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Model calculation
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Room temperature strain-induced Landau levels in graphene on a wafer-scale platform

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We considered a minimal tight-binding model on the honeycomb lattice with nearest-neighbor hoppings and a sublattice symmetry–breaking Semenoff (35) mass term M$H=−t∑(cA†(r)cB(r′)+H.c.)+M(∑rcA†(r)cA(r)−∑r′cB†(r′)cB(r′))$(3)where creates an electron in the pz orbital at lattice site r (r′) on the sublattice A (B) of the honeycomb lattice, t = 2.7 eV, and the nearest-neighbor distance is a0 = 0.142 nm. We neglected the electron spin and thus considered effectively spinless fermions.

We constructed a flake in the shape of an equilateral triangle of side length L ∼ 56 nm. The use of armchair edges avoids the zero-energy edge modes appearing for zigzag edges (3). We applied the simplest strain pattern respecting the triangular symmetry of the problem at hand, namely, the pattern introduced by Guinea et al. (10), which gives rise to a uniform (out-of-plane) pseudomagnetic field$B=4u0ħβea0ẑ$(4)where β ≈ 3.37 in graphene (33), and the corresponding displacement field is given by(5)

The hopping parameter renormalization induced by this displacement field was calculated using the simple prescription(6)where (xij, yij) ≡ rirj is the vector joining the original (unstrained) sites i and j and$ϵij=12[∂jui+∂iuj]$(7)is the strain tensor corresponding to the (in-plane) displacement field u. Outside the strained region (which we took as a triangle of slightly smaller length LS ∼ 48 nm), we allowed the strain tensor to relax: $ϵ→e−r22σ2ϵ$, where r is the perpendicular distance to the boundary of the strained region and σ ∼ 1 nm. We defined the length scale of the homogeneous magnetic field B to be the diameter of the largest inscribed circle in the triangle of side LS: $λ≡LS/3∼28$ nm. We stress here that our simulated flakes are much smaller than the experimentally observed triangular features of size ∼300 nm. The fact that we nevertheless reproduced the experimental features underlines how the number of observable LLs is limited by the length scale of the homogeneous pseudomagnetic field λ, rather than by the size L of the nanoprisms themselves. This length scale could be caused by the more complicated strain pattern present in the nanoprisms or be induced by disorder.

We then diagonalized the Hamiltonian (Eq. 3) with hopping parameters given by Eq. 6 to obtain the full set of eigenstates |n〉 with energies En and computed the momentum-resolved, retarded Green’s function using the Lehmann representation$GαR(k,ω)=∑n∣〈n|cα†(k)|0〉∣2ω−(En−E0)−iη$(8)where α = A, B is a sublattice (band) index and η ∼ 20 meV is a small broadening parameter comparable to the experimental resolution.

We then computed the one-particle spectral function$A(k,ω)=−1π∑αIm[GαR(k,ω)]$(9)which is proportional to the intensity measured in ARPES (modulo the Fermi-Dirac distribution and dipole matrix elements). We note that using a finite system introduces two main effects in the momentum-resolved spectral function: the appearance of a small finite-size gap at the Dirac points (in the absence of a magnetic field) and a momentum broadening of the bands.

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