On Symmetry Properties of The Corrugated Graphene System
Abstract
:1. Introduction
2. Basic Physics of The Corrugated Graphene
3. Symmetries
3.1. The Operator
- at the boundary between Region I and II
- at the boundary between Region II and III
- at the boundary between Regions II and III
- at the boundary between Regions I and II
3.2. The Operator
- 1.
- Quantum numbers: , . These quantum numbers determine the available set of the wave functions: , ,. 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 []. In this case, in the rippled graphene region (see Figure 2), there is one open channel, defined by the wave function . The wave function 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 []. In this case, in the rippled graphene region, there is only the transmission channel, defined by the wave function . The wave function 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 . - 2.
- Quantum numbers: , . The available set of the wave functions: , , . 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 []. In this case, in the rippled graphene region (see Figure 2), there is one open channel, defined by the wave function . The wave function 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 []. In this case, in the rippled graphene region for this electron there is only the transmission channel, defined by the wave function . The wave function 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 .
- 1.
- Quantum numbers: , . In this case, the available set includes the following wave functions: , , . The symmetries dictate the following options.
- (a)
- The electron is moving from the left side (flat graphene sheet) with the spin down polarisation []. In the rippled graphene region, there is only the transmission channel, defined by the wave function . The wave function 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 []. In the rippled graphene region, there is only the transmission channel, defined by the wave function . The wave function 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 . - 2.
- Quantum numbers: , . In this case, the available set includes the following wave functions: , , . This situation is completely equivalent to the case discussed at , Point 1.
Thus, at , the symmetry, associated with the operator , 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 and
4. Summary
Author Contributions
Funding
Conflicts of Interest
References
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Pudlak, M.; Smotlacha, J.; Nazmitdinov, R. On Symmetry Properties of The Corrugated Graphene System. Symmetry 2020, 12, 533. https://doi.org/10.3390/sym12040533
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 StylePudlak, 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