### Introduction

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 strain-induced, pseudomagnetic field *B*_{S} or, more generally, gauge-field 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 time-reversal 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 ‘pseudo-quantum 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 10-nm-wide 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 two-dimensional strain field *u*_{ij}(*x*,*y*) leads to a gauge field^{23, }^{24}

where *a* is the lattice constant, *β*=−*∂*ln*t*/*∂*ln*a*≈2 and *t* the nearest-neighbour hopping parameter, and the *x*-axis 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 non-uniform 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}=8*β**c*/*a* (given in units /*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 non-ambitious *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}.

### Figure 1: Designed strain can generate a strictly uniform pseudomagnetic field in graphene.

**a**, Distortion of a graphene disc which is required to generate uniform *B*_{S}. The original shape is shown in blue. **b**, Orientation of the graphene crystal lattice with respect to the strain. Graphene is stretched or compressed along equivalent crystallographic directions 100. Two graphene sublattices are shown in red and green. **c**, Distribution of the forces applied at the disc’s perimeter (arrows) that would create the strain required in **a**. The uniform colour inside the disc indicates strictly uniform pseudomagnetic field. **d**, The shown shape allows uniform *B*_{S} to be generated only by normal forces applied at the sample’s perimeter. The length of the arrows indicates the required local stress.

The lattice distortions in Fig. 1a can be induced by in-plane 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 QHE-like 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 time-reversal 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 three-dimensional strain on spin–orbit coupling in semiconductor heterostructures, which can lead to quasi-Landau 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 strain-induced 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 pseudo-QHE. 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 pseudo-QHE 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 non-adjacent 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).

### Figure 2: Stretching graphene samples along 100 axes always generates a pseudomagnetic field that is fairly uniform at the centre.

**a**, Distribution of *B*_{S} for a regular hexagon stretched by its three sides oriented perpendicular to 100. Other examples are given in the Supplementary Information.
**b**, Normalized density of states for the hexagon in **a** with *L*=30 nm and *Δ*_{m}=1%. The black curve is for the case of no strain and no magnetic field. The peak at zero *E* is due to states at zigzag edges. The blue curve shows the Landau quantization induced by magnetic field *B*=10 T. The pseudomagnetic field with *B*_{S}≈7 T near the hexagon’s centre induces the quantization shown by the red curve. Comparison between the curves shows that the smearing of the pseudo-Landau levels is mostly due to the finite broadening *Γ*=2 meV used in the tight-binding calculations (*Γ* corresponds to submicrometre mean free paths attainable in graphene devices). The inhomogeneous *B*_{S} plays little role in the broadening of the first few pseudo-Landau levels (see Supplementary Fig. S4).

To verify that the non-uniform *B*_{S} in Fig. 2a leads to well-defined 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 low-energy 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 non-uniform *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 zero-*E* 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 non-uniform *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 micrometre-sized hexagons, the first few pseudo-Landau 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 quasi-uniform *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 quasi-uniform *B*_{S} can be created by depositing graphene over triangular trenches (Supplementary Information).

To probe the pseudo-Landau 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 Corbino-disc 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*_{xy}=0) (refs 15, 16, 17, 18, 19, 20). At the same time, the edge transport can lead to longitudinal resistivity *ρ*_{xx}=*h*/4*e*^{2}*N* where *N* is the number of spin-degenerate Landau levels at the Fermi energy. This non-zero quantized *ρ*_{xx} has the same origin as in so-called dissipative QHE, where two edge states with opposite spins propagate in opposite directions^{29}. In spin-based topological insulators, the edge transport is protected by slow spin-flip rates^{15, }^{16, }^{29}. In our case, atomic-scale 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*_{xx} we may expect highly resistive metallic edge states, similar to the case discussed in ref. 29. The suppression of the edge-state ballistic transport does not affect the pseudo-Landau 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 pseudo-Landau gaps in the bulk. In the Corbino geometry, the edge-state mixing is irrelevant, and we expect two-probe *ρ*_{xx} to be a periodic function of gate voltage and show an insulating behaviour between pseudo-Landau levels. Furthermore, the outer contact can be used to cover perimeter regions, in which *B*_{S} is non-uniform. 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 in-plane 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 pseudo-Landau gaps and QHE are completely attainable and will be realized sooner rather than later.

### Figure 3: Energy gaps can be opened in strained graphene superlattices.

**a**, Strain with triangular symmetry can be created by depositing graphene on profiled surfaces (dimensions in the plot are given in nanometres). The corrugations shown in **a** result in the distribution of pseudomagnetic field plotted in **b**. *B*_{S} varies in the range ±0.5 T (red to violet) with the periodicity half that in **a**. **c**, Low-energy bands induced by the periodic strain. The bands are symmetric with respect to zero *E*.