Comment on “Landau quantized dynamics and spectra for group VI dichalcogenides, including a model quantum wire” [AIP Advances 7, 065316 (2017)]

Horing, Norman J. M.

doi:10.1063/1.5100527

This comment elaborates on the analysis of the Landau-quantized Dichalcogenide quantum wire eigen-energy dispersion relation¹ of Eq. (51). We consider the case in which the width of the quantum wire “ $w$ ” is the dominantly small length parameter and β = ∫dxU(r) is approximately the product of “ $w$ ” with the potential well depth U₀, such that $| β | \sqrt{e B} = w U_{0} \sqrt{e B}$ is much smaller than the difference of neighboring energy/frequency poles of the “non-wire” Green’s function G of Eq. (3). The determinental dispersion relation is given by:

\begin{matrix} d e t (I_{2 D} - β G^{'} (p_{y})) & = & 1 - β [G_{11}^{'} + G_{22}^{'}] \\ + β^{2} [G_{11}^{'} G_{22}^{'} - G_{21}^{'} G_{12}^{'}] = 0, \end{matrix}

(1)

which may be written out more fully as

0 = 1 - β \sqrt{\frac{e B}{π}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n!} [) \frac{ω_{+}}{D_{n +}} + \frac{ω_{-}}{D_{n -}}]) f_{n} (p_{y})

(2)

\begin{matrix} + β^{2} \frac{e B}{π} \sum_{n = 0}^{\infty} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{n} {(- 1)}^{m}}{n! m!} \frac{1}{D_{n +} D_{m -}} \\ \times \{) ω_{+} ω_{-} f_{n} (p_{y}) f_{m} (p_{y}) - [) Λ_{+}^{o p} (p_{y}) f_{n} (p_{y})]) [) Λ_{-}^{o p} (p_{y}) f_{m} (p_{y})]) \}) . \end{matrix}

(3)

To obtain a root of this eigen-energy dispersion relation for small β (as defined above), a denominator factor D_n+ or D_n− in the (β∑_n)-terms on the right of Eq. (3) must be of order β to make these terms (which contain the prefactor β) of order O(β⁰) = O(1). In conjunction with this, considering the (β²∑_n∑_m)-terms on the right of Eq. (3), most of the D_n+D_m− denominator factors will have one of order β, say D_n+ = O(β), and the other of order D_m− = O(β⁰) = O(1) due to a different energy/frequency pole location. Because of the prefactor β², such terms will be small, of order O(β²/β) = O(β)): However, there is a subset of the (∑_m)-terms having index m = n − 1_ν with D_m− = D_n+, so that $D_{n +} D_{m -} = D_{n +}^{2} = O (β^{2})$ for m = n − 1_ν, and these second-order-pole terms are competitive in generating a root with terms of the form

β^{2} \sum_{n} \sum_{m} \Rightarrow β^{2} \sum_{\begin{array}{c} n = 0 \\ m = n - 1_{ν} \end{array}}^{\infty} = β^{2} O (1 / β^{2}) = O (β^{0}) = O (1) .

(4)

Neglecting the other small terms of $β^{2} \sum_{n} \sum_{m \neq n - 1_{ν}} \sim O (β)$ and focusing on the root generated by the n = n′ term of ∑_n, the other n ≠ n′ terms are similarly small of order O(β), so the dispersion relation, as it pertains to the n′-root, takes the approximate form (± → +, for example; similar results apply for ± → −)

\begin{matrix} 0 & = & 1 - \frac{β}{D_{n^{'} +}} \sqrt{\frac{e B}{π}} () \frac{{(- 1)}^{n^{'}}}{n^{'}!} ω_{+} f_{n^{'}} (p_{y}) \\ + \frac{{(- 1)}^{n^{'} - 1_{ν}}}{(n^{'} - 1_{ν})!} ω_{-} f_{n^{'} - 1_{ν}} (p_{y}))) - \frac{β^{2}}{D_{n^{'} +}^{2}} \frac{e B}{π} \frac{1}{n^{'}! (n^{'} - 1_{ν})!} \\ \times \{) ω_{+} ω_{-} f_{n^{'}} (p_{y}) f_{n^{'} - 1_{ν}} (p_{y}) - [) Λ_{+}^{o p} (p_{y}) f_{n^{'}} (p_{y})]) \\ \times [) Λ_{-}^{o p} (p_{y}) f_{n^{'} - 1_{ν}} (p_{y})]) \}) . \end{matrix}

(5)

This quadratic equation for D_n′+, which is defined as

D_{n^{'} +} = (ω - E_{s_{z}} \mp^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}) (ω - E_{s_{z}} \pm^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}),

(6)

may be solved as

D_{n^{'} +} = \frac{- β}{2} Φ_{n^{'} +} \pm^{''} \frac{β}{2} \sqrt{Φ_{n^{'} +}^{2} - 4 Ψ_{n^{'}}},

(7)

where we have used the notation

Φ_{n^{'} +} = - \sqrt{\frac{e B}{π}} (\frac{{(- 1)}^{n^{'}}}{n^{'}!} ω_{+} f_{n^{'}} (p_{y}) + \frac{{(- 1)}^{n^{'} - 1_{ν}}}{(n^{'} - 1_{ν})!} ω_{-} f_{n^{'} - 1_{ν}} (p_{y}))

(8)

and

\begin{matrix} Ψ_{n^{'}} & = & - \frac{e B}{π} \frac{1}{n^{'}! (n^{'} - 1_{ν})!} \{) ω_{+} ω_{-} f_{n^{'}} (p_{y}) f_{n^{'} - 1_{ν}} (p_{y}) \\ - [) Λ_{+}^{o p} (p_{y}) f_{n^{'}} (p_{y})]) [) Λ_{-}^{o p} (p_{y}) f_{n^{'} - 1_{ν}} (p_{y})]) \}) . \end{matrix}

(9)

As D_n′ in Eq. (7) is already of order O(β), the quantity $ω_{\pm} = ω - E_{s_{z}} \pm g$ in equations 8 and 9 is to be taken to order O(β⁰) = O(1), given by value of ω at the nearest vanishing of D_n′+, namely $ω = E_{s_{z}} \pm^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}$ ⁠, unshifted by the presence of the quantum wire, so that $ω_{\pm} = ω - E_{s_{z}} \pm g \Rightarrow \pm^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}} \pm g$ ⁠. Maintaining the evaluation of the wire-shifted-root to order O(β), the evaluation of the factor on the right of Eq. (6)other than that of the “nearest vanishing of D_n′+” must be taken to the order O(β⁰) in the same way: For example, if the nearest vanishing of D_n′+ is given by $ω = E_{s_{z}} \pm^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}$ ⁠, then $ω - E_{s_{z}} \pm^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}} \Rightarrow \pm^{'} 2 \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}$ ⁠. In this context, Eq. (6) must be written as $D_{n^{'} +} = (ω - E_{s_{z}} \mp^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}) (\pm^{'} 2 \sqrt{g^{2} + ϵ_{n^{'} +}^{2}})$ ⁠, and the n′-root ω is obtained by employing this in Eq. (7), with the result to order O(β) as

\begin{align} ω = E_{s_{z}} \pm^{'} \sqrt{g^{2} + ϵ_{n^{'} +}^{2}} \mp^{'} \frac{β}{4 \sqrt{g^{2} + ϵ_{n^{'} +}^{2}}} (Φ_{n^{'} +} \mp^{''} \sqrt{Φ_{n^{'} +}^{2} - 4 Ψ_{n^{'}}}) . \end{align}

(10)

Similar remarks apply to the case ± → −.

The insensitivity of the dispersion relation to the sign ±′ is a manifestation of the symmetry of the positive and negative branches of the Dichalcogenide energy spectrum above and below the shifted (spin-split) energy origin $E_{s_{z}}$ ⁠.

Finally, we note that Eq. (60) was transcribed improperly. It should read as (see Appendix A, Eq. (A2))

[) \frac{i \partial}{\partial τ} - \frac{(p_{x}^{2} + p_{y}^{2})}{2 M_{\pm}} + \frac{M_{\pm} Ω_{c \pm}^{2}}{8} (\frac{\partial^{2}}{\partial p_{x}^{2}} + \frac{\partial^{2}}{\partial p_{y}^{2}})]) A_{\begin{array}{c} 11 \\ 22 \end{array}}^{'} (\vec{p}, τ) = 0,

(11)

where τ is a “time” variable conjugate to Ω_∓ (not ω) under Fourier transformation. The arguments of $A_{11}^{'}$ ⁠, $A_{22}^{'}$ ⁠, A two lines above Eq. (60) should involve τ rather than T as: $A_{11}^{'} (\vec{p}, τ), A_{22}^{'} (\vec{p}, τ), A^{'} (\vec{p}, τ)$ ⁠. Correspondingly, the “time” arguments of Eqns. (62) and (63) should be τ rather than T. (Also, “m” in Eq. (62) and “ $M$ ” in Eq. (63) should both be “ $M_{\pm}$ ”.) Finally, the superscript on $G_{\begin{array}{c} 11 \\ 22 \end{array}} (R, ω) \Rightarrow G_{\begin{array}{c} 11 \\ 22 \end{array}}^{'} (R, ω)$ in the middle of Eq. (68) should be a prime.

REFERENCE

1.

N. J. M.

Horing

, “

Landau quantized dynamics and spectra for group VI dichalcogenides, including a model quantum wire

,”

AIP Advances

7

,

065316

(

2017

).

https://doi.org/10.1063/1.4986221

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Crossref

2019

Author(s)

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Comment on “Landau quantized dynamics and spectra for group VI dichalcogenides, including a model quantum wire” [AIP Advances 7, 065316 (2017)]

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