Weyl semimetal is a three-dimensional Dirac material whose low energy dispersion is linear in momentum. Adding a quadratic (Schrödinger) term to the Weyl node breaks the original particle-hole symmetry and also breaks the mirror symmetry between the positive and negative Landau levels in present of magnetic field. This asymmetry splits the absorption line of the longitudinal magneto-optical conductivity into a two peaks structure. It also results in an oscillation pattern in the absorption part of the Hall conductivity. The two split peaks in Reσxx (or the positive and negative oscillation in Imσxy) just correspond to the absorptions of left-handed (σ) and right-handed (σ+) polarization light, respectively. The split in Reσxx and the displacement between the absorption of σ+ and σ are decided by the magnitude of the quadratic term and the magnetic field.

Weyl semimetal, a three-dimensional (3D) analog to graphene, has a pair of non-degenerate bands crossing at isolated points (Weyl nodes) within the Brillouin zone.1–4 Around the Weyl nodes, the dispersion is linear and the low energy physics is described by the 3D two-component Dirac Hamiltonian H = ± ħ vFσk, where vF is the Fermi velocity, σ is the vector of the Pauli matrices, k is the momentum, and ± defines the chirality of the Weyl node. Weyl semimetals can only exist in system with either time-reversal or inversion symmetry broken.5–8 When both of the symmetries are present, the Weyl nodes must be degenerate,5,9 and the system is thus described by four-component Dirac Hamiltonian. Such a material is referred to 3D Dirac semimetal. Na3Bi10,11 and Cd3As212–14 have been predicted and verified to be 3D Dirac semimetals. Recently a series of experiments have also realized Weyl semimetal state in TaAs.15–18 A Zeeman field B along the primary axes will break the time-reversal symmetry of the Dirac semimetal and split the Dirac node (degenerate Weyl nodes with different chirality) into two chiral Weyl nodes in the momentum space along the Zeeman field’s direction.9 

When a material is subjected to an external magnetic field, Landau levels directly related to the nature of the fermion will form in the electronic density of state. The absorption lines in the optical conductivity, which result from the transitions between these Landau levels, thus provide a useful method to investigate the nature of the fermion. Researches have pointed out that there is a series of asymmetric peaks lying on top of the linear background in the optical conductivity of Weyl semimetal in present of magnetic field.19 For two-dimensional (2D) Dirac materials like graphene and the surface state of 3D topological insulator, the results are similar but the background is flat.20–22 Adding a finite Schrödinger term to the surface state of 3D topological insulator breaks the particle-hole symmetry and splits the magneto-optical absorption line into two peaks structure.22,23 In fact, particle-hole asymmetry also exists in Weyl semimetals and Dirac semimetals.10,12 We often neglect the quadratic term in momentum because the effective mass is very large. If we consider larger energy region and larger magnetic field, it will be more practical to include the quadratic term.

Besides, it is easier to measure the optical conductivity in Weyl semimetal than only the surface of a 3D topological insulator as it needs to remove the effect of the bulk. We wonder whether a small quadratic term in Weyl semimetal will influence the magneto-optical absorption lines or not. In addition, owing to one more dimension, the dispersive nature of the Landau levels in Weyl semimetal may produce richer results than 2D case. So in this contribution, we first calculate the energies of the Landau levels and the density of state for single Weyl node with quadratic term in momentum in present of magnetic field. And then we study in detail the absorption part of the magneto-optical conductivity of the system. Our results can be applied to 3D fermion system with both linear (Dirac) and quadratic (Schrödinger) terms.

We begin with the Hamiltonian for single isolated Weyl node with isotropic dispersion and including the quadratic term in momentum with effective mass m

H 0 = ħ 2 k 2 2 m + ħ v F σ k ,
(1)

Because of the lack of data for Weyl semimetal, we refer to the four-component Hamiltonian for Dirac semimetal Na3Bi10 and Cd3As212 

H Γ k = ϵ 0 k + M k A k + 0 B k A k M k B k 0 0 B k M k A k B k 0 A k + M k ,

where ϵ 0 k = C 0 + C 1 k z 2 + C 2 k x 2 + k y 2 , k± = kx ± iky, and M k = M 0 + M 1 k z 2 + M 2 k x 2 + k y 2 . For the sake of simplicity, we neglect the large out-of-plane anisotropy, and adopt the parameters for Na3Bi,10 C2 = 8.4 eV ⋅ Å2 and A = 2.46 eV ⋅ Å to obtain the effective mass and Fermi velocity, respectively. So we have m = 0.45me (me is the electron mass) and vF = 3.74 × 105 m/s in our following numerical calculation. In present of a uniform magnetic field B along z direction, the Hamiltonian can be obtained through kΠ = k + eA/ħ, with A the vector potential and in Landau gauge A = B y , 0 , 0 . Introducing the raising and lowering operators

a = l B 2 Π x + i Π y , a = l B 2 Π x i Π y ,
(2)

with the magnetic length l B = ħ / e B , which satisfy the commutation rule [a, a] = 1, we can have the Hamiltonian

H = ħ 2 m l B 2 a a + 1 2 + ħ 2 k z 2 2 m + 2 ħ v F / l B k z l B / 2 a a k z l B / 2 .
(3)

Then we can solve the eigenvalue problem = . For n = 1, 2, 3... and s = + / −, we have the eigen-energies

E n s = n ε 0 + ħ 2 k z 2 2 m + s n ε 1 2 + ħ v F k z ε 0 / 2 2 ,
(4)

with ε 0 = ħ 2 m l B 2 = ħ e B m related to the equivalent space of the Landau levels in pure Schrödinger fermion system, and ε 1 = 2 ħ 2 v F 2 / l B 2 = v F 2 ħ e B related to the Landau levels in pure Dirac fermion system. Just like Ref. 22, we can introduce P = ε 1 2 / ε 0 2 to measure the “Diracness” of the system. When P → ∞ (or 0), the system is pure Dirac (or Schrödinger) system. The corresponding eigenstates have the form

ψ n s = n 0 C n s + s n 1 C n s 0 ,
(5)

with n the usual Fock state, and

C , n s = 1 2 1 + ħ v F k z ε 0 / 2 s n ε 1 2 + ħ v F k z ε 0 / 2 2 .
(6)

For special case n = 0, we have

E 0 = ħ 2 k z 2 2 m + ε 0 2 ħ v F k z ,
(7)

and

ψ 0 = 0 0 1 .
(8)

Figure 1(a) shows the Landau levels’ energies at kz = 0 as a function of the magnetic field. Owing to the large mass m = 0.45me we consider here, our system is more “Dirac” than “Schrödinger” like. But the mirror symmetry between positive and negative branches is visibly broken for large magnetic field and high index Landau levels. Figure 1(b) show the dispersive Landau levels along kz direction. Unlike 2D case, the dispersive structure can lead to more possible optical transitions (indicated by vertical arrows) and richer results. Using these eigen-energies, we calculate the density of state in magnetic field

D ħ ω = 1 2 π l B 2 d k z 2 π δ ħ ω E 0 + n s δ ħ ω E n s .
(9)

The results are displayed in Fig. 2. We also calculate the density of state for system without magnetic field, D 0 ħ ω = m π 2 ħ 3 v F 2 + ħ ω / m v F 2 + 2 ħ ω / m v F and D 0 ħ ω , m = = ħ ω 2 2 π 2 ħ v F 3 for comparison. We can see that the density of state for the Landau level system is a series of peaks on top of the zero magnetic field background. Compared with Fig. 2(b), (a) shows that the quadratic term strikingly breaks the particle-hole symmetry, and it has a stronger effect in the negative energy region than the positive one. In 2D case, the n = 0 Landau level lies exactly at zero energy for pure Dirac system or near zero-energy even consider the quadratic term.22 Thus, there is a peak near zero energy for the 2D density of state. In 3D case, however, the density of state around ħ ω = 0 is rather small though not exactly zero because the n = 0 Landau level is dispersive in kz instead of a constant.

FIG. 1.

(a) Landau level energies Ens as a function of the magnetic field at kz = 0 for various values of Landau index n (as labeled). (b) Landau level dispersion along the kz direction for B = 5 z ˆ Tesla. The vertical arrows indicate the allowed inter-Landau level transition for different chemical potential (gray dotted line), (b1) μ is between the n = 0 and 1 Landau level, (b2) μ is between the n = 1 and 2 Landau level. The blue and red circles with different direction arrowheads represent right-handed (σ+) and left-handed (σ) circularly polarized light, respectively. Here, the quadratic term has effective mass m = 0.45me. Positive branch and n = 0 Landau level are plotted as solid lines, while the negative branch levels are as dashed lines.

FIG. 1.

(a) Landau level energies Ens as a function of the magnetic field at kz = 0 for various values of Landau index n (as labeled). (b) Landau level dispersion along the kz direction for B = 5 z ˆ Tesla. The vertical arrows indicate the allowed inter-Landau level transition for different chemical potential (gray dotted line), (b1) μ is between the n = 0 and 1 Landau level, (b2) μ is between the n = 1 and 2 Landau level. The blue and red circles with different direction arrowheads represent right-handed (σ+) and left-handed (σ) circularly polarized light, respectively. Here, the quadratic term has effective mass m = 0.45me. Positive branch and n = 0 Landau level are plotted as solid lines, while the negative branch levels are as dashed lines.

Close modal
FIG. 2.

The density of state (DOS) as a function of the energy for single Weyl node (a) with mass m = 0.45me and (b) without mass term. The red solid lines are DOS for magnetic field B = 5 T, while the black dashed lines are DOS without magnetic field just for comparison. The insert in (a) is the same DOS for larger energy region.

FIG. 2.

The density of state (DOS) as a function of the energy for single Weyl node (a) with mass m = 0.45me and (b) without mass term. The red solid lines are DOS for magnetic field B = 5 T, while the black dashed lines are DOS without magnetic field just for comparison. The insert in (a) is the same DOS for larger energy region.

Close modal

The magneto-optical conductivity of the Landau level system in the clean limit can be obtained through the Kubo formula

σ α β ħ ω = i ħ 2 π l B 2 n , n , s , s d k z 2 π f E n s f E n s E n s E n s × ψ n s j α ψ n s ψ n s j β ψ n s ħ ω + E n s E n s + i 0 + ,
(10)

where f E n s = 1 exp E n s μ / k B T + 1 is the usual Fermi distribution function with chemical potential μ and temperature T. The current operator jα is related to the velocity operator vα and is given as follows.

j x = e v x = e H ħ Π x = e ħ 2 m l B a + a + e v F σ x , j y = e v y = e H ħ Π y = i e ħ 2 m l B a a + e v F σ y .
(11)

And the matrix elements are

ψ n s j α ψ n s = λ 1 e ħ 2 m l B s s C n s C n s n + C n s C n s n + 1 + s e v F C n s C n s δ n , n + 1 + λ 2 e ħ 2 m l B s s C n s C n s n 1 + C n s C n s n + s e v F C n s C n s δ n , n 1 ,
(12)
ψ 0 j α ψ n s = λ 2 e ħ 2 m l B C n s n + s e v F C n s δ 0 , n 1 ,
(13)

with λ1 = λ2 = 1 if α = x, and λ1 = − λ2 = − i if α = y. So the optical selection rule nn′ = ± 1 still exits for the Landau level transition. As we are only interested in the absorptive part of the optical conductivity, i.e., the real part of the longitudinal conductivity Reσxx and the imaginary part of the transverse one Imσxy, by using the matrix elements above and 1 x + i 0 + = P 1 x i π δ x , we can derive

Re σ x x ħ ω Im σ x y ħ ω = ħ 4 π l B 2 s d k z f E 0 f E 1 s E 0 E 1 s ψ 0 j x ψ 1 s 2 δ ω + E 0 E 1 s ± δ ω E 0 + E 1 s ħ 4 π l B 2 s , s n = 1 d k z f E n s f E n + 1 , s E n s E n + 1 , s ψ n s j x ψ n + 1 , s 2 δ ω + E n s E n + 1 , s ± δ ω E n s + E n + 1 , s ,
(14)

where

ψ n s j x ψ n + 1 , s 2 = e ħ 2 m l B s s C n s C n + 1 , s n + C n s C n + 1 , s n + 1 + s e v F C n s C n + 1 , s 2 ,
(15)
ψ 0 j x ψ 1 s 2 = e ħ 2 m l B C 1 s + s e v F C 1 s 2 .
(16)

We consider low temperature T = 4 K and numerical calculate Reσxx and Imσxy in present of magnetic field B = 5 T with chemical potential falling between n = 0 and positive n = 1 Landau level at kz = 0 (just as the gray dotted line shown in Fig. 1(b1)). The results are shown in Fig. 3(a). And the gray dotted line is the well-known conductivity Re σ x x 0 = e 2 ω 24 π ħ v F θ ħ ω / 2 μ for 3D Dirac fermion system24 without magnetic field as background. Comparing the results of (a1) with (a2), we find that except the first two increasing platforms related to the n = 0 Landau level transitions, all other peaks in Reσxx are split into two peaks if considering the quadratic mass term. Each peak corresponds to certain inter-Landau level transition, shown schematically by the vertical arrows in Fig. 1(b1) and typed next to the peak of Reσxx. And Imσxy is no longer zero, but shows a striking oscillation with a positive peak first and then a negative one at the same position of the two split peaks in Reσxx. These results are similar to that for the surface state of 3D topological insulator,22,23 and can be directly explained from the symmetry broken between the positive and negative branches of Landau level by the quadratic term. Because of this asymmetry, the transition from negative n to positive n + 1 Landau level require more energy than the transition from negative n + 1 to positive n Landau level. This energy difference will split the original degenerate peaks for pure Dirac system into two.

FIG. 3.

(a) The real part of the longitudinal conductivity Reσxx (black solid line) and the imaginary part of the transverse one Imσxy (green dashed line), as well as (b) the real part of circular polarized light optical conductivity Reσ+ (blue solid line) and Reσ (red dashed line), in unit of e 2 2 π ħ l B as a function of the photo energy ħ ω with magnetic field B = 5 T and chemical potential μ is between the n = 0 and 1 Landau level. The gray dotted lines in (a) are for Reσxx without magnetic field and are used for comparison. (1) is for fermion with mass m = 0.45me while (2) is without mass term.

FIG. 3.

(a) The real part of the longitudinal conductivity Reσxx (black solid line) and the imaginary part of the transverse one Imσxy (green dashed line), as well as (b) the real part of circular polarized light optical conductivity Reσ+ (blue solid line) and Reσ (red dashed line), in unit of e 2 2 π ħ l B as a function of the photo energy ħ ω with magnetic field B = 5 T and chemical potential μ is between the n = 0 and 1 Landau level. The gray dotted lines in (a) are for Reσxx without magnetic field and are used for comparison. (1) is for fermion with mass m = 0.45me while (2) is without mass term.

Close modal

For further understanding, we also calculate the optical conductivity for circularly polarized light. It is defined as σ± = σxx ± xy for right-handed (σ+) and left-handed (σ) polarization. So the absorptive part is Re σ ± ħ ω = Re σ x x ħ ω Im σ x y ħ ω . The results are shown in Fig. 3(b). There is no peak splitting behavior. However, the peaks of Reσ+ and Reσ with quadratic mass term are displaced in energy. Compared with the results without mass term, the peaks of Reσ+ are shifted to higher energy while the peaks of Reσ to lower energy. The combination of Reσ+ and Reσ just results in the two peaks structure in Reσxx shown in (a1). So we know that the two kinds of transition correspond to different handedness of polarizations. Right-handed polarization (σ+) light only cause transitions from negative n to positive n + 1 Landau level, while left-handed polarization (σ) only cause transitions from negative n + 1 to positive n Landau level, shown by blue and red vertical arrows in Fig. 1(b), respectively. That also explains why there is no split in the first two increasing platforms in (a1) because each platform only relates to one kind of transition.

We also notice that the position of the first two increasing platforms in Fig. 3(a) is totally decided by the chemical potential’s position owing to the Pauli exclusion principle. And as the Landau levels are dispersive in kz, the absorption line has increasing platforms instead of an abrupt peak in 2D case. To further study the chemical potential’s influence, we have similar figures like Fig. 3 but with chemical potential falling between positive n = 1 and 2 Landau level at kz = 0 (gray dotted line shown in Fig. 1(b2)) in Fig. 4. We notice that the patterns at high energy remain the same as Fig. 3, but are quite different at low energy. Owing to the Pauli block, transition from negative n = 2 to positive n = 1 Landau level does not exist everywhere but only for larger kz (refer to arrow transitions in Fig. 1(b2)). So the corresponding peak disappears and becomes a platform at higher energy in Reσxx and Reσ. As for Imσxy, the first positive oscillation peak disappears. Transition from negative n = 1 to n = 0 (or n = 0 to positive n = 1) Landau level requires more energy, so the platforms shift to higher energy. However, Transition from n = 0 to positive n = 1 can also occur with a little energy. There is also an extra intra-branch transition, form positive n = 1 to 2 Landau level, which also needs a little energy (refer to the two blue short arrows in Fig. 1(b2)). And both of the transitions need right-handed polarization light. Therefore, there is a very large absorption peak at lower energy region in Reσxx and Reσ+.

FIG. 4.

(a) Reσxx (black solid line) and Imσxy (green dashed line), as well as (b) Reσ+ (blue solid line) and Reσ (red dashed line), in unit of e 2 2 π ħ l B as a function of the photo energy ħ ω with magnetic field B = 5 T and chemical potential μ is between the n = 1 and 2 Landau level. The gray dotted lines in (a) are for Reσxx without magnetic field and (2) is the results without mass term used for comparison.

FIG. 4.

(a) Reσxx (black solid line) and Imσxy (green dashed line), as well as (b) Reσ+ (blue solid line) and Reσ (red dashed line), in unit of e 2 2 π ħ l B as a function of the photo energy ħ ω with magnetic field B = 5 T and chemical potential μ is between the n = 1 and 2 Landau level. The gray dotted lines in (a) are for Reσxx without magnetic field and (2) is the results without mass term used for comparison.

Close modal

For chemical potential much higher than the magnetic energy, the Landau level quantization is no longer important, and this is the semiclassical limit. If we are interested in energy range much less than the chemical potential, only intra-branch transitions are involved. For En,+,kz=0 < μ < En+1,+,kz=0, with n ≫ 1, we then have the cyclotron resonance energy around

ħ ω c = E n + 1 , + , k z = 0 E n , + , k z = 0 ε 0 + ε 1 n + 1 n ε 0 + ε 1 2 n .
(17)

And the definitions of ε0 and ε1 are given after Eq. (4). For pure Dirac system without quadratic mass term m , we have ħ ω c ε 1 2 n . So in semiclassical limit, the cyclotron resonance energy will displace to higher energy if consider the quadratic term. And the displacement is related to the Landau level’s space for pure Schrödinger fermion system which is determined by the magnitude of the effective mass and the magnetic field. Figure 5(a) gives the semiclassical cyclotron resonance energy as a function of the chemical potential for both with and without mass term. And (b) is the real part of the longitudinal conductivity Reσxx in semiclassical limit. It shows a single resonance absorption peak at low energy which contains most of the spectral weight. Although μ lying between En,+,kz=0 and En+1,+,kz=0, this resonance peak results from a series of transitions from El,+ to El+1,+ with l 0 , n instead of one owing to the dispersive relation along kz of the Landau levels. As this resonance peak appears at very low energy, its line shape is easy to be influenced by temperature.

FIG. 5.

(a) The semiclassical cyclotron resonance energy as a function of the chemical potential μ = E n , + , k z = 0 for system with mass m = 0.45me (red open stars) or without mass term (black open circles). (b) The real part of the longitudinal conductivity Reσxx in unit of e 2 2 π ħ l B as a function of the photo energy ħ ω in semiclassical limit with large chemical potential μ = 0.16 eV. The red solid line is with m = 0.45me and μ lying between n = 21 and 22 Landau levels. The black dashed line is without mass term and μ lying between n = 32 and 33 Landau levels. Both (a) and (b) are for magnetic field B = 5 T.

FIG. 5.

(a) The semiclassical cyclotron resonance energy as a function of the chemical potential μ = E n , + , k z = 0 for system with mass m = 0.45me (red open stars) or without mass term (black open circles). (b) The real part of the longitudinal conductivity Reσxx in unit of e 2 2 π ħ l B as a function of the photo energy ħ ω in semiclassical limit with large chemical potential μ = 0.16 eV. The red solid line is with m = 0.45me and μ lying between n = 21 and 22 Landau levels. The black dashed line is without mass term and μ lying between n = 32 and 33 Landau levels. Both (a) and (b) are for magnetic field B = 5 T.

Close modal

All results considered above are for m = 0.45me and B = 5 T at low temperature T = 4 K. As we know, for smaller effective mass and larger magnetic field, the “Diracness” parameter P becomes smaller, so the system is more Schrödinger like. Under such circumstance, the asymmetry of the positive and negative branches of Landau level is more obvious (refer to Fig. 1(a)). So the peaks split in Reσxx and the displacement between Reσ+ and Reσ will become more dramatic. This can also be used to estimate the magnitude of contribution between the linear (Dirac) and quadratic (Schrödinger) terms.

In summary, we have added a quadratic term in momentum to the linear dispersion of single Weyl node. And we find that this quadratic term breaks the mirror symmetry between the positive and negative branches of Landau level in present of magnetic field. Because of this asymmetry, the transition from negative n to positive n + 1 Landau level is no longer have the same energy as the transition from negative n + 1 to positive n. So it leads to a peak splitting in the absorption line of longitudinal conductivity and an oscillation pattern in the absorption part of the Hall conductivity. And the two split peaks in Reσxx correspond to the absorption of left-handed and right-handed polarization light, respectively. So the absorptions for circular polarized light still have single peak structure, but there is a displacement in energy for right-handed and left-handed polarization compare with the results without mass term. Besides, the dispersive nature of the Landau levels often leads to increasing platform, whose position is decided by the chemical potential, instead of abrupt peak at the beginning of the absorption line. For very large chemical potential, intra-branch transitions between the Landau levels become important. And the absorption line becomes a single resonance peak at low energy, whose position is shifted to higher energy by the quadratic term. Our work provides an optical method to study the linear (Dirac) and small quadratic (Schrödinger) contribution to the 3D fermion system, which can be realized experimentally in Weyl semimetal or 3D Dirac semimetal material.

This work is supported by the State Key Laboratory of Optoelectronic Materials and Technologies of Sun Yat-sen University.

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