Entropy-Seebeck ratio as a tool for elementary charge determination

The gist: The ratio of differential entropy per particle to the Seebeck coefficient converges to the elementary charge across an energy window in gapped armchair silicene nanoribbons, establishing a thermodynamic alternative for estimating charge carriers.

Theoretical Framework

The investigation explores the relationship between the Seebeck coefficient (S) and differential entropy per particle (DEP, s) as a tool for characterizing charge carriers in two-dimensional systems using armchair silicene nanoribbons as a model platform The Seebeck coefficient captures transport properties through the energy dependence of the electronic transmission, while the DEP is directly connected to the system’s electronic entropy.

The paper establishes a direct link between s and S by comparing their definitions derived from thermodynamic and transport perspectives. Specifically, the comparison of Eq. (3) and Eq. (4) generates an explicit correspondence between s and S, telling us that one can compute/measure a pure value of e through the ratio s(µ, T)/S(µ, T)=e. This relationship holds whenever the spectral weights involved in s and S are proportional or vanish within the same energy window.

System and Model

The systems under study are rectangular armchair silicene nanoribbons, electrically contacted to semi-infinite leads of the same material. The electronic states in the central region are described by a single π-orbital tight-binding Hamiltonian. Electronic transport is treated within the nonequilibrium Green’s function formalism combined with decimation techniques.

Key aspects of the model include:

**)&HC = X i εic † i ci − t X ⟨i,j⟩ c † i cj + h.c., where the on-site energy εi at site i and the second term represents the nearest-neighbor hopping with amplitude t = 1.6 eV. The ribbon is metallic when N = 3p + 2 (p ∈ Z), and semiconducting otherwise. The example shown corresponds to N = 10, i.e., a semi-Fig. 1. The energy range is limited to ε ≤ 5 eV. The metallic 8-ASiNR shows linear dispersion about zero energy, resulting in a finite DOS and a single transmission channel about ε = 0. In contrast, the 10-ASiNR exhibits a bandgap around zero energy. The density of states (DOS) is given by D(ε) = − 1/π lim η→0+ Tr[Im G(ε + iη)]. For metallic ribbons, S vanishes around the energy region where the electronic transmission T = 1. In contrast, semiconducting ribbons exhibit pronounced dip–peak structures centered at µ = 0. These large Seebeck responses arise from the presence of an energy gap combined with temperature. The band gap is defined by N = 3p and N = 3p + 1 armchair families. The DOS and transmission are nearly zero within this energy window. The ratio s/S ≈ e holds for semiconducting armchair ribbon systems with N = 3p or N = 3p + 1 width condition. This alignment originates from the presence of an electronic gap, where T(ε) and D(ε) nearly vanish. The ratio s/S ≈ e is a consequence of specific spectral conditions in gapped systems that perfectly align transport and thermodynamic quantities within the ribbon gapped regime. This provides a practical route to estimate the elementary charge and gain insight into how clean a band gap is when the ratio shows a value of e. The authors selected a chemical potential of µ = −0.15 eV for numerical stability within the energy gap. Numerical computations at T < 200 K proved to be highly unstable. The solid line represents a linear fit to the data, from which the estimated value of the elementary charge is found to be approximately 2% lower than the real value. The ratio linearly decreases with temperature within the plateau for a fixed value of µ ≠ 0. All these responses demonstrate that s/S provides a direct estimate of the elementary charge at/around room temperature. The results show that s/S is mirror symmetric about µ = 0 eV, fluctuating between negative and positive values outside the band gap energy region, and stabilizing within it with a near flat plateau of value ≃ e. The ratio linearly changes with temperature within the plateau for a fixed value of µ ≠ 0. All these responses demonstrate that s/S provides a direct estimate of the elementary charge at/around room temperature. The ratio linearly decreases with temperature, deviating from its perfect value of 1e as T increases. These behaviors for the ratio s/S show that it is also possible to estimate charge fluctuations present in gapped 2D materials at finite temperatures. The ratio exhibits symmetric positive peaks near µ ≃ ±0.44 eV (N = 10), µ ≃ ±0.4 eV (N = 12), associated with a positive ratio of the first oscillations appearing just after the dip-peak curves for s and S. As µ approaches the conduction and valence band edges, symmetric negative minima appear due to the opposite signs for s and S. As µ goes to zero, the ratio s/S starts to increase, and a nearly flat plateau is reached for both ribbon widths.

1. Confidence in elementary charge determination through spectral analysis: The improved system can determine a pure value of e by calculating the ratio s/S converges to the elementary charge e across that energy window, fulfilling the Kelvin formula S = s/e, specifically by analyzing systems where s/S matches with e.

2. Predictive characterization of band gaps: The system can identify the cleanness of electronic band gaps as s/S matches with e, allowing it to distinguish between metallic and semiconducting regimes based on the calculated ratio.

3. Thermodynamic proxy for transport properties: The AI can use s as a robust and purely thermodynamic proxy for the Seebeck coefficient S in 2D systems with well-defined electronic gaps, enabling it to characterize charge carriers using a transport-thermodynamic framework.

4. Fluctuation analysis at finite temperatures: The improved system can calculate the ratio s/S for two ribbon widths N = 10 and N = 12, both as a function of chemical potential µ to assess its reliability for estimating the elementary electron charge at room temperature, revealing charge fluctuations present in gapped 2D materials at finite temperatures.

5. Material optimization via width control: The system can optimize thermoelectric performance by understanding that the reduction of the ribbon width—and hence the number of active conducting armchair channels—broadens the µ range over which the ratio s/S ≈ e holds.

In this work, we investigate the relationship between the Seebeck coefficient (S), and the differential entropy per particle (DEP, s), as a tool for characterizing charge carriers in two-dimensional systems. Using armchair silicene nanoribbons as a model platform, we analyze how both quantities and their ratio depend on chemical potential at room temperature. While the Seebeck coefficient captures transport properties through the energy dependence of the electronic transmission, the DEP is directly connected to the system's electronic entropy, offering a direct thermodynamic alternative for estimating S. We evaluate these transport-thermodynamic properties considering diverse ribbon widths, defining metallic and semiconducting regimes. We find both quantities S and s, are highly interconnected within the ribbon's band gap energy region, and their ratio s/S converges to the elementary charge e across that energy window, fulfilling the Kelvin formula S=s/e. On the contrary, s/S is undefined for gapless ribbons in the energy window of the first transmission channel. These results establish the ratio between the DEP and the Seebeck coefficient as a reliable and complementary probe for the determination of the elementary charge, and to identify the cleanness of electronic band gaps as s/S matches with e.