Comment: Energy Spectrum of a Graphene Quantum Dot in a Perpendicular Magnetic Field
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Comment: Energy Spectrum of a Graphene Quantum Dot in a PerpendicularMagnetic Field
S. Schnez , K. Ensslin , M. Sigrist , and T. Ihn Solid State Physics Laboratory, ETH Z¨urich, 8093 Z¨urich, Switzerland Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland
In a recent comment [1], Falaye et al. claim that there are certain flaws in our publication [2]. Wepoint out that our results, in particular the analytic derivation of the energy spectrum of a circulargraphene quantum dot exposed to a perpendicular magnetic field, are correct and equivalent to theresult of Falaye et al. . A misleading notation error is corrected.
PACS numbers: 73.23.-b, 73.63.-b, 73.63.Kv
Falaye et al. claim [1] that there are certain flaws inour publication [2], in particular that the wave functionsgiven by Eq. 5 in Ref. [2] cannot be normalized and that,correspondingly, the implicit equation Eq. 6 describingthe energy spectrum is incorrect. We note the following: • The mathematical derivation based on our ansatz as described in Ref. [2] is correct. As a matter offact, the results of Falaye et al. , who use the conflu-ent hypergeometric function instead of the gener-alized Laguerre polynomials, are equivalent to ourresults. The parameter a in the generalized La-guerre polynomials L ( a, b, x ) can take real values,not only integers as in Ref. [1]. This is beyond thedefinition in Ref. [3], but well-defined and used to-day (also implemented in e.g. Mathematica). • Our definition of the quantum number n differsfrom the definition in Ref. [1]. They do not denotethe same quantity. • Using a recursion theorem for the generalized La-guerre polynomials [3], the energy spectrum Eq. 6in Ref. [2] can be written in a more compact formas (as pointed out by Falaye et al. ) L (cid:18) k l B − m − , m + 1 , R l B (cid:19) − τ kl B R/l B · L (cid:18) k l B − m − , m, R l B (cid:19) = 0 . (1) The use of the parameter m in Eq. 11 of our publication[2] is incorrect. Rather, it should read E = ± v F p e ~ B ( m + 1 + p ) , (2)where m is the previously defined quantum number and p is an integer with p > − ( m + 1). This follows fromthe fact that Eq. 6 in Ref. [2] or Eq. 1 above, respec-tively, can be simplified to (Γ( α )Γ( − α )) − = 0 in thelimit R/l B → ∞ with α := k l B / − m −