exception, see Ref. [21] dealing with rather low mobility devices, µ ≃ 1.000 cm 2 /Vs). Fig.4(a) shows the low-field magneto-resistance of a Hall bar device with µ ≃ 60.000 cm 2 /Vs, for different values of V g around V g = 8 V. A narrow dip in conductivity (width ≃ 1 mT or less) is seen around B = 0 T, originating from weak localization. Aperiodic conductance fluctuations due to random interference are also visible, which we suppress by averaging measurements taken for slightly different V g values [22]. "Ensembleaveraged" curves obtained in this way around three different V g values are shown in Fig. 4(b). The data are fit to existing theory [18], from which we extract τ iv , the phase coherence time τ φ , and the time τ * needed to break effective single-valley time reversal symmetry. At T = 250 mK τ φ is much larger than τ iv , which is why weak-localization is observed. More importantly throughout the density range investigated τ and τ * nearly coincide (which, as we discuss below, gives important indications as to the origin of the dominant disorder), and τ iv ≫ τ by at least one order of magnitude, (and by nearly two at low n). This last observation implies that intra-valley scattering due to long-range potentials is the process limiting µ, a conclusion that -in conjunction with previous measurements on graphene-on-SiO 2 [20,21]-holds at least in the mobility range between 1.000 and 80.000 cm 2 /Vs.
The two sources of long-range disorder that have been proposed theoretically to play a role in graphene are potentials originating from a low density n imp of charged impurities [11,23,24] at the substrate surface, and strain present in graphene [25]. Different aspects of our data consistently indicate that strain dominates [26]. A first indication comes from the observation that τ ≃ τ * . This finding is naturally explained by strain, which generates random pseudo-magnetic fields [27] that not only scatter charge carriers, but also break the effective time reversal symmetry in a single valley [18,19] on a comparable time scale. On the contrary, for long range potentials generated by charges on the substrate, τ is determined by the Fourier components with k ≈ k F , whereas τ * is determined by random fluctuations in the potential difference between the A and B atoms in the individual unit cells of graphene, i.e. by the Fourier component with k ≃ 1/a [28]. Through Fermi golden rule, a potential generated by charge impurities leads to τ * ≃ τ nimpa 2 >> τ , incompatible with the experimental observations. The evolution of µ upon annealing also points to the effect of strain. As discussed above, repeated annealing at low temperature (≃ 200 C) in an inert atmosphere systematically reduces µ by one order of magnitude. These annealing processes have no significant chemical effect, and therefore are not expected to change the density of charge at the surface of hBN by one order of magnitude (as it would be needed to explain the changes in µ [11]). On the contrary, they do lead to visible mechanical deformations, compatible with strain causing a decrease in mobility. Finally, having µ limited by strain-induced pseudo-magnetic fields also explains why the use of high-ǫ substrates does not lead to a very large increase in mobility : a high-ǫ substrate can screen electrostatically the scalar potentials induced by strain, but not the effect of a pseudo-magnetic field.
Following these considerations, we need to determine whether strain can explain the observed relation between 1/µ and n * (see Fig. 3a,b). We do not attempt to discuss theoretically the full universality, and focus on the case of low dielectric constant (ǫ 0 ) substrates, such as hBN and SiO 2 , which concerns the majority of the experimental data. We treat graphene as an elastic membrane [25,29,30], and consider strain associated to random ripples, described by the correlation function for the height fluctuations h(q)h(−q) = A/|q| 4 (A = T q /κ, with T q quenching temperature of the height fluctuations, and κ bending rigidity of graphene. Strain due to in-plane deformations also contributes, but does not change our conclusions). From this expression, the correlation functions for the random scalar and vector (gauge) potential V s (q) and A(q) -which describe how strain couples to electrons in graphene-are calculated (see Ref. [30] and [28]). The scalar and the gauge potential scatter the electrons with comparable (on low-ǫ substrates) rates (1/τ s and 1/τ g , respectively), and limit µ. The magnitude of the charge fluctuations n * , on the contrary, is determined by the scalar potential only. We calculate 1/τ s and 1/τ g using Fermi golden rule, and obtain the total scattering time as 1/τ = 1/τ s + 1/τ g . Specifically [28] : and where A ⊥ (q) is the component of A perpendicular to q, N (E F ) = kF 2π vF is the one-valley density of states at the Fermi energy, ǫ(q) = (ǫ 0 + 1)/2 + 4e 2 k F /v F |q| is the dielectric function including the substrate contribution, and k F ,v F , and E F are the Fermi momentum, velocity, and energy. We extract the mobility from µ = σ/ne = 2 e 2 h EF τ ne (the factor of 2 accounts for the two valleys). To calculate the magnitude of charge fluctuations we use the relation n(r) = 1 π ( Vs(r) vF ) 2 between local charge density and potential, from which : Since the correlation functions of all the potentials are determined by h(q)h(−q) , µ and n * are related, and we find [28] : with g 1 = 4 eV and g 2 = 2.3 eV quantifying the strength of electron-phonon coupling in graphene, µ L = 9.4 eV/Å 2 and λ L = 3.3 eV/Å 2 Lamé coefficients [31], and vF e 2 = 2.2 (a is the lattice constant of graphene and the logarithm appears when cutting off the integrals at large q-values, at q = 1/a).
Eq. (4) shows that the relation between 1/µ and n * is linear (the deviations caused by the logarithm are within the fluctuations in the data) as found experimentally, and it only depends on fundamental constants and on the elastic properties of graphene (i.e. the intrinsic properties of graphene). It is therefore fully compatible with the universality found experimentally. With the value of all the constants known (and using n * = 10 11 cm −2 as a characteristic value to calculate the logarithm) we obtain 1 µ ≈ h e n * × 0.076 (5) which reproduces the slope of the relation measured experimentally with a remarkable accuracy (see Fig. 3a,b), without having introduced any free parameter. This last result shows that strain does not only explain the qualitative aspects of our observations, but also account for the measured relation between µ and n * at a quantitative level.
We conclude that our observations provide compelling evidence for the role of mechanical deformations in graphene as the dominant source of disorder, that both cause scattering of charge carriers, and spatial fluctuations in their density. The important implication of this finding is the remarkable universality of the behavior of grapheneon-substrate : it suffices to provide one quantity (equivalently, n * , µ, the magnitude of the mechanical deformations) to characterize in a rather complete manner the low-energy transport properties of a graphene device. AFM gratefully acknowledges support by the SNF and by the NCCR QST. FG acknowledges support from the Spanish Ministry of Economy (MINECO) through Grant No. FIS2011-23713 and the European Research Council Advanced Grant (contract 290846). Both AFM and FG acknowledge funding from the EU under the Graphene Flagship.