Comment: Energy Spectrum of a Graphene Quantum Dot in a Perpendicular Magnetic Field

TL;DR

This paper defends the accuracy of an analytic derivation of the energy spectrum for a graphene quantum dot in a magnetic field, clarifying previous misunderstandings and correcting notation errors.

Contribution

It confirms the correctness of the original energy spectrum derivation and addresses critiques, reinforcing the validity of the analytic approach.

Findings

Original derivation is correct and equivalent to previous results

Notation errors are identified and corrected

Analytic expressions for energy spectrum are validated

Abstract

In a recent comment (arXiv:1607.06081), Falaye et al. claim that there are certain flaws in our publication (Phys. Rev. B, 78, 195427 (2008)). We point out that our results, in particular the analytic derivation of the energy spectrum of a circular graphene quantum dot exposed to a perpendicular magnetic field, are correct and equivalent to the result of Falaye et al. A misleading notation error is corrected.

Equations

$$\begin{array}{rl}& L\left(\frac{k^2{l}_B^2}{2}-m-1,m+1,\frac{R^2}{2l_B^2}\right)\\ & -\tau \frac{k{l}_B}{R/l_B}\cdot L\left(\frac{k^2{l}_B^2}{2}-m-1,m,\frac{R^2}{2l_B^2}\right)=0.\end{array}$$

$$E=\pm v_F\sqrt{2e\hslash B(m+1+p)},$$

Full text

Falaye et al. claim Falaye16 that there are certain flaws in our publication Schnez08 , in particular that the wave functions given by Eq. 5 in Ref. Schnez08 cannot be normalized and that, correspondingly, the implicit equation Eq. 6 describing the energy spectrum is incorrect. We note the following:

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The mathematical derivation based on our ansatz as described in Ref. Schnez08 is correct. As a matter of fact, the results of Falaye et al., who use the confluent hypergeometric function instead of the generalized Laguerre polynomials, are equivalent to our results. The parameter $a$ in the generalized Laguerre polynomials $L(a,b,x)$ can take real values, not only integers as in Ref. Falaye16 . This is beyond the definition in Ref. Abramowitz , but well-defined and used today (also implemented in e.g. Mathematica).

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Our definition of the quantum number $n$ differs from the definition in Ref. Falaye16 . They do not denote the same quantity.

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Using a recursion theorem for the generalized Laguerre polynomials Abramowitz , the energy spectrum Eq. 6 in Ref. Schnez08 can be written in a more compact form as (as pointed out by Falaye et al.)

[TABLE]

The use of the parameter $m$ in Eq. 11 of our publication Schnez08 is incorrect. Rather, it should read

[TABLE]

where $m$ is the previously defined quantum number and $p$ is an integer with $p>-(m+1)$. This follows from the fact that Eq. 6 in Ref. Schnez08 or Eq. 1 above, respectively, can be simplified to $\left(\Gamma(\alpha)\Gamma(-\alpha)\right)^{-1}=0$ in the limit $R/l_{B}\to\infty$ with $\alpha:=k^{2}l_{B}^{2}/2-m-1$. This is fulfilled for $\alpha=\pm p$ and $p$ being an integer. The later restriction of $p>-(m+)$ is then required to make the radicand non-negative.