Distorted AB-Pair Graphene Calculation Formulas

This project uses a real-space distorted six-site supercell. The formulas below show the full calculation route: distorted hopping law, 6x6 Bloch Hamiltonian, Brillouin-zone integration, and the numerical approximations used to build the plots.

1. Distorted Hopping Law

t(d) = t₀ exp[-β (d / a₀ - 1)]

The nearest-neighbor hopping is recalculated from each distorted bond length after the AB-pair coordinates are displaced.

2. Six-Site Bloch Hamiltonian

H_ij(k) = ∑_R t_ij(R) exp[i k · (R + r_j - r_i)]

E_n(k) = eigenvalues of H(k), n = 1, ..., 6

E_g = min_k [E₄(k) - E₃(k)]

3. Supercell Geometry

T₁ = (2.598076, 1.500000)

T₂ = (-2.598076, 1.500000)

4. Distorted Nearest-Neighbor Bonds

d = 0.760000, t(d) = 2.245212
d = 1.139122, t(d) = 0.625726

5. Band-Path Sampling

k(s) = (1 - s) k_a + s k_b, 0 ≤ s ≤ 1, path = G → K → M → G

For each sampled k-value, the code diagonalizes the 6x6 matrix H(k) and sorts the six eigenvalues.

6. Brillouin-Zone DOS Integral and Approximation

D(E) = (1 / A_BZ) ∑_n=1⁶ ∫_BZ δ(E - E_n(k)) d²k

δ(x) ≈ L_η(x) = [η / π] / (x² + η²)

D(E) ≈ (1 / N_k) ∑_m ∑_n=1⁶ L_η(E - E_n(k_m))

7. Reciprocal-Space Map and Numerical Gap

k_mn = u_m B₁ + v_n B₂, (u_m, v_n) on a uniform grid clipped to the first Brillouin zone

E_g^(num) ≈ min_m [E₄(k_m) - E₃(k_m)]

8. DOS Broadening

The same Lorentzian broadening is used for all six bands when generating DOS curves from the sampled reciprocal-space eigenvalues.

Run parameters
pair_shift = 0.120000
t₀ = 1.000000
β = 3.370000

Gap result from this run
Direct gap from reciprocal-space sampling = 1.987520