# On Symmetry Properties of The Corrugated Graphene System

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## Abstract

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## 1. Introduction

## 2. Basic Physics of The Corrugated Graphene

## 3. Symmetries

#### 3.1. The Operator ${\widehat{S}}_{t}={\tau}_{y}\otimes i{\sigma}_{y}C$

- at the boundary between Region I and II$$\begin{array}{c}{\mathsf{\Psi}}_{-{k}_{x},-{k}_{y}}^{-}(x)-{r}_{L}{(\varphi )}_{\uparrow}^{\uparrow *}{\mathsf{\Psi}}_{{k}_{x},-{k}_{y}}^{-}(x)+{r}_{L}{(\varphi )}_{\downarrow}^{\uparrow *}{\mathsf{\Psi}}_{{k}_{x},-{k}_{y}}^{+}(x)=\\ i{a}_{+}^{*}{B}_{+}^{*}({m}_{+},{k}_{y}){\mathsf{\Phi}}_{-{m}_{+},-{k}_{y}}^{+}(\theta )+i{b}_{+}^{*}{B}_{+}^{*}(-{m}_{+},{k}_{y}){\mathsf{\Phi}}_{{m}_{+},-{k}_{y}}^{+}(\theta )+\\ +i{a}_{-}^{*}{B}_{-}^{*}({m}_{-},{k}_{y}){\mathsf{\Phi}}_{-{m}_{-},-{k}_{y}}^{-}(\theta )+i{b}_{-}^{*}{B}_{-}^{*}(-{m}_{-},{k}_{y}){\mathsf{\Phi}}_{{m}_{-},-{k}_{y}}^{-}(\theta )\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}x=Rcos{\theta}_{0},\phantom{\rule{4pt}{0ex}}\theta =-\varphi /2\phantom{\rule{0.166667em}{0ex}};\end{array}$$
- at the boundary between Region II and III$$\begin{array}{c}i{a}_{+}^{*}{B}_{+}^{*}({m}_{+},{k}_{y}){\mathsf{\Phi}}_{-{m}_{+},-{k}_{y}}^{+}(\theta )+i{b}_{+}^{*}{B}_{+}^{*}(-{m}_{+},{k}_{y}){\mathsf{\Phi}}_{{m}_{+},-{k}_{y}}^{+}(\theta )+\\ +i{a}_{-}^{*}{B}_{-}^{*}({m}_{-},{k}_{y}){\mathsf{\Phi}}_{-{m}_{-},-{k}_{y}}^{-}(\theta )+i{b}_{-}^{*}{B}_{-}^{*}(-{m}_{-},{k}_{y}){\mathsf{\Phi}}_{{m}_{-},-{k}_{y}}^{-}(\theta )=\\ {t}_{L}{(\varphi )}_{\uparrow}^{\uparrow *}{\mathsf{\Psi}}_{-k}^{-}(x)-{t}_{L}{(\varphi )}_{\downarrow}^{\uparrow *}{\mathsf{\Psi}}_{-k}^{+}(x)\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}x=-Rcos{\theta}_{0},\phantom{\rule{4pt}{0ex}}\theta =\varphi /2\phantom{\rule{0.166667em}{0ex}}.\end{array}$$

- at the boundary between Regions II and III$$\begin{array}{c}{\mathsf{\Psi}}_{-{k}_{x},-{k}_{y}}^{-}(x)+{r}_{R}{(\varphi )}_{\downarrow}^{\downarrow}{\mathsf{\Psi}}_{{k}_{x},-{k}_{y}}^{-}(x)+{r}_{R}{(\varphi )}_{\uparrow}^{\downarrow}{\mathsf{\Psi}}_{{k}_{x},-{k}_{y}}^{+}(x)=\\ {\tilde{a}}_{+}{\mathsf{\Phi}}_{{m}_{+},-{k}_{y}}^{+}(\theta )+{\tilde{b}}_{+}{\mathsf{\Phi}}_{-{m}_{+},-{k}_{y}}^{+}(\theta )+{\tilde{a}}_{-}{\mathsf{\Phi}}_{{m}_{-},-{k}_{y}}^{-}(\theta )+{\tilde{b}}_{-}{\mathsf{\Phi}}_{-{m}_{-},-{k}_{y}}^{-}(\theta )\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}x=Rcos{\theta}_{0}\phantom{\rule{0.166667em}{0ex}},\theta =\varphi /2;\end{array}$$
- at the boundary between Regions I and II$$\begin{array}{c}{\tilde{a}}_{+}{\mathsf{\Phi}}_{{m}_{+},-{k}_{y}}^{+}(\theta )+{\tilde{b}}_{+}{\mathsf{\Phi}}_{-{m}_{+},-{k}_{y}}^{+}(\theta )+{\tilde{a}}_{-}{\mathsf{\Phi}}_{{m}_{-},-{k}_{y}}^{-}(\theta )+{\tilde{b}}_{-}{\mathsf{\Phi}}_{-{m}_{-},-{k}_{y}}^{-}(\theta )=\\ {t}_{R}{(\varphi )}_{\downarrow}^{\downarrow}{\mathsf{\Psi}}_{-{k}_{x},-{k}_{y}}^{-}(x)+{t}_{R}{(\varphi )}_{\uparrow}^{\downarrow}{\mathsf{\Psi}}_{-{k}_{x},-{k}_{y}}^{+}(x)\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{1.em}{0ex}}x=-Rcos{\theta}_{0}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\theta =-\varphi /2\phantom{\rule{0.166667em}{0ex}}.\end{array}$$

#### 3.2. The Operator ${\widehat{S}}_{ch}={\tau}_{x}\otimes {\sigma}_{y}$

- $E>{\lambda}_{x}.$
- 1.
- Quantum numbers: $\alpha =+1$, $s=-1$. These quantum numbers determine the available set of the wave functions: ${\mathsf{\Psi}}_{k}^{+}(x)$, ${\mathsf{\Psi}}_{-k}^{-}(x)$,${\mathsf{\Phi}}_{3,m}(\theta )$. If the corresponding symmetries are responsible for the transport properties, there are only the following options.
- (a)
- The electron is moving from the left side of our structure (flat graphene sheet) with the spin up polarisation [${\mathsf{\Psi}}_{k}^{+}(x)\equiv |\uparrow +k\rangle $]. In this case, in the rippled graphene region (see Figure 2), there is one open channel, defined by the wave function ${\mathsf{\Phi}}_{3,m}$. The wave function ${\mathsf{\Psi}}_{-k}^{-}(x)\equiv |\downarrow -k\rangle $ describes the reflection with the electron spin–flip.
- (b)
- The electron is moving from the right side (flat graphene sheet) with the spin down polarisation [${\mathsf{\Psi}}_{-k}^{-}(x)\equiv |\downarrow -k\rangle $]. In this case, in the rippled graphene region, there is only the transmission channel, defined by the wave function ${\mathsf{\Phi}}_{3,m}$. The wave function ${\mathsf{\Psi}}_{k}^{+}(x)\equiv |\uparrow +k\rangle $ describes the reflection with the electron spin–flip.

As a result, we expect the equivalence between the left/right transmission probabilities with the opposite spin polarisations. Indeed, this expectation is consistent with Equation (33), obtained from the different arguments at ${k}_{y}=0$. - 2.
- Quantum numbers: $\alpha =-1$, $s=+1$. The available set of the wave functions: ${\mathsf{\Psi}}_{k}^{-}(x)$, ${\mathsf{\Psi}}_{-k}^{+}(x)$, ${\mathsf{\Phi}}_{1,m}(\theta )$. In this case, the symmetries dictate the following options.
- (a)
- The electron is moving from the left side (flat graphene sheet) with the spin down polarisation [${\mathsf{\Psi}}_{k}^{-}(x)\equiv |\downarrow +k\rangle $]. In this case, in the rippled graphene region (see Figure 2), there is one open channel, defined by the wave function ${\mathsf{\Phi}}_{1,m}(\theta )$. The wave function ${\mathsf{\Psi}}_{-k}^{+}(x)\equiv |\uparrow -k\rangle $ describes the reflections with the electron spin–flip.
- (b)
- The electron is moving from the right side (flat graphene sheet) with the spin up polarisation [${\mathsf{\Psi}}_{-k}^{+}(x)\equiv |\uparrow -k\rangle $]. In this case, in the rippled graphene region for this electron there is only the transmission channel, defined by the wave function ${\mathsf{\Phi}}_{1,m}(\theta )$. The wave function ${\mathsf{\Psi}}_{k}^{-}(x)\equiv |\downarrow +k\rangle $ describes the reflections with the electron spin–flip.

Again, we expect the equivalence between the left/right transmission probabilities with the opposite spin polarisations. Indeed, this expectation is consistent with Equation (35), obtained from different arguments at ${k}_{y}=0$.

- $E<{\lambda}_{x}.$
- 1.
- Quantum numbers: $\alpha =-1$, $s=+1$. In this case, the available set includes the following wave functions: ${\mathsf{\Psi}}_{-k}^{+}(x)$, ${\mathsf{\Psi}}_{k}^{-}(x)$, ${\mathsf{\Phi}}_{2,m}(\theta )$. The symmetries dictate the following options.
- (a)
- The electron is moving from the left side (flat graphene sheet) with the spin down polarisation [${\mathsf{\Psi}}_{k}^{-}(x)\equiv |\downarrow +k\rangle $]. In the rippled graphene region, there is only the transmission channel, defined by the wave function ${\mathsf{\Phi}}_{2,m}(\theta )$. The wave function ${\mathsf{\Psi}}_{-k}^{+}(x)\equiv |\uparrow -k\rangle $ describes the reflections with the electron spin–flip.
- (b)
- The electron is moving from the right side (flat graphene sheet) with the spin up polarisation [${\mathsf{\Psi}}_{-k}^{+}(x)\equiv |\uparrow -k\rangle $]. In the rippled graphene region, there is only the transmission channel, defined by the wave function ${\mathsf{\Phi}}_{2,m}(\theta )$. The wave function ${\mathsf{\Psi}}_{k}^{-}(x)\equiv |\downarrow +k\rangle $ describes the reflections with the electron spin–flip.

Again, the expected equivalence between the left/right transmission probabilities with the opposite spin polarisations is consistent with Equation (35), obtained from different arguments at ${k}_{y}=0$. - 2.
- Quantum numbers: $\alpha =+1$, $s=-1$. In this case, the available set includes the following wave functions: ${\mathsf{\Psi}}_{k}^{+}(x)$, ${\mathsf{\Psi}}_{-k}^{-}(x)$, ${\mathsf{\Phi}}_{3,m}(\theta )$. This situation is completely equivalent to the case discussed at $E>{\lambda}_{x}$, Point 1.

Thus, at ${k}_{y}=0$, the symmetry, associated with the operator ${\widehat{S}}_{ch}$, determines the following transport properties through the rippled graphene piece: (i) at the transmission, it preserves the electron spin polarisation, while forbids the spin–flip; and (ii) the reflection occurs only with the spin–flip.

#### 3.3. The Relation Between the Operators ${\widehat{S}}_{t}$ and ${\widehat{S}}_{ch}$

## 4. Summary

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The corrugated graphene system. There are two flat surfaces: Region I, defined in the intervals $-\infty <x<-Rcos{\theta}_{0}$; and Region III, defined in the intervals $Rcos{\theta}_{0}<x<\infty $. A ripple is modelled by an arc of a circle (Region II) of radius R, defined as $-Rcos{\theta}_{0}<x<Rcos{\theta}_{0}$. At ${\theta}_{0}=0$, the ripple is a half of the nanotube, while at ${\theta}_{0}=\pi /2$ the ripple does not exist. The angle $\varphi =\pi -2{\theta}_{0}$. Here, we have $-\infty <y<\infty $. We keep the translational invariance along the y-axis, which is chosen as the symmetry and the quantisation axis.

**Figure 2.**The spectrum in Equation (4) (${k}_{y}=0$) versus the magnetic quantum number m. The non-quantised values $\pm {m}_{s=\pm 1}$ at the energy $E=0.2$ eV (thin horizontal line that mimics the Fermi energy) are indicated at the crossing of the energy branches with different s. Symbols ${E}_{1},{E}_{2},{E}_{3},{E}_{4}$ are used to guide the eyes on the formal solutions (straight lines) defined by Equation (36), irrespective of the sign of the quantum number s. In contrast, there is anticrossing of the energy branches with the same quantum number $s=+1$ for the pair $({\mathcal{E}}_{1},{\mathcal{E}}_{2})$ at $E>0$. Similar anticrossing occurs at $E<0$, when the pair (${\mathcal{E}}_{3},{\mathcal{E}}_{4}$) has the same quantum number $s=+1$. These anticrossings are caused by the term ${\lambda}_{y}$ in the Hamiltonian in Equation (1), which creates the energy gaps $2{\lambda}_{y}$ near the energy $E=\pm {\lambda}_{x}$ (see [13,17]). The following parameters are used: $R=10$ Å, $\delta =0.01$, $p=0.1$, $\gamma =(4.5\xb71.42)\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}\xb7$ Å, ${\gamma}^{\prime}=\frac{8}{3}\gamma $. These parameters define the values of the spin–orbit coupling strengths: ${\lambda}_{y}=\delta {\gamma}^{\prime}/4R=0.0043$ eV, ${\lambda}_{x}=\gamma (1/2+2\delta p)/R=0.32$ eV (see Section 2).

**Figure 3.**The same as in Figure 2. Solid lines are associated with the positive energy states ($E>0$), while the negative energies are denoted by dashed lines (see text). Once the energy changes the sign, it affects the sign of the corresponding quantum number s. There are anticrossing of the energy branches with the same quantum number $s=+1$ for the pair $({\mathcal{E}}_{1},{\mathcal{E}}_{2})$ at $E>0$ and for the pair (${\mathcal{E}}_{3},{\mathcal{E}}_{4}$) at $E<0$.

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**MDPI and ACS Style**

Pudlak, M.; Smotlacha, J.; Nazmitdinov, R.
On Symmetry Properties of The Corrugated Graphene System. *Symmetry* **2020**, *12*, 533.
https://doi.org/10.3390/sym12040533

**AMA Style**

Pudlak M, Smotlacha J, Nazmitdinov R.
On Symmetry Properties of The Corrugated Graphene System. *Symmetry*. 2020; 12(4):533.
https://doi.org/10.3390/sym12040533

**Chicago/Turabian Style**

Pudlak, Mihal, Jan Smotlacha, and Rashid Nazmitdinov.
2020. "On Symmetry Properties of The Corrugated Graphene System" *Symmetry* 12, no. 4: 533.
https://doi.org/10.3390/sym12040533