Abstract
Among many remarkable qualities of graphene, its electronic properties attract particular interest owing to the chiral character of the charge carriers, which leads to such unusual phenomena as metallic conductivity in the limit of no carriers and the halfinteger quantum Hall effect observable even at room temperature^{1,2,3}. Because graphene is only one atom thick, it is also amenable to external influences, including mechanical deformation. The latter offers a tempting prospect of controlling graphene’s properties by strain and, recently, several reports have examined graphene under uniaxial deformation^{4,5,6,7,8}. Although the strain can induce additional Raman features^{7,8}, no significant changes in graphene’s band structure have been either observed or expected for realistic strains of up to ∼15% (refs 9, 10, 11). Here we show that a designed strain aligned along three main crystallographic directions induces strong gauge fields^{12,13,14} that effectively act as a uniform magnetic field exceeding 10 T. For a finite doping, the quantizing field results in an insulating bulk and a pair of countercirculating edge states, similar to the case of a topological insulator^{15,16,17,18,19,20}. We suggest realistic ways of creating this quantum state and observing the pseudomagnetic quantum Hall effect. We also show that strained superlattices can be used to open significant energy gaps in graphene’s electronic spectrum.
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Main
If a mechanical strain Δ varies smoothly on the scale of interatomic distances, it does not break the sublattice symmetry but rather deforms the Brillouin zone in such a way that the Dirac cones located in graphene at points K and K′ are shifted in the opposite directions^{2}. This is reminiscent of the effect induced on charge carriers by magnetic field B applied perpendicular to the graphene plane^{2,12,13,14}. The straininduced, pseudomagnetic field B_{S} or, more generally, gaugefield vector potential A has opposite signs for graphene’s two valleys K and K′, which means that elastic deformations, unlike magnetic field, do not violate the timereversal symmetry of a crystal as a whole^{12,13,14,21,22}.
On the basis of this analogy between strain and magnetic field, we ask the following question. Is it possible to create such a distribution of strain that it results in a strong uniform pseudomagnetic field B_{S} and, accordingly, leads to a ‘pseudoquantum Hall effect (QHE)’ observable in zero B? The previous attempts to engineer energy gaps by applying strain^{5,6,7} seem to suggest a negative answer. Indeed, the hexagonal symmetry of the graphene lattice generally implies a highly anisotropic distribution of B_{S} (refs 21, 22). Therefore, the strain is expected to contribute primarily in the phenomena that do not average out in a random magnetic field, such as weak localization^{13,14}. Furthermore, a strong gauge field implies the opening of energy gaps owing to Landau quantization, (>0.1 eV for B_{S}=10 T), whereas no gaps were theoretically found for uniaxial strain as large as ≈25% (ref. 4). The only way to induce significant gaps known so far is to spatially confine carriers (δ E≈0.1 eV requires 10nmwide ribbons)^{1,2}. Contrary to these expectations, we have found that by applying stresses with triangular symmetry it is possible to generate a uniform quantizing B_{S} equivalent to tens of Tesla so that the corresponding gaps exceed 0.1 eV and are observable at room temperature.
A twodimensional strain field u_{i j}(x,y) leads to a gauge field^{23,24}
where a is the lattice constant, β=−∂lnt/∂lna≈2 and t the nearestneighbour hopping parameter, and the xaxis is chosen along a zigzag direction of the graphene lattice. In the following, we consider valley K, unless stated otherwise. We can immediately see that B_{S} can be created only by nonuniform shear strain. Indeed, for dilation (isotropic strain), equation (1) leads to A=0 and, for the uniform strain previously considered in refs 4, 5, 6, to A=const, which also yields zero B_{S}.
Using polar coordinates (r,θ), equation (1) can be rewritten as
which yields the pseudomagnetic field
In the radial representation, it is easy to show that uniform B_{S} is achieved for the following displacements:
where c is a constant. The strain described by (2) and its crystallographic alignment are shown in Fig. 1a,b, respectively. This yields uniform B_{S}=8β c/a (given in units ℏ/e≡1). For a disc of diameter D, which experiences a maximum strain Δ_{m} at its perimeter, we find c=Δ_{m}/D. For nonambitious Δ_{m}=10% and D=100 nm, we find B_{S}≈40 T, the effective magnetic length and the largest Landau gap of ≈0.25 eV. Note that distortions (2) are purely shear and do not result in any changes in the area of a unit cell, which means that there is no effective electrostatic potential generated by such strain^{23}.
The lattice distortions in Fig. 1a can be induced by inplane forces F applied only at the perimeter and, for the case of a disc, they are given simply by
where μ is the shear modulus. Figure 1c shows the required force pattern. It is difficult to create such strain experimentally because this involves tangential forces and both stretching and compression. To this end, we have solved an inverse problem to find out whether uniform B_{S} can be generated by normal forces only (Supplementary Information, part I). There exists a unique solution for the shape of a graphene sample that enables this (see Fig. 1d).
A strong pseudomagnetic field should lead to Landau quantization and a QHElike state. The latter is different from the standard QHE because B_{S} has opposite signs for charge carriers in valleys K and K′ and, therefore, generates edges states that circulate in opposite directions. The coexistence of gaps in the bulk and counterpropagating states at the boundaries without breaking the timereversal symmetry is reminiscent of topological insulators^{15,16,17,18,19,20} and, in particular, the quantum valley Hall effect in ‘gapped graphene’^{20} and the quantum spin Hall effect induced by strain^{16}. The latter theory has exploited the influence of threedimensional strain on spin–orbit coupling in semiconductor heterostructures, which can lead to quasiLandau quantization with opposite B_{S} acting on two spins rather than valleys. Weak spin–orbit coupling allows only tiny Landau gaps <1 μeV (ref. 16), which, to be observable, would require temperatures below 10 mK and carrier mobilities higher than 10^{7} cm^{2} V s^{−1}. Our approach exploits the unique strength of pseudospin–orbit coupling in graphene, which leads to δ E>0.1 eV and makes the straininduced Landau levels realistically observable.
The two cases shown in Fig. 1 prove that by using strain it is possible to generate a strong uniform B_{S} and observe the pseudoQHE. They also prove the general concept that if the strain is applied along all three 〈100〉 crystallographic directions to match graphene’s symmetry this prevents the generated fields from changing sign. However, it is a difficult experimental task to generate such a complex distribution of forces as shown in Fig. 1. Below we develop the found concept further and show that the pseudoQHE can be observed in geometries that are easier to realize, even though they do not provide a perfectly uniform B_{S}.
Let us consider a regular hexagon with side length L and normal stresses applied evenly at its three nonadjacent sides and along 〈100〉 axes (Fig. 2a). Our numerical solution for this elasticity problem shows that B_{S} has a predominant direction (positive for K and negative for K′) and is fairly uniform close to the hexagon’s centre. Assuming L=100 nm and Δ_{m}=10%, we find for Fig. 2a that B_{S} varies in the range ±22 T but is ≈20 T over most of the hexagon’s central area. For other L and Δ, we can rescale the plotted values of B_{S} by using the expression B_{S}∝Δ_{m}/L. We have also examined other geometries and always found a nearly uniform distribution of B_{S} near the sample’s centre (see Supplementary Information).
To verify that the nonuniform B_{S} in Fig. 2a leads to welldefined Landau quantization, we have calculated the resulting density of states D(E). To this end, we have used the scaling properties of the Dirac equation, which allows us to extrapolate the lowenergy spectrum of small lattices to larger systems. The scaling approach is unfortunately limited to sizes L≈30 nm (Supplementary Information, part V). Figure 2b plots our results for Δ_{m}=1% (B_{S}≈7 T at the hexagon centre) and compares them for the case of the same hexagon in B=0 and 10 T but without strain. In the absence of strain or B, the peak at zero E is due to the states localized at the edges^{2}. This peak increases with increasing strain, and its development is better seen in D(E) calculated at the centre of the hexagon (Supplementary Fig. S5). We can also see that both nonuniform B_{S} and uniform B generate Landau levels, and the qualities of the induced quantization are fairly similar. In Fig. 2b and Supplementary Fig. S5, the width of the zeroE peak and the remnant density of states between zero and adjacent peaks are determined by the finite broadening (∼2 meV) introduced in the calculation of Green’s function, whereas the next two levels are slightly broadened by nonuniform B_{S}. In general, the influence of the inhomogeneity in B_{S} on the zero level should be minimal because field inhomogeneity does not lead to broadening of this level^{25}. We emphasize that the smearing of the Landau levels in Fig. 2b is mostly due to the small sample size used in calculations and, for micrometresized hexagons, the first few pseudoLandau levels should be well resolved in experiment (Supplementary Fig. S4).
To create the required strain, we can generally think of exploiting the difference in thermal expansion of graphene and a substrate^{11} and apply temperature gradients along 〈100〉 axes. For the case of quasiuniform B_{S}, there are many more options available, including the use of suspended samples and profiled substrates. For example, a graphene hexagon can be suspended by three metallic contacts attached to its sides, similar to the technique used to study suspended graphene^{26,27}, and the strain can then be controlled by gate voltage. Alternatively, a quasiuniform B_{S} can be created by depositing graphene over triangular trenches (Supplementary Information).
To probe the pseudoLandau quantization, we can use optical techniques, for example Raman spectroscopy, which should reveal extra resonances induced by B_{S} (ref. 28). This technique should allow detection of pseudomagnetic field locally, within submicrometre areas. We can also use transport measurements in both standard and Corbinodisc geometries. In the former case, the counterpropagating edge states imply that contributions from two valleys cancel each other and no Hall signal is generated (ρ_{x y}=0) (refs 15, 16, 17, 18, 19, 20). At the same time, the edge transport can lead to longitudinal resistivity ρ_{x x}=h/4e^{2}N where N is the number of spindegenerate Landau levels at the Fermi energy. This nonzero quantized ρ_{x x} has the same origin as in socalled dissipative QHE, where two edge states with opposite spins propagate in opposite directions^{29}. In spinbased topological insulators, the edge transport is protected by slow spinflip rates^{15,16,29}. In our case, atomicscale disorder at the edges is likely to mix the countercirculating states on a submicrometre scale (Supplementary Information, section VI). Therefore, instead of quantization in ρ_{x x} we may expect highly resistive metallic edge states, similar to the case discussed in ref. 29. The suppression of the edgestate ballistic transport does not affect the pseudoLandau quantization in graphene’s interior, where intervalley scattering is very weak^{13,30} and should not cause extra level broadening. Highly resistive edges should in fact make it easier to probe pseudoLandau gaps in the bulk. In the Corbino geometry, the edgestate mixing is irrelevant, and we expect twoprobe ρ_{x x} to be a periodic function of gate voltage and show an insulating behaviour between pseudoLandau levels. Furthermore, the outer contact can be used to cover perimeter regions, in which B_{S} is nonuniform. This should improve the quality of quantization.
Finally, we point out that the developed concept can be used to create gaps in bulk graphene. Imagine a macroscopic graphene sheet deposited on top of a corrugated surface with a triangular landscape (Fig. 3a). In the following calculations, we have fixed the graphene sheet at the landscape’s extrema and enabled the resulting inplane displacements to relax^{21,22} (at the nanoscale, graphene should then be kept in place by van der Waals forces). The resulting pseudomagnetic superlattice is plotted in Fig. 3b whereas Fig. 3c shows the resulting energy spectrum. Close to zero E, there is a continuous band of electronic states, in agreement with the fact that the zero level is insensitive to the field’s inhomogeneity^{25}. At higher E, there are multiple gaps with δ E>100 K. The relatively small gaps are due to the weak shear strain induced in this geometry (Δ_{m}<0.1%). By improving the design of strained superlattices, it must be possible to achieve much larger gaps. We believe that the suggested strategies to observe the pseudoLandau gaps and QHE are completely attainable and will be realized sooner rather than later.
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Acknowledgements
This work was supported by the EPSRC (UK), FOM (The Netherlands), the Royal Society, Office of Naval Research and Air Force Office of Scientific Research. F.G. also acknowledges support by MICINN (Spain) through grants FIS200800124 and CONSOLIDER CSD200700010, and by the Comunidad de Madrid, through CITECNOMIK.
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Guinea, F., Katsnelson, M. & Geim, A. Energy gaps and a zerofield quantum Hall effect in graphene by strain engineering. Nature Phys 6, 30–33 (2010). https://doi.org/10.1038/nphys1420
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DOI: https://doi.org/10.1038/nphys1420
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