Energy spectrum and dynamical properties of Dirac fermions in 3D SnTe(001) surface state under combined exchange and strain effects

3.1. Energy levels and energy gap in crystalline 3D SnTe(001) surface state with exchange effect:

The Hamiltonian of SnTe deposited on a ferromagnetic material is diagonalized in Eq. (5). It is important to note that the resulting energy dispersion relation is anisotropic due to the different in Fermi velocities along the x- and y-directions, with v₁ = 1.3 eV Å and v₂ = 2.4 eV Å. Consequently, the energy levels must be analyzed along both directions accordingly.

According to the above equation, if n = 0, δ = 0, and M = 0 equals zero. In the isotropic case, immediately. The dispersion equation in Eq. 5 was reduced to the graphine dispersion relation.

According to the above equation, in the isotropic case where n = 0, δ = 0, and M = 0, the dispersion equation in Eq. (5) simplifies directly to the graphene-like dispersion relation ($E_{\mu }^v=\mu {\text{vk}}_{\text{x}}$), characterized by the emergence of two gapless Dirac cones at k_x=0, k_y=0. However, in the anisotropic case (v₁≠v₂), and due to the presence of$n$ and$\delta$, the two copies of Dirac cones shifted accordingly to$kx=\pm \frac{\sqrt{n^2+{\delta }^2}}{v_1}$. Moreover, at k=0, a bandgap of$\Delta \mathcal{E} = \pm \sqrt{n^2+{\delta }^2 }$appears. Fig. 1(a) demonstrates the energy levels versus${\text{k}}_{\text{x}}$, when$M=0$ -black curve- conduction state intersects the valence state at$\wedge = \left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ and${\wedge }^{\prime }= \left(-\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$. The energy at$\wedge$,${\wedge }^{\prime }$ equal zero, at these points, Dirac fermions are massless. It can be concluded that the perturbation terms ($n$ and$\delta$) induce a horizontal displacement of the Dirac points, causing them to shift and separate from each other.

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Fig. 1.

Electronic band structure of SnTe(001) near X1 point at different values of exchange effect: black line$M=0$, red line$M=0.02$, blue line$M=0.04$ (a) in x-direction where$k_y=0$ (b) in y-direction where$k_x=0$.

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When a ferromagnetic substrate is present, the exchange effect causes the Dirac cones to separate vertically, resulting in the opening of an energy gap. The size of this energy gap depends on the magnitude of the exchange effect, denoted as M, and on the position of the bandgap. Fig. 1(a) shows that by including M, the two valence bands and the two conduction bands separate from each other, and the degeneracy of these states is broken. From Fig. 1(a), we can see that there is a small shift in the position of$\wedge$ and${\wedge }^{\prime }$ points. The new locations of these points are$\left(\pm \frac{1}{v_1}\sqrt{n^2+{\delta }^2-\frac{n^2{M}^2}{n^2+{\delta }^2}},0\right),$ which is dependent on the value of the exchange effect (M).

From Fig. 1(b), which represents the band structure along the y-direction, it can be seen that even in the absence of Magnetization, there is an energy gap although if which equals$2\delta$when$M=0.$ This gap doesn’t change significantly by the magnetization. From Fig. 1(b), even in the absence of M=0, there are two saddle points in y-direction at points:$s_1=\left(0,n/v_2\right)$ and$s_2=\left(0,-n/v_2\right)$. When the exchange effect is included (M> 0), the positions of the two saddle points are shifted. The new positions of these saddle points are$\left(0,\pm \frac{\sqrt{n^2 - M^2}}{v_2}\right)$, which is exchange field dependent.

From the differences between Fig. 1(a) and (b), it’s obvious that the energy is anisotropic and the fermions are much massive in the y-direction. To quantify how the energy gap changes in both directions, the energy band gap is calculated from Eq. (5). By taking the minimum difference between the lower conduction band and the higher valence band, using the following equation:

In Fig. 2, the dependence of the difference between the energy band gap on the exchange effect is elucidated in both x- and y-directions. In the x-direction, the energy band gap vanishes (E_g=0) when the exchange effect is zero (M=0). Then, E_g increases as the exchange effect M increases. This result is expected because the exchange effect splits the conduction band from the valence band, thereby opening the energy band gap.

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Fig. 2.

Energy band gap with exchange effect in x-direction and in y-direction.

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However, in the y-direction, the energy band gap already exists without the exchange effect and remains constant, and its value equals$2\delta$ until$M = \sqrt{{\delta }^2+n^2}$ Beyond this point, as the exchange effect increases, the energy band gap also increases. At this stage, the energy band gap becomes equal in both x and y directions. This occurs because beyond this critical point, the saddle points$S_1$ and$S_2$ are removed, along with$\wedge$ and${\wedge }^{\prime }$. Consequently, the energy gap occurs at$\Gamma$ point. Fig. 3 demonstrates the lower conduction level and the higher valence level in both x- and y-directions when$M = \sqrt{{\delta }^2+n^2}$. In both directions, the band gap is located at the$\Gamma$ point. Our results are consistent with the previous experimental and theoretical studies that the exchange effect produced by the contact between TI/ferrosubstrate leads to a splitting in the Dirac cone (Eremeev et al., 2013; Hirahara et al., 2017; Otrokov et al., 2017a,b). For example, Otrokov et. al. show experimentally that there is a Dirac cone gap opening of 100 meV in MnBi₂Se₄/TCI heterostructure (Otrokov et al., 2017b).

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Fig. 3.

Lower conduction level and higher valance level when$\text{M }=\sqrt{{\delta }^2+{\text{n}}^2}$ in both x-direction and y-direction.

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It’s obvious from Fig. 2, when$M = \sqrt{{\delta }^2+n^2}$ the energy band gap is equal in both directions, and the saddle points$S_1$,$S_2$ and the$\wedge$ ,${\wedge }^{\prime }$points are removed. The energy gap occurs at k=0.

3.2. Group velocity of fermions in SnTe(001) surface state with exchange effect

Group velocity can be calculated analytically using equation$v_g=\frac{1}{\hslash }\overrightarrow{{\nabla }_{\text{k}}}\text{E}\left(\text{k}\right)$.

Fig. 4(a) represents the x-component of the group velocity of conduction levels with$k_x$, and${\text{k}}_{\text{y}}=0$, at different values of the exchange effect (M). In the absence of the exchange effect (M=0), the group velocity of the first conduction is either +/-v1, which gives strong evidence that the Dirac fermions behave linearly with$k_x$. The group velocity has a negative value of velocity -1.3 ev\A (denoted$-v_1$), in the following values of${\text{k}}_{\text{x}}$:${\text{k}}_{\text{x}}<-\sqrt{{\text{n}}^2+{\delta }^2}$ and$0<{\text{k}}_{\text{x}}<\sqrt{{\text{n}}^2+{\delta }^2}$. While the group velocity of the electrons is along the positive x-direction if k_x satisfy the following conditions:$-\sqrt{{\text{n}}^2+{\delta }^2}<{\text{k}}_{\text{x}}<0$ or if${\text{k}}_{\text{x}}>\sqrt{{\text{n}}^2+{\delta }^2}$ . From Fig. 4(a), it can be seen that the group velocity in the second conduction band has a value of -1.3 ev\A for$k_x<0$ and 1.3 ev/A for$k_x>0$.

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Fig. 4.

The x-component and y-component of the group velocity (v_x and v_y) of Dirac fermions in SnTe(001) as a function of momentum at different values of exchange effect M.

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When the exchange effect switches, the group velocities along the x-axis for the first conduction band are not zero at Dirac points because of the band gap opening in the band structure. Also, it can be seen that the maximum velocity it reached is smaller than v1. The group velocity switches its sign from positive to negative values in the${\text{k}}_{\text{x}}$ falls in the range$\left(+\frac{1}{v_1}\sqrt{n^2+{\delta }^2-\frac{n^2{M}^2}{n^2+{\delta }^2}},0\right).$However, for the upper state, the maximum and minimum values of the group velocity remain$v_1$ and$-v_1$, respectively, but the value of the group velocity along the curve decreases and converges to the ± v1 at high momenta.

Fig. 4(b) represents the y-component of the group velocity of conduction levels with${\text{k}}_{\text{y}}$, and$k_x=0$, at different values of the exchange effect, for M=0, the group velocity of the first conduction level flips from a negative value when${\text{k}}_{\text{y}}<-\text{ n}/{\text{v}}_2$ to positive value when$-\text{n}/{\text{v}}_2<{\text{k}}_{\text{y}}<0$. Subsequently, the sign of group velocity reverses to minus when$0<{\text{k}}_{\text{y}}<\text{n}/{\text{v}}_2$, and it becomes positive when${\text{k}}_{\text{y}}>\text{n}/{\text{v}}_2$. Note that the maximum and minimum values of group velocity are$v_2$ and$-v_2$. In the second conduction band the sign of the group velocity flips from negative when${\text{k}}_{\text{y}}<0$, to positive when${\text{k}}_{\text{y}}>0$.

When the exchange effect switches, the group velocity for both lower and upper bands has similar behavior as the x-direction due to the band gap opening and the curvature of the bands.

The presence of saddle points offers opportunities for manipulating the electronic properties of topological insulators through external fields such as electric fields, magnetic fields, or strain. By tuning the position or energy of saddle points, one can control the group velocity and the functionality of the material to be utilized for different applications. Because of the importance of the$\wedge$,${\wedge }^{\prime }$ Dirac points and saddle points$S_1$and$S_2$ the group velocity will be studied in detail at these points in the first and second conduction states.

Fig. 5 demonstrates the behavior of the group velocity in the x-direction of Dirac fermions with momentum$\left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$, and the group velocity in the y-direction of Dirac fermions with momentum$\left(0,n/v_2\right)$, against the exchange effect M, in the first and the second conduction states. The group velocity in the x-direction of Dirac fermions at$\left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ , the point in the first conduction state has its minimal value at M=0. As M increases (either parallel or antiparallel to the host spin directions), the group velocity increases as shown in the figure, up to critical exchange fields$\pm \sqrt{n^2+{\delta }^2}$ then starts to decrease slightly. On the other hand, in the second conduction state, a group of velocities at$\text{Λ}$ approaches its maximum value$v_1$ as M gets closer to 0, then as the magnetization deviates from zero. The group velocity decreases as shown in Fig. 5. Similar behavior can be noticed for the group velocity along the y-direction. The figure shows that the group velocities for upper and lower conduction/valence bands are found to behave in opposite to each other as the magnetization vector of the ferromagnet is applied along the material.

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Fig. 5.

Group velocity of Dirac fermions at surface state in SnTe (001) as a function of exchange effect, both in the x-direction at$\Lambda = \left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ point, and y-direction at$S_1=\left(0,n\text{/}{v}_2\right)$point.

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3.3. Effective mass of fermions in SnTe(001) surface state with exchange effect

The effective mass is defined as the mass that appears in the equation of motion for electrons in a solid. It characterizes the curvature of the energy bands, which determines the electrons’ behavior, such as their mobility and conductivity. Since the energy dispersion relation is also available analytically, the effective mass can be calculated analytically in both the x and y directions using equation${\text{m}}^{\text{*}}=\frac{{\hslash }^2}{{\nabla }_{\text{k}}^2\text{E}(\text{k})}$ as the following:

From Eqs. (7 and 8), it’s obvious that the effective mass m_x and m_y of electrons in the SnTe(001) surface state with exchange effect (M) is anisotropic, note that n and$\text{δ}$ parameters that arise from symmetries of the surface state play a critical role in the properties of the material. Due to the presence of electron-hole symmetry, we focus on the conduction band only.

Fig. 6(a) and (b) show the dependence of the effective mass m_x and m_y of the electron on the exchange effect M.

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Fig. 6.

Reciprocal of effective mass for Dirac fermions in SnTe(001) surface state in both (a) x-direction and (b) y-direction with momentum.

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When M=0, the effective mass in the x-direction becomes zero at Dirac cones, as shown in Fig. 6(a). This implies that the Dirac fermions with momentum$\left(\mp \frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ are massless. However, they become massive when there is an exchange effect due to the presence of a ferromagnetic substrate, for example.

Including an exchange term affects the electrons in both the first and second conduction states. In the first conduction state, as mentioned, the Dirac fermions become massive, and the minimum value of the effective mass shifts. This shift occurs due to the displacement of the$\text{Λ}$ and$\text{Λ}{\prime }^{}$ points, and the minimum value occurs at$\left(\pm \frac{1}{v_1}\sqrt{n^2+{\delta }^2-\frac{n^2{M}^2}{n^2+{\delta }^2}},0\right)$ momentum. This means that the exchange effect can be used to control the effective mass of fermions. The exchange effect plays an important role in switching the effective mass of the electron from zero to a given value, as presented in Fig. 6.

In addition, within a certain range of exchange effects, there exists a range of momentum where the effective mass in the first conduction state becomes negative. This implies that the electrons behave like holes, as they respond to the field opposite to how a free electron would. This electron-hole symmetry plays a crucial role that leading to topological phase change in the material.

In the second conduction band without the exchange effect, the fermions are highly massive. When the exchange effect is included (M$\ne 0)$, the effective mass decreases, and the minimum value of effective mass occurs at k=0.

The Dirac fermions in the y-direction have opposite behavior; they are massive even when the exchange effect is zero, and the minimum value of effective mass occurs at saddle points$S_1$ and$S_2$. When the exchange effect is introduced, the fermions become even more massive, and the minimum value of effective mass shifts accordingly to$\left(0,\pm \frac{\sqrt{n^2 - M^2}}{v_2}\right)$. Similar to the x-direction, there is a range of M values corresponding to the effective mass of Dirac electrons, which exhibits negative mass and behaves like holes at certain momenta in the first conduction state. Our results are inconsistent with experimental findings of Anqi Zhang et al. that support the magnetoelectrical transport properties of SnTe thin films deposit on magnetic substrates can be tuned by an external magnetic field by examining the Linear magnetic resistance in SnTe/EuIG heterostructures with applied magnetic field (Zhang et al., 2022). In the x-direction, there is a range of momenta where the sign of the effective mass in the first conduction state becomes negative. This occurs when$-\frac{{\text{n}}^2+{\text{δ}}^2}{\text{n}}\le \frac{1}{m_x}<0$ and$0<\frac{1}{m_x} <\frac{n^2+{\delta }^2}{n}$. Within this interval of$m_x$, there exists a specific range of momentum for which the effective mass in the x-direction is negative ($m_x$ <0). To determine this interval, the equation Eq. (107.a)$1/m_x=0$ must be solved. Due to the complexity$1/{\text{m}}_{\text{x}}$, numerical methods can be employed to solve this equation for each value of the exchange effect, and then fit a curve to obtain a proper formula.

Fig. 7 illustrates the momentum values corresponding to the first conduction state, where the sign of the effective mass in the x-direction switches at each value of exchange effect.

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Fig. 7.

Critical value of k_x as function M. a) the value of k_x at which m_x changes from positive to negative b) the value of k_x at which m_x changes from negative to positive.

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In Fig. 7(a), the momentum values are plotted against the exchange effect, for instance, where the sign of the x-direction effective mass changes from positive to negative. In Fig. 7(b), the momentum values are plotted for cases where the sign of the x-direction effective mass shifts from negative to positive.

The dependence of the k on the exchange effect M is given by the relation$\text{k}(\text{M})={\text{aM}}^2+{\text{bM}}^4+{\text{cM}}^6+{\text{dM}}^8+\text{e }$ which serves as the fitted function for the curves. The fitting parameters (a,b,c,d, and e) have been listed in Table 1.

At each value of the exchange effect M, electrons with momentum k₁(M)$<$ k$<$ k₂(M) exhibits a negative effective mass in the x-direction and behaves similarly to holes.

In the y-direction, there is a range of momenta where the sign of the effective mass in the first conduction state becomes negative. This occurs when$-\text{n }\le \frac{1}{{\text{m}}_{\text{y}}}<0$ and$0<\frac{1}{{\text{m}}_{\text{y}}}<\text{n}$. Within this interval of$m_y$, there exists a specific range of momentum for which the effective mass in the y-direction is negative. To determine this interval, the equation$1/m_y=0$ must be solved. Applying the same fitting procedure, we understand the behavior dependence of${\text{m}}_{\text{y}}$ on k.

Fig. 8 illustrates the momentum values corresponding to the first conduction state, where the sign of the effective mass in the y-direction switches at each value of exchange effect. In Fig. 8(a), the momentum values are plotted against the exchange effect for instances where the sign of the y-direction effective mass changes from positive to negative.

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Fig. 8.

Momentum when the sign of the effective mass in y-direction flips with exchange effect.

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In Fig. 8(b), the momentum values are plotted for cases where the sign of the y-direction effective mass shifts from negative to positive. This behavior can be well explained through the fitting functional relation$\text{k}\left(\text{M}\right)=\text{a*ln}\left(\text{b}\frac{\left|\text{c*M}\right|}{{\text{M}}^2}\right)+{\text{d*M}}^2+{\text{e*M}}^4+{\text{f*M}}^6+\text{g}$. The fitting parameter values have been given in Table 2.

At each value of the exchange effect M, electrons with momentum k₁(M)< k < k₂(M) exhibit a negative effective mass in the y-direction and behave similarly to holes.

Because of the importance of the$\wedge$,${\wedge }^{\prime }$ points and saddle points$S_1$and$S_2$ the effective mass studied at these points in the first and second conduction states with exchange effect have been shown in Fig. 9. The effective mass in x-direction of Dirac fermions at$\left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ momentum in the first conduction state decreases when$M<0$, until M=0, at this point the effective mass in x-direction equal zero, representng the minimum value of effective mass of fermions with$\left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ momentum in the first conduction state, when$M>0$ the effective mass increases. In the second conduction state, Dirac fermions at the Λ point are highly massive, and the effective mass in the x-direction remains approximately constant with the exchange effect.

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Fig. 9.

Reciprocal of the effective mass of Dirac fermions at surface state in SnTe (001) as a function of exchange effect both the x-direction at$\Lambda = \left(\frac{\sqrt{n^2+{\delta }^2}}{v_1},0\right)$ point, and y-direction at$S_1=\left(0,n/v_2\right)$ point.

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The effective mass in the y-direction of Dirac fermions, which has momentum$\left(0,n/v_2\right)$ in the first conduction state, effective mass decreases when$M<0$, reaching its minimum value at M=0. However, even at M=0, the fermions remain massive, unlike in the x-direction where fermions become massless. As M becomes positive, the effective mass increases again. In the second conduction state, Dirac fermions at the saddle point are highly massive, and the effective mass in the y-direction remains approximately constant with the exchange effect.