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Article Content
Highlights
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Rotation breaks the degeneracy of Landau levels.
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Sagnac effect alters thermodynamic properties of a rotating 2DEG.
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modifies magnetization and MCE behavior.
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Clockwise rotation with may induce cooling.
Abstract
This work investigates the impact of the Sagnac effect on the thermodynamic properties of a non-interacting two-dimensional electron gas (2DEG) in a rotating sample under the influence of a uniform magnetic field. We derive an analytical expression for the energy spectrum using an effective Hamiltonian incorporating inertial forces; we apply canonical ensemble statistical mechanics to evaluate thermodynamic quantities. The results show that rotation modifies the energy levels, the application of a magnetic field leads to the formation of Landau levels further altered by rotation and gravitational mass, and thermodynamic quantities (internal energy, specific heat, free energy, entropy, magnetization, and magnetocaloric effect) exhibit a strong dependence on these parameters. In particular, the difference between effective mass and gravitational mass influences magnetization and the magnetocaloric effect, with negative rotations potentially inducing a cooling effect when these masses are distinct. We conclude that rotational effects and effective mass properties are crucial for understanding the thermodynamics of electronic systems under magnetic fields, with implications for thermal modulation in semiconductor materials.
Graphical abstract
Introduction
The Sagnac Effect [1], [2], [3] is a fundamental physical phenomenon with profound implications for both theoretical physics [4], [5], [6] and technological applications [7], [8]. Its sensitivity to rotation, gravity, and the properties of matter makes it a valuable tool for investigating crucial questions in fundamental physics, such as the nature of dark matter [9] and gravitomagnetic effects [10], while also serving as the basis for high-precision rotational sensors used in various fields [11], [12]. The continuous development of Sagnac interferometers, both optical [13] and matter-wave-based [14], promises significant advancements in our understanding of the universe and sensing technologies.
The Sagnac effect is closely related to inertial forces, as it can be used to study and determine them. Inertial forces, such as the Coriolis and Euler forces, manifest in rotating or non-uniformly rotating reference frames. Inertial effects play a crucial role in quantum mechanics [15], [16], [17], [18]. The well-established analogy between inertial forces on massive particles and electromagnetic forces on charged particles provides an insightful perspective [19]. The Coriolis force, for instance, acts on a particle with mass in a manner analogous to the magnetic force on a charged particle. In contrast, the centrifugal force affects the particle within a rotating reference frame. For a spinless particle, the combined action of Coriolis and centrifugal forces in the quantum Hamiltonian results in a coupling between the particle’s angular momentum and the system’s rotation [19], [20]. Even in the absence of the centrifugal force, the Landau levels in the system still exhibit coupling between the particle’s angular momentum and rotation. Furthermore, if the rotation changes steadily over time, the Euler force must also be accounted for in the analysis.
Among the quantum systems where rotational and electromagnetic effects become particularly relevant, the two-dimensional electron gas (2DEG) stands out as a model system with rich physics and technological potential [21]. A 2DEG is a system in which electrons are confined to move in only two spatial dimensions, typically formed at the interface of semiconductor heterostructures such as GaAs/AlGaAs, or in devices like metal–oxide–semiconductor field-effect transistors (MOSFETs) [22]. The high mobility of these systems enables the observation of quantum phenomena such as the Quantum Hall Effect [23], making them essential for the study of low-dimensional systems and strongly correlated electrons [24]. Under strong magnetic fields, these systems exhibit remarkable quantum properties, including conductance quantization and the formation of Landau levels [25]. They have shown significance in optoelectronic applications as well [26]. Developing semiconductor nanomaterials and exploring quantum effects at the nanoscale are driving significant advancements in high-performance devices such as efficient solar cells, color micro-LEDs, advanced sensors, and innovative 3D displays [27].
When considering electrons in materials, an intriguing question arises regarding the distinction between three types of mass: inertial, gravitational, and effective mass. Inertial mass reflects an object’s resistance to changes in its motion, while gravitational mass relates to the gravitational attraction between bodies. According to the equivalence principle in general relativity, inertial and gravitational masses are equivalent. In contrast, the effective mass, which describes an electron’s response to electromagnetic forces within a material, can deviate from both inertial and gravitational masses, depending on the material’s structure [28]. The effective mass is derived from the curvature of the material’s energy band structure and can vary significantly, largely independent of gravitational effects. Despite this variability in the effective mass due to electromagnetic interactions, the gravitational mass of electrons remains constant, regardless of the medium in which they are situated.
According to [28], both effective mass and gravitational mass should be considered within the Schrödinger equation. For electrons in semiconductors, these masses differ, which can lead to notable shifts in energy levels and, consequently, in the physical properties of the system under study [29]. Although [28] does not explicitly frame the discussion, the work conceptually addresses the distinction between these three masses, emphasizing that while gravitational mass is associated with an accelerated reference frame, effective mass governs electron behavior within materials. Notably, the equivalence principle does not directly compare gravitational and effective masses; a conceptually distinct test would be required to evaluate them.
Our aim is to investigate thermodynamic properties and the magnetocaloric effect (MCE) for a two-dimensional electron gas (2DEG) subject to inertial effects associated with the Sagnac effect. Since two models are presented in the literature, we will analyze both contexts as mentioned above. We will follow the general model, for which , obtaining the equivalent results simply by considering these masses equal.
In our understanding, although the arguments presented in Ref. [28] are strong, we still believe that the effective theory for the case should be investigated through first-principles calculations. However, experiments and/or simulations involving thermodynamic properties could shed light on this matter.
This work is organized as follows. In Section 2, we derive the energy levels for a rotating 2DEG under a uniform magnetic field applied perpendicular to the plane of rotation. In Section 3, we introduce the thermodynamic properties to be analyzed. Section 4 presents the results and a detailed discussion of their implications. Finally, our conclusions are summarized in Section 5.
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Section snippets
Energy levels for a rotating 2DEG with a uniform magnetic field applied perpendicular to the rotating plane
Ref. [28] addresses the distinction between inertial, gravitational, and effective mass. Inertial mass, tied to an object’s resistance to motion, differs from gravitational mass, which relates to gravitational attraction, as the Equivalence Principle describes. While gravitational and inertial masses are equivalent in general relativity, effective mass pertains to how electrons respond to electromagnetic forces in materials. For semiconductors, the effective mass is derived from the
Partition function
Although the system is a two-dimensional electron gas (2DEG), using the Boltzmann partition function is justified due to the statistical distribution of energy states modified by rotation and the magnetic field. However, it can still be treated within the canonical formulation of statistical mechanics. Furthermore, the approach assumes that electron interactions are negligible, making the application of the Boltzmann distribution valid. It is also worth noting that, in this study, we consider
Results and discussions
In all subsequent analyses, we consider the thermodynamic quantities mentioned above as functions of temperature () for magnetic fields T, T, T, and T. For each of these cases, curves corresponding to different values of the rotation frequency ( and ) are presented alongside the non-rotating case (). Initially, we will analyze the scenario where , followed by the case where . Note that the case T is not displayed for the latter scenario
Concluding remarks
This work investigated the thermodynamic properties of a two-dimensional electron gas (2DEG) in a rotating medium under the Sagnac effect and a uniform magnetic field. The focus was on analyzing the impact of rotation and the distinction between effective mass () and gravitational mass () on energy levels and the system’s thermodynamic responses.
The results show that rotation, even in the absence of a magnetic field, breaks the degeneracy of electronic states, and its presence significantly
CRediT authorship contribution statement
Cleverson Filgueiras: Writing – original draft, Supervision, Software, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. Moises Rojas: Writing – review & editing, Visualization, Validation, Investigation. Vinicius T. Pieve: Software, Investigation. Edilberto O. Silva: Writing – review & editing, Visualization, Validation, Software, Investigation.
Funding
This work was partially supported by the Brazilian agencies, CNPq, FAPEMIG, and FAPEMA: C. Filgueiras and M. Rojas acknowledge FAPEMIG Grant No. APQ 02226/22. C. Filgueiras acknowledges CNPq Grant No. 310723/2021-3, and M. Rojas acknowledges CNPq Grant No. 317324/2021-7. Edilberto O. Silva acknowledges the support from grants CNPq/306308/2022-3, FAPEMA/UNIVERSAL-06395/22, and FAPEMA /APP-12256/22. V. T. Pieve thanks for the master’s scholarship provided by FAPEMIG.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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