Random Strain Fluctuations as Dominant Disorder Source for High-Quality On-Substrate Graphene Devices

We perform 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 with a long-ranged disorder potential and provides clear indications that random pseudomagnetic fields due to strain are the dominant scattering source. Spatially resolved Raman spectroscopy measurements confirm the role of local strain fluctuations, since the linewidth 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 pseudomagnetic 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 ${\mathrm{SiO}}_2$ and ${\mathrm{SrTiO}}_3$, 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.

Popular Summary

Graphene is a honeycomb lattice entirely made of carbon only one atom thick; it is the first system discovered to be a perfect two-dimensional crystal. However, even graphene, with all of its ideal properties, is unavoidably deformed when isolated and placed on a supporting substrate. For instance, graphene may not lie perfectly flat but instead form “bumps.” Furthermore, its atoms may be randomly displaced from their ideal positions in the honeycomb lattice because of the force exerted by the atoms in the substrate. Scientists call these types of mechanical deformations “strain.” We show that when graphene exfoliated from graphite is placed on a supporting hexagonal boron nitride substrate, the random strain that appears in the carbon honeycomb lattice reduces the speed with which electrons can move inside the material.

Scientists have puzzled for a long time about the mechanism limiting the speed of electrons in real-world graphene devices. Many possibilities have been proposed, such as missing atoms or chemical impurities, structural defects in the honeycomb lattice, and unintentional “dirt.” All of these mechanisms can play a role in limiting the speed of electrons, which is why the situation has remained unclear for a long time. We use both statistical analyses and experimental verification and provide ample evidence that random strain fluctuations in the material set a limit on the electron speed for temperatures as low as 250 mK. We find that electron scattering is local (i.e., within the same valley) and is due to long-range potentials. The way in which strain affects electrons in graphene is very similar to the effect of a magnetic field. In other words, random strain generates a fluctuating pseudomagnetic field, which randomly deflects electrons and limits their speed. Since the amount of strain can strongly depend on the substrate or on the way in which the graphene layer is mounted on the substrate, the electron speed or mobility can vary significantly. We find that our results can also be applied to graphene mounted on ${\mathrm{SiO}}_2$ and ${\mathrm{SrTiO}}_3$ substrates.

Future electronic applications will necessitate a high electron speed. To optimize applications, therefore, we will need to focus our efforts on selecting substrates and assembly procedures that minimize strain.

Figure 1

(a) Optical microscope image of a monolayer graphene flake on top of a 30-nm-thick hBN crystal before (left) and after (right) depositing metal contacts (the scale bars are $5\mu \mathrm{m}$). (b) Longitudinal resistance $R_{xx}$ as a function of $V_g$ and $B$ showing quantum Hall states originating from the lifting of the single-particle degeneracy already at $B\simeq 8\mathrm{T}$. (c) Hall (blue line) and longitudinal (green line) conductivity as a function of $V_g$ measured at $B=15\mathrm{T}$, showing full degeneracy lifting of Landau level $N=0$, 1. (d) Resistance of a graphene device whose edge was aligned to that of the hBN substrate, showing the emergence of satellite Dirac peaks (well developed for negative $V_g$ and less pronounced for positive $V_g$). All measurements have been taken at $T=250\mathrm{mK}$.

Figure 2

Conductivity $\sigma$ of a graphene monolayer on hBN as a function of carrier density $n$ in linear (a) and double-logarithmic (b) scale, measured after fabrication (blue line), after a first annealing at $150°\mathrm{C}$ (green line), and after a second annealing at $250°\mathrm{C}$ (red line). Panel (b) also illustrates the procedure to extract the value of $n^{*}$. (c) The blue full circles represent the low-temperature mobility $\mu$ (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 Refs. [1, 5] (orange triangles) and from Ref. [2] (green triangle). The green diamonds and red squares are from devices realized in our laboratory on ${\mathrm{SiO}}_2$ and ${\mathrm{SrTiO}}_3$ substrates, respectively.

Figure 3

(a) Same data as those of Fig. 2 plotted as $1/\mu \text{−}{n}^{*}$, showing an overall linear relation. (b) Average inverse mobility as function of $n^{*}$ (obtained as indicated in the main text), showing clearly the linearity of the relation. In (a) and (b) the dashed lines are a linear fit to the data, $(1/\mu )=(h/e)n^{*}\times 0.118$. (c) minimum conductivity ${\sigma }^{*}=n^{*}e\mu$, calculated from the estimated carrier density fluctuations $n^{*}$ and mobility $\mu$, and plotted as a function of measured minimum conductivity. The excellent overall agreement (the dashed line has slope 1) confirms the correctness of the procedures used to extract $n^{*}$ and $\mu$ from the measurements.

Figure 4

(a) $B$ and $V_g$ dependence of the resistivity measured at $T=250\mathrm{mK}$. (b) The circles represent magnetoconductivity curves $\mathrm{\Delta }\sigma (B)$ that have been ensemble averaged, by averaging traces in a range of gate voltages around $V_g=-7$ (blue circles), 7 (green circles), and 30 V (red circles), to suppress sample-specific fluctuations. The continuous lines are fit to the theory of weak localization in graphene. (c) Characteristic times extracted at 250 mK for different values of carrier density, either from the fit of weak-localization curves (${\tau }_{\varphi }$, ${\tau }_{iv}$, and ${\tau }_{*}$) or from the conductivity ($\tau$). The elastic scattering time $\tau$ is always at least 1 order of magnitude smaller than the intervalley scattering time ${\tau }_{iv}$.

Figure 5

(a) Correlation of the inverse mobility and the average full width at half maximum of the Raman 2D peak ${\mathrm{\Gamma }}_{2\mathrm{D}}$ for a number of graphene flakes on different substrates. (b) Raman maps of two graphene flakes resting on two different substrates (left, hBN; right, ${\mathrm{SiO}}_2$) highlighting the different values of the spatially resolved ${\mathrm{\Gamma }}_{2\mathrm{D}}$ (same color scale). The white scale bars are $2\mu \mathrm{m}$. (c) Histograms of ${\mathrm{\Gamma }}_{2\mathrm{D}}$ for the two examples shown in (b). These histograms are used to extract the data points illustrated in (a).

Figure 6

Correlation of the inverse mobility and $(n^{*}{)}^2$ for a number of different bilayer graphene flakes on different substrates.