A versatile model with three-dimensional triangular lattice for unconventional transport and various topological effects

ABSTRACT The finite Berry curvature in topological materials can induce many subtle phenomena, such as the anomalous Hall effect (AHE), spin Hall effect (SHE), anomalous Nernst effect (ANE), non-linear Hall effect (NLHE) and bulk photovoltaic effects. To explore these novel physics as well as their connection and coupling, a precise and effective model should be developed. Here, we propose such a versatile model—a 3D triangular lattice with alternating hopping parameters, which can yield various topological phases, including kagome bands, triply degenerate fermions, double Weyl semimetals and so on. We reveal that this special lattice can present unconventional transport due to its unique topological surface states and the aforementioned topological phenomena, such as AHE, ANE, NLHE and the topological photocurrent effect. In addition, we also provide a number of material candidates that have been synthesized experimentally with this lattice, and discuss two materials, including a non-magnetic triangular system for SHE, NLHE and the shift current, and a ferromagnetic triangular lattice for AHE and ANE. Our work provides an excellent platform, including both the model and materials, for the study of Berry-curvature-related physics.

The plane-wave cutoff energy was set to 550 eV.A Monkhorst-Pack k-point mesh [4] with a size of 13×13×13 was used for the BZ sampling.The crystal structure was optimized until the forces on the ions were less than 0.001 eV/ Å.The surface spectrum was calculated by using the Wannier functions and the iterative Green's function method [5][6][7].The nonequilibrium Keldysh Green's function and Landauer-Bttiker formula [8] as implemented in the numerical package KWANT [9] were employed to calculate the electronic conductance G, i.e.G 12 = (2e 2 /h)Tr(Γ 2 G r Γ 1 G a ), where G 12 represents the conductance from Lead1 to Lead2, G r/a are the retarded and advanced Green's functions of the central lattice region, and Γ 1/2 are the line-width functions that couple Lead1 and Lead2 terminals to the central region, respectively.The anomalous Hall conductivity was calculated via: where f nk is the Fermi distribution function and Ω n (k) = ∇ k × i n, k|∇ k |n, k is the Berry curvature [10].According to the Kubo formula, the intrinsic spin Hall conductivity (SHC) can be written as [11][12][13] σ spinz xy (ω) = where n and m are band indexes, n and m are the eigenvalues, ĵspinz x = 1 2 {ŝ z , vx } is the spin current operator and ŝz = 2 σz is the spin operator, vy = 1 ∇ y H(k) is the velocity operator, and the frequency ω and η are set to zero in the dc clean limit.It is noted that if we replace ĵspinz x in Eq.S2 by −ev x , Eq. S2 is exactly Eq.S1.The Berry curvature dipole that describes nonlinear Hall effect is defined as [14] The shift current was calculated by Wannier interpolation method [15] based on the formula [16]:

Symmetry analysis
In this section, we will give a symmetry analysis of why the Weyl points locate at the high-symmetry points in our model and take K and Γ points as examples.The k •p model H is subjected to the D 3 little group at K, with two generators, C 3z and C 2x .The symmetry constraints are given by where q is measured from K. In the basis of the 2D irreducible representation E for D 3 , we find that to linear order in q, the effective model takes the form of the 2D Weyl model, where ν 1 and ν 2 are the Fermi velocities in the xy plane and z direction, respectively, and σ x/y/z are the Pauli matrices acting in the space of the two basis states.Thus, the low-energy electrons indeed resemble 2D Weyl fermions.
As for Γ point, it not only have the D 3 little group, but also is a time-reversal invariant momentum.The time-reversal symmetry constraint is given by where q is measured from Γ.In the basis of the 2D irreducible representation E for D 3 , we find that to leading order in q, the effective model takes the form where f 0 (q) = a 0 + a 1 (q 2 x + q 2 y ) + a 2 q 2 z , f x (q) = −2b 1 q x q y + b 2 q x q z , f y (q) = c 0 and f z = −b 1 (q 2 x − q 2 y ) + b 2 q y q z .The reduced Hamiltonian is in the Weyl form, linear in q x , q y , and q z at K but has the leading second-order terms of q x , q y , and q z at Γ. Similar results can be obtained at H and A points.These results indicate that two Weyl points with topological charges ±1 can be found at K and H and double Weyl points with topological charges ±2 can be found at Γ and A. The topological charge 2 of double Weyl points are protected by the combination of the screw symmetry and the time-reversal symmetry.x and z are taken as the a and c axes of the crystal.For an electric field E = (E x , 0, E z ), the nonlinear current density j (2) is given as . According to the Ohm's law, we have the second-order non-linear electric field , where j is the current amplitude and θ is the angle measured from c axis, we have the first order electric field and its longitudinal component is E = j(ρ a sin 2 θ + ρ c cos 2 θ).If the non-linear Hall effect is measured in the y direction, the angle resolved non-linear response takes the similar form as Eq.( 4) in the main text.If the non-linear Hall effect is measured in xy plane, the transverse component of the second-order electric field can be written as , where γ 1 is the resistance anisotropy as γ 1 = ρ a /ρ c .Thus, angle resolved non-linear response through the second-order non-linear susceptibilities can be obtained as: (S10) According to the global factor sinθ, it can be inferred that the non-linear Hall response is maximal when the driving current is applied along the x direction.Compared to the case with the SOC caused by an out-of-plane electric field in Fig. 2(h), where the AHC and ANC are measured in the xy plane, in the presence of an in-plane electric field induced SOC, the AHC and ANC are measured in the xz and yz planes.

Material candidates
Besides the two 3D triangular materials discussed in the main text, some other 3D triangular materials are also shown in Fig. S3.

3 τ 2 2
Non-linear Hall effect for the space groups of No. 144 and 145In general, the second-harmonic part of the non-linear Hall current related to BCD can be represented byj 2ω a = χ abc E ω b E ω c ,where χ abc is the non-linear tensor and the E is an external electric field.In the TR invariant system, χ abc = acd D bd e (1+iωτ ) , where τ is the relaxation time and is the Levi-Civita symbol.The space groups of No. 144 and 145 correspond the point group of 3. BCD with symmetry constraint has the non-zero matrix elements D xx , D yy , D zz and D yx = −D xy , so that possible non-linear tensor elements are χ yxz = −χ zxy = d 1 , χ zyx = −χ xyz = d 2 , χ yzx = −χ xzy = d 3 and χ zxx = −χ xxz = χ zyy = −χ yyz = d 4 .Thus, for an electric field in the x direction, we have j (2) z ∼ D xy E x E x , i.e. the non-linear Hall current can be measured in the z direction.Based on the symmetry analysis, the non-linear susceptibility has four independent non-zero elements d i (i = 1, 2, 3, 4).Here, the coordinates

Anomalous
FIG. S2.The band structures of magnetic double Weyl semimetal without and with SOC from an in-plane electric field based on Fig. 1(h), where Zeeman splitting and SOC strength are taken as m = 0.1 and λ = 0.05.(b) The AHC and ANC as a function of Fermi level.