The paper discusses the electronic properties of graphene, a two-dimensional (2D) material composed of a single layer of carbon atoms. The authors, K.S. Novoselov, A.K. Geim, and others, report that graphene exhibits unique behaviors not typically observed in conventional materials due to its Dirac-like electron behavior. Key findings include: 1. **Integer Quantum Hall Effect (IQHE) Anomaly**: The IQHE in graphene occurs at half-integer filling factors, deviating from the usual integer values. 2. **Minimum Conductivity**: Graphene's conductivity never falls below a minimum value corresponding to the conductance quantum \( e^2/h \), even at low carrier concentrations. 3. **Cyclotron Mass**: The cyclotron mass \( m_c \) of massless carriers in graphene is described by \( E = m_c c_*^2 \), where \( c_* \approx 10^6 \text{m/s} \). 4. **Shubnikov-de Haas Oscillations (SdHO)**: SdHO in graphene exhibit a phase shift of \( \pi \) due to Berry's phase, and the fundamental SdHO frequency \( B_F \) is linearly dependent on carrier concentration \( n \). The authors also explore the high-field limit, where SdHO evolve into the quantum Hall effect (QHE). They observe that the QHE plateaus in graphene correspond to half-integer filling factors, a phenomenon not seen in conventional systems. This behavior is attributed to the unique properties of massless Dirac fermions in graphene, including the existence of both electron- and hole-like Landau states at zero energy. Additionally, the paper discusses the zero-field behavior of graphene, noting that the conductivity never falls below a well-defined value, independent of temperature. This minimum conductivity is an intrinsic property of 2D systems described by the Dirac equation, and it is robust across different graphene samples. Overall, the study highlights the distinctive electronic properties of graphene, which are governed by the Dirac equation rather than the Schrödinger equation, and provides insights into the rich physics of quantum electrodynamics and related phenomena.The paper discusses the electronic properties of graphene, a two-dimensional (2D) material composed of a single layer of carbon atoms. The authors, K.S. Novoselov, A.K. Geim, and others, report that graphene exhibits unique behaviors not typically observed in conventional materials due to its Dirac-like electron behavior. Key findings include: 1. **Integer Quantum Hall Effect (IQHE) Anomaly**: The IQHE in graphene occurs at half-integer filling factors, deviating from the usual integer values. 2. **Minimum Conductivity**: Graphene's conductivity never falls below a minimum value corresponding to the conductance quantum \( e^2/h \), even at low carrier concentrations. 3. **Cyclotron Mass**: The cyclotron mass \( m_c \) of massless carriers in graphene is described by \( E = m_c c_*^2 \), where \( c_* \approx 10^6 \text{m/s} \). 4. **Shubnikov-de Haas Oscillations (SdHO)**: SdHO in graphene exhibit a phase shift of \( \pi \) due to Berry's phase, and the fundamental SdHO frequency \( B_F \) is linearly dependent on carrier concentration \( n \). The authors also explore the high-field limit, where SdHO evolve into the quantum Hall effect (QHE). They observe that the QHE plateaus in graphene correspond to half-integer filling factors, a phenomenon not seen in conventional systems. This behavior is attributed to the unique properties of massless Dirac fermions in graphene, including the existence of both electron- and hole-like Landau states at zero energy. Additionally, the paper discusses the zero-field behavior of graphene, noting that the conductivity never falls below a well-defined value, independent of temperature. This minimum conductivity is an intrinsic property of 2D systems described by the Dirac equation, and it is robust across different graphene samples. Overall, the study highlights the distinctive electronic properties of graphene, which are governed by the Dirac equation rather than the Schrödinger equation, and provides insights into the rich physics of quantum electrodynamics and related phenomena.