Comment on "Effective of the q-deformed pseudoscalar magnetic field on the charge carriers in graphene"

TL;DR

This paper critiques a recent study on q-deformed pseudoscalar magnetic barriers in graphene, highlighting errors in the original analysis that compromise its main findings.

Contribution

It clarifies the correct interpretation of the potential and corrects the erroneous calculations in the previous work.

Findings

Identified misinterpretation of the potential in the original paper

Corrected the calculations related to the Dirac-Weyl equation

Showed that the original results are invalid due to these errors

Abstract

We point out a misleading treatment in a recent paper published in this Journal [J. Math. Phys. (2016) 57, 082105] concerning solutions for the two-dimensional Dirac-Weyl equation with a q-deformed pseudoscalar magnetic barrier. The authors misunderstood the full meaning of the potential and made erroneous calculations, this fact jeopardizes the main results in this system.

Equations

$$\overrightarrow{A}=2B_{0}d{\tanh }_q(x/2d){\text{eˆ}}_y,$$

$${\sinh }_{q}a\equiv \frac{{\mathrm{e}}^a-q{\mathrm{e}}^{-a}}{2},{\cosh }_{q}a\equiv \frac{{\mathrm{e}}^a+q{\mathrm{e}}^{-a}}{2},$$

$${\tanh }_{q}a\equiv \frac{{\sinh }_{q}a}{{\cosh }_{q}a}\equiv \frac{1}{{\coth }_{q}a},$$

$${\tanh }_q(a)=\tanh \left(a-\frac{1}{2}\log q\right),$$

$$\frac{d}{da}{\tanh }_{q}a=\frac{q}{{\cosh }_q^{2}a}=1-{\tanh }_q^{2}a.$$

$$\overrightarrow{B}=B_0\frac{q}{{\cosh }_q^2(x/2d)}{\text{eˆ}}_z\text{.}$$

$$\overrightarrow{\sigma }.\left[v_f\left(-i\hslash \overrightarrow{\nabla }+e\overrightarrow{A}\right)\right]\psi (x,y)=E\psi (x,y)$$

$$+\left[\frac{d}{dx}+\left(k_y+\frac{e}{\hslash }{A}_y\right)\right]{\psi }_B(x)$$

$$-\left[\frac{d}{dx}-\left(k_y+\frac{e}{\hslash }{A}_y\right)\right]{\psi }_A(x)$$

$$\frac{d^2}{d{x}^2}{\psi }_{\tau }+\left[{\Xi }^2-W_{\tau }\right]{\psi }_{\tau }=0,$$

$$W_{\tau }=\frac{e^2}{{\hslash }^2}{A}_y^2+\frac{2e}{\hslash }{k}_y{A}_y+\tau \frac{e}{\hslash }\frac{d{A}_y}{dx}+k_y^2,$$

$${\Xi }^2=\frac{E^2}{{\hslash }^2{v}_f^2}.$$

$$\left[\frac{d}{dx}+\left(k_y+\frac{e}{\hslash }{A}_y(x)\right)\right]{\psi }_{-}$$

$$\left[\frac{d}{dx}-\left(k_y+\frac{e}{\hslash }{A}_y(x)\right)\right]{\psi }_{+}$$

$${\psi }_{+}(x)$$

$${\psi }_{-}(x)$$

$$\mathcal{F}={\int }^{x}d\xi \left(k_y+\frac{e}{\hslash }{A}_y(\xi )\right).$$

$${\int }_{-\infty }^{+\infty }dx\left(|N_{+}{|}^2{\mathrm{e}}^{2\mathcal{F}}+|N_{-}{|}^2{\mathrm{e}}^{-2\mathcal{F}}\right)<\infty .$$

$${\psi }_{-}^0$$

$${\psi }_{+}^0$$

$$\psi^{0}(x)=N_{-}\mathrm{e}^{-k_{y}x}\left(\cosh_{q}\frac{x}{2d}\right)^{-4d^{2}/l_{B}^{2}}\left(\begin{array}[]{c}0\\ 1\end{array}\right)\,.$$

$$W_{\tau }(x)=\frac{1}{l_B^2}\left(4\frac{d^2}{l_B^2}-\tau \right){\tanh }_q^2\left(\frac{x}{2d}\right)+4\frac{k_{y}d}{l_B^2}{\tanh }_q\left(\frac{x}{2d}\right)+k_y^2+\tau \frac{1}{l_B^2}.$$

$$E_n=\pm \frac{\hslash v_f}{2d}\sqrt{\left(8\frac{d^2}{l_B^2}-n\right)n\left[1-{\left(\frac{2k_{y}d}{n-4d^2/l_B^2}\right)}^2\right]},n=1,2,\mathrm{3...}{n}_{\max },$$

$$n<n_{\max }\left(=4\frac{d^2}{l_B^2}\right),\left|k_y\right|⩽\frac{1}{2d}\left|n-4\frac{d^2}{l_B^2}\right|.$$

$$\psi^{E\neq 0}(x,y)=N\mathrm{e}^{-ik_{y}y}(1+z)^{\eta}z^{\mu}\left(\begin{array}[]{c}\frac{i\hslash v_{f}}{Ed}\left[g(z)F\left(a,b,c;z\right)+\frac{ab}{c}F\left(a+1,b+1,c+1;z\right)\right]\vskip 6.0pt plus 2.0pt minus 2.0pt\\ F\left(a,b,c;z\right)\end{array}\right),$$

$$a=\mu +\eta -4\frac{d^2}{l_B^2},b=\mu +\eta +4\frac{d^2}{l_B^2}+1,c=2\mu +1,$$

$$\mu =d\sqrt{{\left(k_y-2\frac{d}{l_B^2}\right)}^2-{\Xi }^2},\eta =d\sqrt{{\left(k_y+2\frac{d}{l_B^2}\right)}^2-{\Xi }^2}.$$

Full text

In Iranianos , Eshgi and Mehraban studied the dynamics of the charge carriers in graphene in presence of a $q$-deformed pseudoscalar magnetic barrier. Such barrier is represented by a inhomogeneous background magnetic field which is associated to a vector potential with a hyperbolic profile as follows

[TABLE]

where $B_{0}~{}$and $d~{}$are constant and  $q$ is a deformation parameter. Note that the expression (1) is being characterized by a $q$-deformed hyperbolic functions, which are based on a $q$-deformation of the usual hyperbolic functions Arai , and are denoted by (we assume $0<q\leqslant 1$)

[TABLE]

which are related as $\cosh_{q}^{2}a-\sinh_{q}^{2}a=q$. Thereby, the $q$-tangent hyperbolic function is defined via a simple analogy to the usual hyperbolic functions:

[TABLE]

which can be re-expressed using (2) as

[TABLE]

and whose derivative

[TABLE]

Such $q$-deformed hyperbolic functions were used in Iranianos to obtain what the authors believed to be general exact expressions for the energies and their associated wavefunctions for the proposed system. With these results, they also address the scattering regime to calculate the reflection and transmission coefficients by using the Riemann’s equation. Unfortunately, due to a incorrect manipulation of the expressions (3) and (5) into the Dirac equation, the results found in Iranianos would not be correct.

It is the aim of this Comment, to point out and correct these mistakes. With that purpose in mind, we will adopt the notation used in Iranianos and we begin with the correct expression for the background magnetic field $\overrightarrow{B}$ $=\overrightarrow{\nabla}\times\overrightarrow{A}=B(x)$êz, given by

[TABLE]

The figure (1) shows the real behavior of $B(x)/B_{o}~{}$versus $x/d~{}$for different values of $q$. Note that the $q$-parameter does not deform the magnetic field profile, but only displaces it horizontally. That behavior is in clear contradiction with the figure (1) in Iranianos , where is stated that the amplitude of magnetic field decrease with increasing value of $q$. We disagree with that last statement, due that is based on incorrect expression for the magnetic field.

Now, to study the dynamics of the carriers charge in graphene in presence of a background magnetic field, the authors in Iranianos used the so-called two-dimensional Dirac-Weyl equation

[TABLE]

for a given valley degree of freedom. Here, $v_{f}$ $\approx c/300~{}$is the Fermi velocity, $\vec{\sigma}=\left(\sigma_{x},\sigma_{y}\right)$ are the Pauli matrices and $\psi(x,y)~{}$is a two-component spinor, whose transpose is $\psi^{T}(x,y)=$e${}^{ik_{y}y}\left(\psi_{A}~{}i\psi_{B}\right)$. The superscripts $A$ and $B$ in the spinor components designate the triangular sublattices where the electrons are supported on. The eq. (7) represent the version correct of the Dirac-Weyl equation showed in Iranianos , which presents dimensional inconsistencies that are maintained throughout the paper (see eqs. (1)-(4) in Iranianos ).

Substituting $\psi(x,y)$ into equation (7), the Weyl-Dirac equation give rise to two coupled first-order equations for the upper, $\psi_{A}$ and the lower, $\psi_{B}~{}$components of the spinor

[TABLE]

The coupling between the upper and the lower components can be formally eliminated for $E\neq 0$. Using the expression for $\psi_{A}$ obtained from (8) and inserting it in (9) one obtains a second-order differential equation for $\psi_{B}$. In a similar way, using the expression for $\psi_{B}$ obtained from (9) and inserting it in (8) one obtains a second-order differential equation for $\psi_{A}$. Both results can be written in a compact form:

[TABLE]

where $\tau=\pm 1~{}$(the upper values correspond to $\psi_{A}~{}$ and the lower values correspond to $\psi_{B}$),

[TABLE]

and

[TABLE]

These last results tell us that the solutions for this kind of problem can be formulated as a Sturm–Liouville problem for the component $\psi_{+}$ and $\psi_{-}$. Nevertheless, the solutions for $E=0$, excluded from the Sturm-Liouville problem, was not taken into account in Iranianos . Such solutions (so-called isolated solutions or isolated zero modes) can be obtained directly form the first–order equations (8) and (9)

[TABLE]

One can observe that the isolated zero mode for the upper and lower components are given by

[TABLE]

where $N_{+}$ and $N_{-}$ are normalization constants and

[TABLE]

In order to guarantee the normalization condition for the zero mode solutions, the integral must be convergent, i.e.,

[TABLE]

This result clearly shows that the normalization of the zero mode is decided by the asymptotic behavior of $\mathcal{F}$. One can check that it is impossible to have both components different from zero simultaneously as physically acceptable solutions. So, with the vector potential proposed in (1), the zero mode solutions adopt the explicit form

[TABLE]

where $l_{B}\equiv\sqrt{\hslash/eB_{0}}~{}$is the magnetic lenght. In order to check the normalization condition (17), the integral can be convergent only for $N_{+}=0$ and $2d/l_{B}>k_{y}$. Therefore, the isolated solution is given by

[TABLE]

With regards to the energy spectrum and the corresponding eigenstates for $E\neq 0$, the authors in Iranianos obtained exact bounded solutions from the second–order differential equations (10) with the effective potential $W_{\tau}~{}$given by eq. (9) in Iranianos . Nevertheless, such potential is dimensionally and structurally wrong, this due to that the starting point was a incorrect Dirac-Weyl equation (eq. (1) in Iranianos ) and also because a careless manipulation of the $q$ -deformed hyperbolic functions. Here we show the correct expression for the effective potential in the form of a deformed Rosen-Morse potential Others ; milpas :

[TABLE]

As seen in figure (2), the potential $W_{\tau}(x)$ for $\tau=-1~{}$ is characterized for have two maximum values in $W_{-}^{\max}(x\rightarrow\pm\infty)~{}=(k_{y}\pm 2d/l_{B}^{2})^{2}~{}$and one minimum value in $W_{-}^{\min}=\left(k_{y}^{2}l_{B}^{4}-l_{B}^{2}-4d^{2}\right)/\left(4d^{2}l_{B}^{2}+l_{B}^{4}\right)$. Note that $W_{-}^{\max}~{}$and $W_{-}^{\min}$ not depend on the deformation parameter $q$, which one, anecdotally, does not deform the potential and only displaces it horizontally (the same conclusion was reached for $\tau=+1$). Analitically, such behavior for $W_{\tau}~{}$can be proven after replacing (4) in (22), being now evident that the deformed Rosen-Morse potential depend on the deformation parameter $q~{}$only by a translation. In other words, the parameter $q~{}$is not necessary to know how many bound-state solutions exists. This behavior is reflected in the expression for the energy spectrum

[TABLE]

which is $q$-independent, as expected. Note that $n~{}$and $k_{y}~{}$are restricted in order to satisfy the square integrability condition:

[TABLE]

The eigenfunctions associated to (23) can be obtained from the second–order differential equation (10) for only one component of the Dirac spinor, in our case we choose $\psi_{-}~{}$(=$\psi_{B}$). The expression for $\psi_{+}~{}$(=$\psi_{A}$) can be directly built replacing in (8), the solution previously obtained for $\psi_{-}~{}$. In this way, by defining a new variable $z=\left[1+\tanh_{q}\left(x/2d\right)\right]/2$, the general set complete of solutions can be written as

[TABLE]

where $N~{}$is the normalization constant, $F\left(a,b,c;z\right)~{}$is the hypergeometric function with

[TABLE]

and $g(z)=\left[\left(\mu-2d^{2}/l_{B}^{2}\right)(1-z)-(\eta-2d^{2}/l_{B}^{2})z-k_{y}d\right]$ with

[TABLE]

Normalizable polynomial solutions are obtained by putting $a=-n$, which allows to rewrite the hypergeometric function $F\left(-n,b,c;z\right)~{}$as Jacobi polynomials $P_{n}^{(c-1,-n+b-c)}(z)$. Such mapping is shown in detail in milpas , where the authors also studied the dynamics of the carriers in graphene subjected to an inhomogeneous magnetic field with a vector potential $A_{y}=2B_{0}d\tanh(x/2d)$, which is the same from (1 ) for $q=1$. In such limit, our results are consistent to those found in milpas .