Electronic transport in two-dimensional strained Dirac materials under multi-step Fermi velocity barrier: transfer matrix method for supersymmetric systems

Abstract

In recent years, graphene and other two-dimensional Dirac materials like silicene, germanene, etc. have been studied from different points of view: from mathematical physics, condensed matter physics to high energy physics. In this study, we utilize both supersymmetric quantum mechanics (SUSY-QM) and transfer matrix method (TTM) to examine electronic transport in two-dimensional Dirac materials under the influences of multi-step deformation as well as multi-step Fermi velocity barrier. The effects of multi-step effective mass and multi-step applied fields are also taken into account in our investigation. Results show the possibility of modulating the Klein tunneling of Dirac electron using strain or electric field.

Graphic abstract

Appendices

Appendix A. The pseudo-spinor rotations

A general rotation $U(\alpha ,\beta ,\gamma )$ of the pseudo-spinor has the form

$$\begin{aligned}&K(w) = U G(w) = \exp \nonumber \\&\quad \left[ \alpha /2 \left( \sigma _y \cos \beta + \sigma _z \sin \beta \cos \gamma + \sigma _x \sin \beta \sin \gamma \right) \right] G(w),\nonumber \\ \end{aligned}$$

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where $\alpha$, $\beta$ and $\gamma$ are the three degrees of freedom (the fourth one is canceled by imposing the constraint $\det U = 1$). Using this rotation, we can transform the equation for K(w) into an equation for G(w)

$$\begin{aligned}&\left\{ -i \sigma _x \partial _w + \sigma _y \left[ {\tilde{k}} + {\tilde{A}}(w) \right] + \sigma _z {\tilde{\Delta }}(w)/2 + {\tilde{\phi }}(w) - {\tilde{\varepsilon }} \right\} \nonumber \\&\quad G(w) = 0, \end{aligned}$$

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with

$$\begin{aligned}&{\tilde{A}}(w) = \bigg [\sinh ^2 \dfrac{\alpha }{2} \sin ^2\beta \left( i q_\phi \sin 2\gamma - q_\mathrm{v} (k+A_0) \cos 2\gamma \right) \nonumber \\&\qquad \qquad + \sinh ^2 \dfrac{\alpha }{2} \left( \dfrac{1}{2} q_v (k+A_0) \cos 2\beta + q_\mathrm{m} \sin 2\beta \cos \gamma \right) \nonumber \\&\qquad \qquad + \sinh \alpha (q_\phi \cos \beta + i q_\mathrm{m} \sin \beta \sin \gamma ) \nonumber \\&\qquad \qquad + \dfrac{1}{4} q_\mathrm{v} (k+A_0) (1 + 3 \cosh \alpha ) \bigg ] p(w), \nonumber \\&{\tilde{\phi }}(w) = \bigg [ \sinh ^2 \dfrac{\alpha }{2} \sin ^2\beta ( q_\phi \cos 2\gamma - i q_\mathrm{v} (k+A_0) \sin 2\gamma ) \nonumber \\&\qquad \qquad + \sinh \alpha (q_\mathrm{v} (k+A_0) \cos \beta +q_\mathrm{m} \sin \beta \cos \gamma ) \nonumber \\&\qquad \qquad + \sinh ^2 \dfrac{\alpha }{2} \left( \dfrac{1}{2} q_\phi \cos 2\beta + i q_\mathrm{m} \sin 2\beta \sin \gamma \right) \nonumber \\&\qquad \qquad + \dfrac{1}{4} q_\phi (1 + 3 \cosh \alpha ) \bigg ] p(w), \nonumber \\&\dfrac{{\tilde{\Delta }}(w)}{2} \nonumber \\&\quad = \bigg [\sinh ^2 \dfrac{\alpha }{2} (q_\mathrm{v} (k+A_0) \sin 2\beta \cos \gamma \nonumber \\&\qquad \qquad -i q_\phi \sin 2\beta \sin \gamma -q_\mathrm{m} \cos 2\beta ) \nonumber \\&\qquad \qquad + \sinh \alpha \sin \beta \left( q_\phi \cos \gamma -i q_\mathrm{v} (k+A_0) \sin \gamma \right) \nonumber \\&\qquad \qquad + \cosh ^2 \dfrac{\alpha }{2} q_\mathrm{m}\bigg ] p(w) \nonumber \\&\qquad \qquad + \sinh \alpha \sin \beta \left( (\phi _0 - \varepsilon ) \cos \gamma - i q_\mathrm{v} q_\mathrm{A} \sin \gamma \right) \nonumber \\&\qquad \qquad + \sinh ^2 \dfrac{\alpha }{2} \sin 2\beta (q_\mathrm{v} q_\mathrm{A} \cos \gamma - i (\phi _0 - \varepsilon ) \sin \gamma ) , \nonumber \\&{\tilde{k}} = + \sinh ^2 \dfrac{\alpha }{2} \nonumber \\&\quad \left[ \dfrac{1}{2} q_\mathrm{v} q_\mathrm{A} \cos 2\beta - \sin ^2\beta (q_\mathrm{v} q_\mathrm{A} \cos 2\gamma -i (\phi _0 - \varepsilon ) \sin 2\gamma )\right] \nonumber \\&\qquad \qquad + (\phi _0 - \varepsilon ) \sinh \alpha \cos \beta + \dfrac{1}{4} q_\mathrm{v} q_\mathrm{A} (1 + 3\cosh \alpha ) , \nonumber \\&{\tilde{\varepsilon }} = - \sinh ^2 \dfrac{\alpha }{2} \nonumber \\&\quad \bigg [ \dfrac{1}{2} (\phi _0 - \varepsilon ) \cos 2\beta + \sin ^2\beta ((\phi _0 - \varepsilon ) \cos 2\gamma \nonumber \\&\qquad \qquad -i q_\mathrm{v} q_\mathrm{A} \sin 2\gamma )\bigg ] \nonumber \\&\qquad \qquad - q_\mathrm{v} q_\mathrm{A} \sinh \alpha \cos \beta - \dfrac{1}{4} (\phi _0 - \varepsilon ) (1 + 3\cosh \alpha ).\nonumber \\ \end{aligned}$$

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To obtain the B-case, we can use the rotation $U_B (\alpha ^B, \beta ^B,\gamma ^B)$ with

$$\begin{aligned} \cosh \dfrac{\alpha ^B}{2}= & {} \dfrac{s_1^B \left[ (k+A_0)q_\mathrm{v} + \sqrt{-Q} \right] - i s_2^B (q_\mathrm{m}+q_\phi )}{2}, \nonumber \\ \cos \beta ^B= & {} i\dfrac{s_2^B \left[ (k+A_0)q_\mathrm{v} - \sqrt{-Q} \right] + i s_1^B (q_\mathrm{m}+q_\phi )}{2 \sinh \dfrac{\alpha ^B}{2}}, \nonumber \\ \cos \gamma ^B= & {} \dfrac{s_1^B \left[ (k+A_0)q_\mathrm{v} + \sqrt{-Q} \right] + i s_2^B (q_\mathrm{m}+q_\phi )}{2 \sin \beta ^B \sinh \dfrac{\alpha ^B}{2}}.\nonumber \\ \end{aligned}$$

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For E-case, the rotation $U_E$ is now determined by

$$\begin{aligned}&\cosh \dfrac{\alpha ^E}{2} \nonumber \\&\quad = \dfrac{i (k+A_0)q_\mathrm{v} (s_1^E+s_2^E) - \sqrt{Q}(s_1^E-s_2^E) + (q_\mathrm{m}+q_\phi )(s_1^E-s_2^E)}{2}, \nonumber \\&\cos \beta ^E \nonumber \\&\quad = i \dfrac{i (k+A_0)q_\mathrm{v} (s_1^E-s_2^E) - \sqrt{Q}(s_1^E+s_2^E) - (q_\mathrm{m}+q_\phi )(s_1^E+s_2^E)}{2 \sinh \dfrac{\alpha ^E}{2}}, \nonumber \\&\cos \gamma ^E \nonumber \\&\quad = \dfrac{i (k+A_0)q_\mathrm{v} (s_1^E+s_2^E) - \sqrt{Q}(s_1^E-s_2^E) - (q_\mathrm{m}+q_\phi )(s_1^E-s_2^E)}{2 \sin \beta ^E \sinh \dfrac{\alpha ^E}{2}}.\nonumber \\ \end{aligned}$$

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Here, the factors s are given by⁵

$$\begin{aligned}&(s_{1,2}^{B})^2 = \dfrac{ \sqrt{\pm 1} }{ \sqrt{\mp 1} } \left[ 2 (q_\mathrm{m}+q_\phi ) \sqrt{Q} \right] ^{-1} \times \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\&\dfrac{ \sqrt{q_\mathrm{v} \left[ (k+A_0) q_\mathrm{v} \mp \sqrt{-Q}\right] \left[ (\varepsilon -\phi _0) (k+A_0) + q_\mathrm{A} (q_\mathrm{m}+q_\phi )\right] + (\varepsilon -\phi _0) q_\mathrm{m} (q_\mathrm{m} + q_\phi )} }{ \sqrt{q_\mathrm{v} \left[ (k+A_0) q_\mathrm{v} \pm \sqrt{-Q}\right] \left[ (\varepsilon -\phi _0) (k+A_0) + q_\mathrm{A} (q_\mathrm{m}+q_\phi )\right] + (\varepsilon -\phi _0) q_\mathrm{m} (q_\mathrm{m} + q_\phi )} },\quad \nonumber \\&(s_{1,2}^{E})^2 = \dfrac{ \sqrt{\pm 1} }{ \sqrt{\mp 1} } \left[ 4 (q_\mathrm{m}+q_\phi ) \sqrt{Q} \right] ^{-1} \times \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\&\dfrac{ \sqrt{q_\mathrm{v} \left[ (k+A_0) q_\mathrm{v} \mp i \sqrt{Q} \right] \left[ (\varepsilon -\phi _0) (k+A_0) + q_\mathrm{A} (q_\mathrm{m} + q_\phi )\right] + (\varepsilon -\phi _0) q_\mathrm{m} (q_\mathrm{m} + q_\phi )} }{ \sqrt{q_\mathrm{v} \left[ (k+A_0) q_\mathrm{v} \pm i \sqrt{Q} \right] \left[ (\varepsilon -\phi _0) (k+A_0) + q_\mathrm{A} (q_\mathrm{m} + q_\phi )\right] + (\varepsilon -\phi _0) q_\mathrm{m} (q_\mathrm{m} + q_\phi )} }.\quad \end{aligned}$$

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Appendix B. The formulae of the probability density and the probability current density

For both the B- and E-cases, the expressions of the probability density $\rho$ and the probability current density $\mathbf {J}$ are given by [54]

$$\begin{aligned}&\rho ^{B,E} (x) = \Psi ^\dagger (x,y) \Psi (x,y) \nonumber \\&\quad = \dfrac{1}{v(x)} G_{B,E}^\dagger (w(x)) U_{B,E}^\dagger U_{B,E} G_{B,E}(w(x)),\nonumber \\&J_x^{B,E} = v(x) \Psi ^\dagger (x,y) \sigma _x \Psi (x,y) \nonumber \\&\quad = G_{B,E}^\dagger (w(x)) U_{B,E}^\dagger \sigma _x U_{B,E} G_{B,E}(w(x)),\nonumber \\&J_y^{B,E} = v(x) \Psi ^\dagger (x,y) \sigma _y \Psi (x,y) \nonumber \\&\quad = G_{B,E}^\dagger (w(x)) U_{B,E}^\dagger \sigma _y U_{B,E} G_{B,E}(w(x)). \end{aligned}$$

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Appendix C. Variable-changing for multi-barrier system

The auxiliary variable is

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We can rewrite the above quantities to show that they satisfy the ansatz (9) with

$$\begin{aligned}&q_\mathrm{v} = -r_\mathrm{v}< 0, \quad q_\mathrm{A} = -r_\mathrm{A}, \quad q_\mathrm{m} = -r_\mathrm{m} \le 0, \quad q_\phi = - r_\phi , \nonumber \\&p(w) = - \left[ 1 + h \sum _{n=0}^{N-1} \Pi \left( \dfrac{w - n L}{A} -\dfrac{1}{2} \right) \right] < 0. \end{aligned}$$

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Appendix D. The increase in the number of transparent peaks when N increases

To explain the increase in the number of transparent peaks when the number N of barriers increases, we re-examine the transfer matrix X. Because, in principle, all $2\times 2$ matrices can be uniquely decomposed into the Pauli matrices $\sigma _j$ ($j=1,2,3$) and the identity matrix ${\mathbb {I}}$, we have

$$\begin{aligned}&X = [M_3 M_2]^N = [A_0 {\mathbb {I}} + A_1 \sigma _1 + A_2 \sigma _2 + A_3 \sigma _3]^N\nonumber \\&= \left[ u_0 + \cos (u) + i \sin (u) \left( \dfrac{u_1}{u} \sigma _1 + \dfrac{u_2}{u} \sigma _2 + \dfrac{u_3}{u} \sigma _3 \right) \right] ^N\nonumber \\ \end{aligned}$$

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where we introduced the vector $\mathbf {u}=(u_1,u_2,u_3)$ whose components satisfy the following relations

$$\begin{aligned} i u_j \sin (u) /u = A_j, \qquad j=1,2,3. \end{aligned}$$

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Here $u=\sqrt{u_1^2+u_2^2+u_3^2}$ is the length of $\mathbf {u}$ and in this situation $\cos (u) = A_0 - u_0$. Then, according to Euler’s identity for matrix, X can be rewritten

$$\begin{aligned} X= & {} \left[ u_0 + e^{i \mathbf {u} \cdot \mathbf {\sigma }} \right] ^N \nonumber \\= & {} \sum _{n=0}^{N} u_0^{N-n} e^{i n \mathbf {u} \cdot \mathbf {\sigma }} \nonumber \\= & {} \sum _{n=0}^{N} u_0^{N-n} \left[ \cos (n u) + i \sin (n u) \left( \dfrac{u_1}{u} \sigma _1 + \dfrac{u_2}{u} \sigma _2 + \dfrac{u_3}{u} \sigma _3 \right) \right] \nonumber \\= & {} X_0(N) + i \left( \dfrac{u_1}{u} \sigma _1 + \dfrac{u_2}{u} \sigma _2 + \dfrac{u_3}{u} \sigma _3 \right) X_\sigma (N) \end{aligned}$$

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where

$$\begin{aligned}&X_0(N) = \sum _{n=0}^{N} u_0^{N-n} \cos (n u) \nonumber \\&\quad = \frac{\cos (N u) - u_0\cos [(N+1) u] - u_0^{N+1} \cos (u) +u_0^{N+2}}{\left( e^{i u} - u_0\right) \left( e^{-i u} - u_0\right) }, \nonumber \\&X_\sigma (N) = \sum _{n=0}^{N} u_0^{N-n} \sin (n u) \nonumber \\&\quad = \frac{\sin (N u) - u_0 \sin [(N+1)u] + u_0^{N+1} \sin (u)}{\left( e^{i u} - u_0\right) \left( e^{-i u} - u_0\right) }. \end{aligned}$$

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Or in matrix form, we have

$$\begin{aligned} X = \begin{pmatrix} X_0 (N) + i X_\sigma (N) u_3/u &{} i X_\sigma (N) u_1/u + X_\sigma (N) u_2/u \\ i X_\sigma (N) u_1/u - X_\sigma (N) u_2/u &{} X_0 (N) - i X_\sigma (N) u_3/u \end{pmatrix}.\nonumber \\ \end{aligned}$$

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Now, we can rewrite transmission probabilities in terms of N for both cases

$$\begin{aligned} T_B(N)= & {} |t_B|^2\nonumber \\= & {} \left| X_0 (N) - X_\sigma (N) \tan \theta _1^B \dfrac{u_3}{u} + i \dfrac{X_\sigma (N)}{\cos \theta _1^B} \dfrac{u_2}{u} \right| ^{-2}, \nonumber \\ T_E(N)= & {} |t_E|^2 \nonumber \\= & {} \left| X_0 (N) - X_\sigma (N) \tan \theta _1^E \dfrac{u_3}{u} - i \dfrac{X_\sigma (N)}{\cos \theta _1^E} \dfrac{u_1}{u} \right| ^{-2}.\nonumber \\ \end{aligned}$$

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We can see that the number N of barriers plays the role of a factor in the phase of the trigonometric functions, making the transmission probability T oscillates more rapidly with respect to the incident angle $\alpha$ (keep in mind that all $u,u_1,u_2,u_3,\theta _1^B,\theta _1^E$ depend on $\alpha$). The obvious consequence is that when N is doubled, the number of transparent peaks is roughly doubled as well, as observed in the Fig. 5.