First-principles study of anomalous Nernst effect in Cr-doped Bi2Se3

We have investigated electronic structures and thermoelectric properties on six quintuple layers of Cr-doped Bi2Se3 as a model of quantized anomalous Hall insulator, Chern insulator. The Chern insulator might be a good transverse thermoelectric material exhibiting a large anomalous Nernst effect by the intrinsic contribution of anomalous Hall effect and large Seebeck effect. Based on rigid band approximation, we have performed first-principles density functional calculations of carrier-dependent anomalous Nernst coefficients. To optimize thermoelectric performance, we discussed the origin of the anomalous Nernst effect, divided into contributions from pure Nernst and Seebeck terms. We found the significant contribution of the Seebeck term in Cr-doped Bi2Se3.


Introduction
Thermoelectric (TE) application has great potential to recover waste/natural heat into electricity via the Seebeck effect.The electric field E in the Seebecek effect device is generated parallel to the temperature gradient (∇T) or longitudinal voltage for power generation.However, it is very challenging to enhance its conversion efficiency due to conflicting parameters between the Seebeck coefficient (S), longitudinal electrical conductivity (σ xx ), and thermal conductivity (κ).A good TE material requires high S and σ xx as well as low κ, which are rarely found in one material like metals or insulators.One of the promising solutions is utilizing topological insulator as TE materials, which does not conduct electricity in their interior due to their bulk band gap but have two-dimensional (2D) conducting surface states protected by time-reversal symmetry. 1)mong the various early discovered topological insulator materials, [2][3][4] Bi 2 Se 3 is already known as an excellent TE material, 5,6) which has a narrow bulk band gap of 0.3 eV (3600 K), an indication of large S, and simplest Dirac cone surface spectrum at the Γ point at RT. 2) In addition, Bi 2 Se 3 belongs to the topological insulator that theoretically predicted 7) to exhibit quantized anomalous Hall effect (QAHE) by introducing magnetic doping, i.e.Cr to make Bi 2 Se 3 ferromagnetic, and the anomalous Hall effect (AHE) has been observed experimentally.8) Such materials hosting intrinsic contribution of AHE and/ or large Seebeck effect are candidates for unconventional TE applications based on anomalous Nernst effect (ANE).9) In contrast to the mechanism of power generation in Seebeck effect devices, in ANE, the TE voltage is generated perpendicular to ∇T. By uilizing its intrinsic magnetic field, the Ettingshausen heat current emerges from the low to hightemperature side in the ANE case, improving conversion efficiency by maintaining temperature differences.Furthermore, due to the orthogonal relationship between electrical and thermal conductivity in ANE, it has several unique benefits compared to Seebeck effect devices, i.e. more simplified in its design with only one ferromagnetic material is used as a thermopile, flexible to any heat source, tunable contact resistance and potential to be used as an energy source for future wearable devices if its anomalous Nernst coefficient (N) higher to 20-30 μ V k −1 .10) Recently, some studies investigated ANE using 3D and 2D materials as the base, e.g.half-Heusler compounds, 11,12) magnetic skyrmion crystal (SkX), 9,13) topological magnet, 14,15) extended to halfmetallic and metallic systems.16,17) The present study investigates the TE properties of six quintuple layers of Cr-doped Bi 2 Se 3 by employing firstprinciples density functional calculations.The carrier concentration dependence of the anomalous Nernst coefficients N and Seebeck coefficients S are estimated by rigid band approximation.We discussed the origin of the ANE in a thin film of Cr-doped Bi 2 Se 3 system divided into the contribution from pure Nernst term related to the asymmetry of the AHC σ xy along the energy axis, and Seebeck term as a result of product between S and Hall angle ratio. 9)On the other hand, studying the ANE is fascinating from the scientific point of view as a sensitive probe to detect topological electronic states apart from Fermi energy.

Models
Bi 2 Se 3 belongs to a rhombohedral crystal structure with space group , with five atoms crystal structure in one unit cell.It comprises five atomic layers arranged along the z-direction, the quintuple layer (QL).Each QL comprises five atoms ordered in the Se-Bi-Se'-Bi-Se sequence, where Se and Se' denote different lattice positions.The Se layer can be related to another Se layer in 1QL Bi 2 Se 3 , by an inversion operation in which Se' atom is located at the inversion center.The bonds between two atomic layers within 1QL are stronger but weaker between neighboring QLs since the former bonds are covalent and the latter are van der Waals type. 1) We calculated the slab model of 6QLs Cr-doped Bi 2 Se 3 using an experimental lattice constant a = b = 4.14 Å 18) and c = 75.0Å.The vacuum region of our unit cell was d vacuum = 20.22Å.We used the experimental atomic positions for the bulk system.For the atomic position of Cr, we just replaced a bismuth atom of the top and bottom QLs.We expected that the Chern insulating state and transport properties would be insensitive to the atomic positions.We evaluate TE properties of 6QLs Cr-doped Bi 2 Se 3 in which a Cr atomic layer substitutes the first Bi layer in the outer QL of our 6QLs Bi 2 Se 3 .The ratio of Cr in our target system of Cr-doped Bi 2 Se 3 is 50%.The top view of the Cr-doped Bi 2 Se 3 structure is shown in Fig. 1(a), while the side view is shown in Fig. 1(b).Blue spheres in Fig. 1(b) indicate the Cr atoms that both are doped on 6QLs Bi 2 Se 3 surfaces so that it can be viewed as 4QLs Bi 2 Se 3 between 2QLs of CrBiSe 3 .In determining the easy axis for magnetization of the Cr atom, the total energy of out-of-plane and in-plane magnetization is compared.We found that the out-of-plane magnetization of the Cr atom is preferable to in-plane magnetization.Besides that, only out-of-plane magnetization generates the QAHE, which is consistent with the experimental result. 8)o observe a ferromagnetic state in Cr-doped Bi 2 Se 3 experimentally, it can be done by surface engineering with modulation doping of 50% Cr-doped on the outer thin-film of Bi 2 Se 3 , combined with Ca compensation doping to lower the Fermi level as previously discussed by Moon et al. 8) On the other hand, to evaluate TE properties based on ANE in Crdoped Bi 2 Se 3 by first-principle calculation, we need 2 Cr dopants located on the top and bottom QL, at first Bi layer, to get Chern insulating states with ferromagnetic spin configurations.If we locate the Cr atom only at the top surface of Bi 2 Se 3 , the bottom surface will have a surface Dirac state.Then, we could not obtain a quantized anomalous Hall insulating state.In addition, in the present implementation of computing anomalous Hall conductivity σ xy , the contributions from top and bottom surfaces to σ xy can not be divided.Thus, we assumed symmetric Cr doping to Bi 2 Se 3 on top and bottom surfaces.

Computational method
First-principles calculations were conducted based on the non-collinear density functional theory to obtain the electronic states 19) of target Cr-doped Bi 2 Se 3 with the OpenMX code. 20,21)The non-collinear density functional theory calculations were performed through the exchange-correlation functional of the generalized gradient approximation with the Perdew-Burke-Ernzerhof functional. 22)We used the norm-conserving pseudopotential method 23) with a cutoff energy for change density of 300 Ry.The spin-orbit interaction was included, and we expanded the wave functions using a linear combination of multiple pseudoatomic orbitals 24,25) with specification Bi8.0-s3p2d2f1, Se7.0-s3p2d2, Cr6.0-s3p2d1, where 8.0, 7.0 and 6.0 are cutoff radii (in bohr) of Bi, Se, and Cr atoms, respectively.The number after s, p, d, and f is the radial function multiplicity of each angular momentum component.A (30, 30, 1) k-point mesh was used for the self-consistent field calculations.After selfconsistent field calculation is achieved, we restart the calculation by increasing the k-point mesh into (301, 301, 1) for the one-shot calculation to get more accurate Fermi energy.We neglect changes in the lattice parameters and atomic positions induced by Cr doping.The spin texture analysis was performed with the post-process diagonalization of the Hamiltonian, 26,27) while the Berry curvature was computed through the Fukui-Hatsugai-Suzuki method. 28,29)e evaluate the anomalous Hall conductivity for our system by conducting a post-processing calculation based on the extension of the Fukui-Hatsugai-Suzuki method to a metallic system 17,28,29), which is implemented in the OpenMX code. 20,21)The Berry curvature Ω n (k) is defined as, where A n is the Berry connection, given by Eq. ( 2) in terms of u n k , which is the periodic part of Bloch states: Anomalous Hall conductivity σ xy dependence on the Fermi energy is required to evaluate the anomalous Nernst conductivity α xy .In insulating systems, anomalous Hall conductivity σ xy is a quantized topological invariant.On the other hand, the anomalous Hall conductivity σ xy is obtained from the Berry curvature Ω n (k) as 30,31) : where N, e, h, f, ε nk and μ are the number of occupation electron, elementary charge, Planck constant, Fermi-Dirac distribution function, indexed band energy n with wave vector k and Fermi energy, respectively.For the insulator, the Eq. 3 should have a quantized value defined by σ xy (ε F ) = e 2 C/h, where C is the "Chern number" = 0, ±1, ±2, ...). 17 To obtain the TE coefficient, we derived from the linear response relation of charge current, which is given as j = σ ij E + α ij (−∇T), where σ ij , α ij , E, and ∇T are the conductivity tensors, TE conductivity tensors, the electric field and temperature gradient, respectively.The Nernst 01SP26-2 © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd coefficients are related to the conductivity tensors, and these relationships can be represented as follows: , 5 where S 0 = α xx /σ xx , N 0 = α xy /σ xx and θ H = σ xy /σ xx are the pure Seebeck coefficient, pure anomalous Nernst coefficient, and Hall angle, respectively.In those equations, there is electrical conductivity, which is defined as n and τ are the group velocity of electrons and relaxation time, respectively.In determining the magnitude of the ANE, we focus on the intrinsic contribution of anomalous Hall conductivity σ xy based on the moderately dirty regime theory 32) and neglect the extrinsic ones caused by impurities or defects.
Finally, we performed electron transport calculations to get σ xx and α xx , i.e. S 0 , using BoltzTraP. 33)We use semi-classical Boltzmann transport theory in the relaxation time approximation based on a smoothed Fourier interpolation of the bands.We investigated the longitudinal TE properties of 6QLs Cr doped Bi 2 Se 3 and constant relaxation time approximation with fixed values of τ = 100 fs, 500 fs, and 1000 fs.We assume the band remains unchanged as we move the Fermi level position to simulate doping.

Results and discussion
Let us first look at the landscape of the electronic structure of 6QLs Bi 2 Se 3 and 6QLs Cr-doped Bi 2 Se 3 .Figure 2 shows the calculated surface electronic structures of Bi 2 Se 3 and Crdoped Bi 2 Se 3 along the K-Γ-M symmetry lines, on which M and Γ are the surface Kramers points for its (111) surface. 34)i 2 Se 3 is known as a non-trivial topological insulator where the surface Dirac states are protected by time reversal symmetry E(k, ↑) = E(−k, ↓).The surface Dirac states are robust against local perturbations.In our calculation, including spin-orbit interaction, a doubly degenerated gapless surface band emerged at the Γ point and crossed the Fermi level.The Dirac states connect the bulk VB and conduction band.For more than 4QLs, since there is no hybridization between top surface wavefunctions and bottom surface wavefunctions of the film Bi 2 Se 3 , the surface Dirac states are doubly degenerated.
The introduction of magnetic impurities to the topological insulator results in significant changes in its electronic structure, for example, in our case of Cr-doped Bi 2 Se 3, as shown in Fig. 2(b).Remarkably, the long-range magnetic order perturbation led to an exchange gap in the gapless Dirac dispersion of the surface states by breaking the timereversal symmetry. 4)The calculated magnetic moment is 3.0 μ B /Cr atom, where μ B is the Bohr magneton.The appearance of a small exchange gap Eg ∼ 0.033 eV at Dirac surface state of Cr-doped Bi 2 Se 3 is beneficial for the Seebeck effect, supposed to enhance the figure of merit ZT as discussed for TE properties of topological insulator. 5)e focus on the surface state of Cr-doped Bi 2 Se 3 to observe the effect of magnetic impurity on the spin textures of Bi 2 Se 3 .To discuss the spin texture due to the doubly degenerate surface states around the Γ point, we projected onto the topmost QL of our Cr-doped Bi 2 Se 3 .
The calculated spin expectation values for z(x) components are visualized in the band structure Fig. 3(b) [Fig.3(c)] on the symmetry lines -M-Γ-M, Fig. 3(a).There is a significant net out-of-plane magnetization (S z ) that appears due to the Zeeman spin splitting caused by the Cr atom shown in Fig. 3(b).The appearance of the out-of-plane magnetization S z (k) = S z (−k) in Cr-doped Bi 2 Se 3 is evidence of timereversal symmetry breaking.Moreover, the upper and lower Dirac bands have opposite spin S z as presented by different colors in Fig. 3(b), which are blue for a spin down and red for a spin up.][37] In addition, Figs.3(e) and 3(f) present the spin textures projected to k x -k y plane of Cr-doped Bi 2 Se 3 , denoting a competition between the in-plane spin component ruled by the spin-orbit interaction and the out-ofplane time-reversal symmetry breaking component due to the effect of magnetic impurity Cr atoms, as previously discussed by Aramberri et al. for the interface between thin film Bi 2 Se 3 and monoatomic layers of Cr. 38) To evaluate the ANE on TE performance, a study on the intrinsic term of anomalous Hall coefficient as a function of Berry curvature in momentum (k) space is needed. 9)and experimentally observed. 40,41)   This indicates the anomalous Nernst conductivity α xy as a reasonable parameter for monitoring between calculation and experimental results. 42)Finally, in terms of the calculated anomalous Nernst coefficient shown in Table I and Fig. 5(b), we found a significant contribution of S 0 than α xy in the doped region.There is also a different sign between α xy and σ xy S 0 , which reinforced each other in Eq. ( 5).Besides that, the small magnitude of α xy originated from constant σ xy which occurred near and at Fermi level, as shown in Fig. 5(a) while large σ xy S 0 is generated due to Zeeman splitting from Cr-Doping.In other words, the significant transverse TE coefficient in this Chern insulator material emerged from the longitudinal TE effect (S 0 ), called the Seebeck-induced effect.Significant changes are observed in the doped region of the Cr-doped Bi 2 Se 3 system, where carrier concentrations are introduced.For the shortest relaxation time (τ = 100 fs), the Nernst coefficient (N) reaches its peak values of approximately 125.66 μV K −1 and −122.83μV K −1 for electron and hole doping, respectively.These magnitudes surpass those obtained for a longer relaxation time of τ = 500 fs and τ = 1000 fs.However, according to the moderately dirty regime theory, τ = 100 fs fails to meet the required condition since |θ H | > 0.333 33 for both electron and hole doping, as indicated in Table I.In this case, τ = 500 fs and τ = 1000 fs offer better adherence to the condition than τ = 100 fs.Nevertheless, experimental data from thin-film Bi 2 Se 3 43) suggests that τ should be around 1000 fs.Hence, the appropriate values for N are 28.55 μV K −1 and −29.0 μV K −1 for the electron and hole-doped regions, respectively.These magnitudes are comparable with a previous study on Chern insulating van der Waals MnBi 2 Te 4 layers. 44)

Conclusions
In summary, the non-collinear density functional calculations were performed to calculate TE properties, especially the ANE of Cr-doped Bi 2 Se 3 6QLs.We showed that Cr doping in the Bi 2 Se 3 system led to a change in its surface state from a topological insulator into a Chern insulator through the opening of a gap around the Dirac point, with Chern number C = −1.We added convincing proof that Cr doping in Bi 2 Se 3 leads to the time-reversal symmetry breaking, shown by the spin expectation value figures visualized in the band structures.The upper and bottom Dirac cones have out-of-plane magnetization (S z ) due to )  When the chemical potential dependence of anomalous Hall conductivity σ xy are obtained, the anomalous Nernst conductivity α xy can be evaluated as follows:31)

Fig. 1 .
Fig. 1.(a) Top and (b) side view of the 6QLs Cr-doped Bi 2 Se 3, which Cr atom substitute first Bi layer, respectively.These figures are drawn by the visualization program VESTA. 45)Green, purple, and blue spheres indicate Se, Bi, and Cr atoms, respectively.Inside the blue square is 1QL of CrBiSe 3 , with an arrow indicating the Cr atom's magnetization.

Figure 4 (
a) shows the calculated Berry curvature Ω(k) of Cr-doped Bi 2 Se 3 bands using a 100 × 100 mesh number.The large Berry curvature Ω(k) is revealed around the Γ-point in the Brillouin zone, which resulting an AHE at 0 K, as can be seen in Fig. 4(b).At T ≈ 0K, the anomalous Hall conductivity σ xy of the Cr-doped Bi 2 Se 3 system has a quantized value with Chern number C = −1, which leads to one plateau around the Fermi energy.This quantized anomalous Hall conductivity is direct evidence of quantized anomalous Hall states, which indicate the changing surface state of Bi 2 Se 3 from a topological insulating state into a Chern insulating state due to magnetic impurity.By the Chern insulating state, the large transverse TE effect may occur.Figures 5(a)-5(b) show the transverse TE properties of Cr-doped Bi 2 Se 3 depending on the chemical potential at T = 20 K.The Curie temperature for long-range ferromagnetic order of Cr-doped Bi 2 Se 3 is around T C = 7 − 20 K, which theoretically predicted

Figure 5 (
a) shows the transverse (longitudinal) electrical conductivity σ xy (σ xx ) of Cr-doped Bi 2 Se 3 dependence on the chemical potential at 20 K.The values of σ xy can be applied to the moderately dirty regime where the intrinsic contribution of anomalous Hall conductivity is dominant, and the maximum Hall ratio should not exceed 0.333 333. 32)(a) (

Fig. 3 .Fig. 4 . 4 ©
Fig. 3. (a) 2D Brillouin zone for 6QLs Bi 2 Se 3 (111) surface.Spin expectation value of highest occupied and lowest unoccupied bands of 6QLs Cr-doped Bi 2 Se 3 along the Γ -M/5 direction for each spin component: (b) S z , (c) S x .The S y is zero.Red and blue arrows represent the opposite spin directions.(d) Side view of Cr-doped Bi 2 Se 3 along with arrows indicating the magnetization of the Cr atom at one surface, drawn by visualization program VESTA.Spin texture projected to k x -k y plane for each band around the Γ (0, 0, 0) point: (e) VB maximum and (f) conduction band minimum.

Figure 5 (
b) shows the chemical potential dependence of the anomalous Nernst conductivity (α xy ) and the product between anomalous Hall conductivity σ xy and pure Seebeck coefficient (σ xy S 0 ) of Cr-doped Bi 2 Se 3 .This relationship explains the intrinsic TE properties independent of relaxation time τ.Looking back at Eq. (5), the anomalous Nernst coefficient can be written as
Zeeman spin splitting caused by Cr doping.The anomalous Nernst coefficient of our Cr-doped Bi 2 Se 3 has a value as high as N = 28.55 μV K −1 and N = −29.03μV K −1 for the electron and hole-doped region, respectively, based on the rigid band approximation.This resultant N originates from pure Seebeck term, by large product of pure Seebeck coefficient S 0 and Hall ratio θ H .For a further investigation to search TE candidates, we can concentrate on the magnitude of α xy , independent of relaxation time τ within the approximation used in this study.In addition, it can be compared between experiments and theory without worries about the delicate quantity of τ and, above all, should pursue larger α xy .
σ xx ( ) e h 2 σ xy ( ) © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd