Graphene, a single layer of carbon atoms arranged in a hexagonal lattice, exhibits extraordinary physical properties that have captivated researchers since its isolation in 2004. Its unique combination of intrinsic strength, electron mobility, and thermal conductivity sets it apart from conventional materials. However, the transition from idealized theoretical predictions to real-world performance reveals complexities arising from quantum confinement, sample size, and material imperfections.

The intrinsic strength of graphene is unparalleled, with a tensile strength exceeding 130 gigapascals (GPa), making it one of the strongest materials ever measured. This exceptional mechanical property stems from the robust sp² covalent bonds between carbon atoms, which form a two-dimensional honeycomb lattice. Theoretical calculations suggest that pristine graphene should approach this upper limit, but experimental measurements often show variability due to defects such as vacancies, grain boundaries, and adatoms. Even at the nanoscale, graphene maintains remarkable strength, though the presence of defects can reduce its fracture toughness. The size of the sample plays a critical role—larger sheets are more susceptible to defect propagation, while smaller flakes may exhibit closer-to-ideal strength due to fewer imperfections.

Electron mobility in graphene is another defining characteristic, with theoretical values reaching up to 200,000 cm²/V·s at room temperature for suspended samples. This high mobility arises from the linear dispersion relation of electrons near the Dirac points, resulting in massless Dirac fermion behavior. However, real-world measurements often report lower values due to substrate interactions, surface adsorbates, and charge inhomogeneities. For instance, graphene on silicon dioxide typically exhibits mobilities between 10,000 to 40,000 cm²/V·s, limited by phonon scattering and Coulomb impurities. The influence of sample size becomes evident when examining nanoribbons or confined structures, where edge states and quantum confinement effects alter the electronic properties. Narrow ribbons exhibit a bandgap due to quantum confinement, which is absent in infinite sheets, demonstrating the delicate balance between dimensionality and electronic behavior.

Thermal conductivity in graphene is equally impressive, with theoretical predictions suggesting values as high as 5,000 W/m·K for defect-free monolayers. Experimental measurements on suspended graphene have confirmed room-temperature thermal conductivities in the range of 3,000 to 4,000 W/m·K, surpassing most known materials. However, substrate interactions, defects, and grain boundaries can significantly reduce this value. For example, supported graphene may exhibit thermal conductivities below 1,000 W/m·K due to phonon leakage into the substrate. The size effect is also critical—larger samples experience increased phonon scattering at grain boundaries, while smaller flakes may approach the intrinsic limit if defect densities are minimized. The interplay between sample dimensions and defect distribution underscores the challenges in achieving theoretical performance in practical applications.

Quantum confinement effects introduce additional complexity when comparing idealized graphene to real-world systems. In infinite sheets, electrons behave as massless Dirac fermions with linear energy dispersion, leading to high mobility and ballistic transport over micrometer-scale distances. However, confinement in one or two dimensions—such as in nanoribbons or quantum dots—alters these properties dramatically. Narrow ribbons develop bandgaps inversely proportional to their width, transitioning graphene from a semimetal to a semiconductor. Quantum dots exhibit discrete energy levels, further modifying optical and electronic responses. These effects highlight the sensitivity of graphene's properties to dimensional constraints, emphasizing the need for precise control over sample geometry in experimental settings.

The contrast between idealized predictions and real-world performance is stark. Theoretical models often assume pristine, defect-free graphene with infinite extent, while actual samples contain impurities, structural defects, and substrate interactions that degrade performance. For instance, while theory predicts near-ballistic transport at room temperature, scattering from charged impurities, surface roughness, and phonons limits mean free paths in practice. Similarly, thermal conductivity is hindered by phonon-boundary scattering and interfacial effects, preventing realization of the full theoretical potential. Even mechanical strength, though still exceptional, falls short of the ideal due to unavoidable defects introduced during synthesis or transfer processes.

Sample size further exacerbates these discrepancies. Larger-area graphene films, necessary for many applications, are more prone to defects such as cracks, wrinkles, and grain boundaries. These imperfections act as scattering centers for electrons and phonons, reducing mobility and thermal conductivity. Smaller samples, while potentially closer to ideal, are often impractical for scalable applications. The trade-off between size and quality presents a significant challenge in harnessing graphene's full potential.

In conclusion, graphene's intrinsic properties—strength, electron mobility, and thermal conductivity—represent the upper limits of what is physically achievable in a two-dimensional material. However, real-world performance is invariably lower due to defects, substrate interactions, and quantum confinement effects. The relationship between sample size and property degradation underscores the importance of material quality and dimensional control in bridging the gap between theory and practice. While graphene remains a material of immense promise, realizing its full potential requires continued advances in synthesis, defect mitigation, and integration techniques.