Thermoelectric properties of ballistic Normal–Weyl semimetal-Normal junction

Weyl semimetals are a new class of topological materials possessing outstanding physical properties. We investigate the thermoelectric properties of a ballistic Weyl semimetal specimen connected to two normal contacts. We introduce a model to evaluate the thermoelectric coefficients of the junction and analyze its features along two distinct directions, one along the chiral axis of the Weyl semimetal and the other perpendicular to it. We demonstrate that the thermoelectric response of this junction depends on whether it is along the chiral axis of the Weyl semimetal or not. Electrical and thermal conductances of this junction reveal considerable dependence on the length and chemical potential of the Weyl semimetal layer. In particular, we observe that, decreasing the chemical potential in the normal contacts enhances the Seebeck coefficient and thermoelectric figure of merit of the junction to substantial values. Hence, we unveil that a ballistic junction of Weyl semimetal can serve as a fundamental segment for application in future thermoelectric devices for thermal energy harvesting.

First, we investigate the electrical conductance of the junction along the chiral axis or z direction in terms of the different parameters characterizing it.Fig. S1 represents the normalized conductances, G n /G 0 with G 0 = (e 2 /h)(µA/8π 2 M), of the junction for electrons incident in two different states in terms of the chemical potential of the normal leads (µ) and distinct values of the other parameters of the junction.As we can see in these figures, the conductance of the first state G 1 shows an overall increase by increasing µ for all values of the other parameters of the junction, whereas the conductance of the second state G 2 vanishes at high chemical potentials after presenting a peak at small values.This behavior originates from the peculiarities of the junction of normal and WSM.Increasing the chemical potential of the normal leads cause the electronic dispersion of the first state in WSM to approach to the dispersion of the normal contacts, and it results in an increase in the conductance.While, for the second state difference in the energy dispersion in the normal contacts and WSM increases, which in turn leads to the suppression of the conductance.
Moreover, in Figs.S1 (d) and (h), we observe that the electrical conductance decreases for both states by increasing the length of the junction as we expect, while figures (c) and (g) show that increasing the chemical potential in WSM layer from negative values to zero leads to enhancement of conductance in both states and this effect is more considerable for the second state at low chemical potentials.In addition, we have analyzed the dependence of the conductance on the inherent parameters of WSM.We find that, by increasing value of γ, figures (a) and (e), the conductance of the first state decreases at high chemical potentials, while the conductance of the second state represents an increment.This feature arises since parameter γ indicates the strength of mixing between two states inside the WSM segment.Therefore, by increasing the value of γ, two states mix increasingly and this leads to decreasing the conductance of the first state and an increment of the conductance of the second state.Meanwhile, by increasing the value of k 0 which denotes the separation of the Weyl nodes we observe a suppression in conductance of the first state at all chemical potentials, figure (b), whereas conductance of the second state exhibits a considerable enhancement by increasing k 0 as it is clear from figure (f).
Now we analyze the behavior of the electrical conductances of N-WSM-N junction for the first and second states when the junction direction is perpendicular to the chiral axis or along the x axis.As we can see in Fig. S2, for the higher chemical potentials of the leads, we observe a behavior similar to the junction along the z axis when γ, k 0 , the chemical potential and length of the WSM layer vary.However, in this case, conductance shows a more profound dependence on the junction parameters in comparison to the junction along z axis.In particular, the dependence of the conductance in both states on γ has been considerably enhanced in the case of the junction along the x axis.We find that there is a chemical potential threshold for the conductance of both states, which increases by increasing the value of γ.Moreover, by increasing γ conductance of both states decrease (increase) at lower (higher) chemical potentials.While we observe an increase (decrease) in the conductance of both states by increasing k 0 at lower (higher) chemical potentials.By increasing values of µ W , a peak is generated in the conductance of both states at very low chemical potentials.Finally, we find that the conductance of both states has identical behavior in both junctions.

Contribution of the Fermi arc surface states
In a finite WSM, edge states manifest themselves as the Fermi arc states on the surface of the relevant Brillouin zone, which connect the projections of the Weyl-fermion Fermi surfaces in momentum space [1].According to the minimal model of WSM, the surface states appear on the terminated faces of the WSM layer along the x and y axes [2].The corresponding Fermi arcs emerge on the relevant surfaces of the Brillouin zone and parallel to the k z direction in the momentum space.It has been shown that these states are described by the Hamiltonians H s,x = γk y and H s,y = γk x [2], respectively.Therefore, the Fermi velocities of these surface states are along the x and y directions, perpendicular to the z axis.As a result, we conclude that these surface states can not contribute to the electrical and thermal currents of the junction along the z axis.In the following, we concentrate on the contribution of the Fermi arc surface states in the thermoelectric coefficients of the junction along the x axis.We consider the Fermi arc surface states on the surfaces of WSM denoted by y = 0 and y = W , where W denotes the thickness of the layer along the y axis.Electrons in the surface states appearing on two opposite faces propagate in opposite directions without backscattering.For an almost thick layer of WSM, 100nm < W < 1µm, we can suppose the following form for the wave function of the surface states on the face y = 0, Applaying this wave function to the bulk Hamiltonian given by Eq. 1 in the main text and imposing the relevant boundary conditions to have zero wave function at this surface, the corresponding values of the decay constant are obtained λ as follows [2], where 0 and E = γk x .Due to the gapless band structure of the bulk states in WSM, the bulk and surface states are completely coupled in the ballistic regime, and we can not express the contribution of these states in the electrical or thermal currents by separate expressions.Nevertheless, in the following, we will argue that in the proposed junction and for a relatively thick WSM sample, W >> λ, the contribution of the surface states in the electrical and thermal conductances is negligible relative to the contribution of the bulk states.To show it, we remember that normal contacts do not support surface states, and only bulk states can propagate in the normal contacts.Thus, the wave function for an incident electron in the normal side along the x axis can be written as, where k y , k z < E/M , and wave vector along the y axis takes discreet values equal to k y = nπ/W .The wave function for the surface state on the y = 0 surface can be written as [2], where λ 1,2 are solutions for the decay constant given by Eq. 2. The amplitude of transition from a bulk state in the normal side to a bulk state in WSM can be written as, by setting k y = nπ/W the result reduces to W/2.However, the amplitude of transition from an incident bulk state in the normal side to a surface state in WSM is given by, Now if we impose the condition of thick sample W >> λ 1 , λ 2 the result reduces to, For small values of E << M, which we are interested, we have, We can see that for the typical values of the parameters considered in our calculation, M = 5eV.nm 2 , γ = 1.0eV.nm,k 0 = 0.5nm − 1, and for a small value of the chemical potential E ≃ µ ≃ 0.05eV , we find that λ 2 2 − λ 2 1 ≃ 0.2i.Thus, we expect that the probability of transition between the bulk states in normal and the surface states in the WSM be of the order of |A b→s | 2 ≃ 0.001.The ratio of the probability of transition from bulk to surface state to the probability of transition from bulk to bulk state is given approximately by |A b→s | 2 /|A b→b | 2 ≃ 0.004/W 2 , which is negligible for a reasonably thick layer of WSM, W ∼ 100 − 1000nm.To illustrate this argument, we have plotted in Fig. S3 the probability of transition from bulk to surface state as a function of the chemical potential of the leads, for the values of the parameters stated above and k y = 0.1 µ/M.This plot shows small values for the transition probability in the interested range of the chemical potential.Moreover, Fig. S4 represents this transition probability in terms of the k y in the full range for µ = 0.05eV .Again, we find small values for this ratio for all possible values of k y .Consequently, we observe that in the proposed junction the contribution of the surface states in the electrical and thermal currents is insignificant relative to the bulk states.On the other hand, depending on the junction direction, some surface states may lie at the interface of the normal and WSM.Once more, it has been shown that due to the localization of these states on the interface and possessing Fermi velocities parallel to the interface, they can not contribute to the scattering processes and, hence, in the longitudinal current (current normal to the interface) [3].Therefore, according to the above argument, we infer that by a good approximation, we can disregard the contribution of the surface states in the thermoelectric properties of the proposed  junction.Besides, due to the gapless spectrum of the bulk and Fermi arc surface states in WSM, and termination of the lower limit of the energy of the incident electrons by the normal contacts, the contribution of the Fermi arc states in the thermoelectric coefficients of the proposed junction will be additive with the contribution of the bulk states.As a result, our main results will not change qualitatively by including the effect of the Fermi arc states, and we expect their contribution cause enhancement of the observed high values of the Seebeck coefficient and thermoelectric figure of merit.

Figure S2 .
Figure S2.Normalized electrical conductances of the first and second states for the junction along x axis as a function of the chemical potential of the normal leads.The values of other parameters are identical with Fig. S1.

Figure S3 .
Figure S3.Probability of transition form a bulk state in the normal contact to a surface state in the WSM layer as a function of the chemical potential of the normal leads for k y = 0.1 µ/M.

Figure S4 .
Figure S4.Probability of transition form a bulk state in the normal contact to a surface state in the WSM layer as a function of k y for µ = 0.05.