Dimensional transport crossovers in thermoelectrics revealed by a simple transport model

Domain: physics
Published: 2025-12-11
Editorial outcome: major_revision

Abstract
Quantum confinement gives low-dimensional materials distinctive electronic behaviour, but assessing their effective band structure dimensionality (D) is difficult. Conventional probes such as angle-resolved photoemission spectroscopy (ARPES) or scanning tunneling microscopy (STM) demand ultraclean surfaces and expensive facilities. We introduce a generalized transport model that functions as an internal dimensionality meter: by tracking how the Seebeck coefficient varies with carrier concentration or temperature, we deduce D from two scaling laws, S/lnT in non-degenerate regimes (e.g. |S|≥200 μVK−1) and S/n−2/D in degenerate regimes (e.g. |S|≤150 μVK−1). Applying this approach to SrTiO3, few-layer Bi2O2Se and Pb1-xSnxTe uncovers temperature-, doping- and alloy-induced crossovers between three-dimensional and lower-dimensional transport. The method offers a rapid, scattering independent framework to design quantum and thermoelectric properties.

Introduction
Low-dimensional materials have ascended to the forefront of materials science by virtue of their extraordinary electronic band structures, which can confer dramatically enhanced properties relative to their bulk analogs1–4. The reduction in dimensionality, from three-dimensional (3D) to two-dimensional (2D), one-dimensional (1D), or even intermediate confined regimes, fundamentally reshapes the density of states (DOS) through quantum confinement, thereby endowing materials with distinctive electronic, optical, and mechanical characteristics5. Such advantages hold particular promise for next-generation electronic devices6, sensors7, and energy conversion technologies8, where performance breakthroughs hinge upon fine control of carrier dynamics at the nanoscale.

A rich array of low-dimensional systems illustrates the transformative impact of reduced dimensionality. For instance, transition metal dichalcogenides (TMDs), such as monolayer Molybdenum disulfide (MoS2) feature direct bandgaps and pronounced excitonic effects, rendering them ideal platforms for high-efficiency optoelectronics9. Likewise, few-layer black phosphorus combines high carrier mobility with in-plane anisotropy, enabling its integration in advanced electronic and photonic architectures10. Even in ostensibly 3D materials, such as strontium titanate (SrTiO3), quantum confinement in thin-film or interfacial configurations drives emergent 2D transport behavior11,12, underscoring the pervasiveness of dimensional tuning across diverse material families. In all these cases, reduced dimensionality amplifies quantum confinement and electron–electron correlations13,14, enabling an unprecedented level of control over material properties. Importantly, the inherent quantum confinement in low-dimensional materials could modify the DOS near band edges, thereby altering carrier mobilities, electrical conductivities, and thermoelectric (TE) properties.

Despite the potential of dimension-driven tuning for engineering high-performance materials, reliably determining the effective dimensionality of the electronic band structure in low-dimensional materials remains nontrivial4. Traditional characterization techniques, such as angle-resolved photoemission spectroscopy (ARPES)15,16 and scanning tunneling microscopy (STM)17,18 have been employed to probe electronic structures. However, these methods often require complex instrumentation, operate under ultra-high vacuum conditions, and demand pristine surface quality, such as atomic-level smoothness19. This has motivated the pursuit of transport-based approaches that can probe electronic dimensionality under more accessible experimental conditions.

Typically, the analysis of Seebeck coefficient S and electrical conductivity σ involves both electron scattering and electronic structure considerations. Conductivity is strongly influenced by scattering mechanisms, whereas the Seebeck coefficient predominantly reflects the entropy per charge carrier—a thermodynamic quantity directly related to electronic dimensionality through the DOS20. However, clearly separating these two contributions is inherently ambiguous, often leading to reliance on assumptions about scattering and electronic structure that rarely align exactly with experimental conditions.

In this work, we introduce a broadly applicable strategy to discern the effective electronic dimensionality independently of any assumptions regarding scattering mechanisms. Our approach relies on measurements of Seebeck coefficient S and electrical conductivity σ as a function of carrier concentration n. By applying a weighted mobility model21–23, we examine how dimensional confinement modifies the electronic transport behavior. Interestingly, our results show that doping, temperature T, and alloying can trigger transitions between 3D-like and lower dimensional-like behavior in various materials. This framework accelerates the design of systems with tailor-made electronic and TE properties for advanced technological applications.

Results
Simplified model development
Low-dimensional materials (1D, 2D) exhibit electronic properties that deviate markedly from those of conventional 3D semiconductors, primarily because of the dimensional dependence of the DOS. This evolution governs the carrier dynamics of the DOS for 3D, 2D, and 1D systems.

Many 3D semiconductors with roughly spherical Fermi surfaces feature a DOS, g3D(E), that scales with the square root of energy E and vanishes at the Fermi level24 (Fig. 1a). Mathematically,
g3D(E) = (2m*)3/2 / (2π2ℏ3) * sqrt(E) (1)
Where m* is the effective mass of charge carriers, ℏ is the reduced Planck constant.

In 2D materials, the Fermi surface usually adopts a cylindrical shape. The DOS in this case is constant and independent of energy, reflecting the reduced dimensionality (Fig. 1b):
g2D(E) = m* / (πℏ2a) (2)
Here, a represents the confinement width perpendicular to the 2D plane. The constant DOS implies a uniform availability of electronic states at different energies near the Fermi level. This constant DOS leads to unique electronic and optical properties that underpin cutting-edge applications in 2D semiconductor quantum wells25 and TMDs9.

In 1D systems, including nanowires and carbon nanotubes, the DOS exhibits a characteristic divergence (i.e., a van Hove singularity26) near the band edge (Fig. 1c)
g1D(E) = 1 / (a1a2) * sqrt(2m*) / (πℏ) * 1 / sqrt(E) (3)
where the factor 1/(a1a2) reflects confinement in two perpendicular directions. This divergence means that minor energy shifts can dramatically change the carrier density, a hallmark of reduced dimensionality exploited in sensitive optoelectronic devices27–31.

Because the DOS changes with dimensionality, the carrier concentration n defined by n = ∫ g(E)f(E)dE also exhibits unique dependencies on the position of the Fermi level EF. Here, f(E) = 1 / (1 + e^((E-EF)/kBT)) represents the Fermi-Dirac distribution function, where kB is the Boltzmann constant. These are captured by the following expressions (see Supplementary Information 1.1 for more details):
n3D = (2m*kBT / 2πℏ2)3/2 * F1/2(η) (4)
n2D = m*kBT / (πℏ2a) * F0(η) (5)
n1D = 1 / (a1a2) * sqrt(2m*kBT / πℏ2) * F-1/2(η) (6)
Here, the reduced Fermi-level η = EF/kBT, and Fi(η) is the Fermi-Dirac integral with the reduced energy ε = E/kBT
Fi(η) = ∫ (ε^i / (1 + e^(ε-η))) dε (7)
These integrals encapsulate how the occupancy of electronic states evolves as a function of T and Fermi-level position. Consequently, the distinct DOS in 3D, 2D, and 1D systems yield different carrier concentration profiles as a function of η.

Boltzmann transport theory, within the relaxation-time approximation, provides a powerful framework for analyzing TE and electronic transport properties in materials of various dimensionalities. Under this approach, the Seebeck coefficient S and electrical conductivity σ can be expressed in terms of the reduced Fermi level η, and the scattering exponents (or confining dimension in 2D/1D)22. Specifically,
S = kB/e * ((s+1)Fs(η) / sFs-1(η) - η) (8)
σ = σE0(T) * sFs-1(η) (9)
Here, σE0(T) is a temperature-dependent factor related to the intrinsic material properties and varies with the dimensionality of the system, s is a parameter corresponding to the system’s dimensionality and scattering mechanism (see Supplementary Information 1.3 and 1.4 for more details), and the Fermi-Dirac integral Fs-1(η) incorporates the energy distribution of carriers.

A practical transport toolkit can be built by exploiting the distinct fingerprints that dimensionality D and scattering physics leave on the Seebeck coefficient S in two complementary regimes.

In the non-degenerate regime (η<0, practically η≤ −1), the Pisarenko slope of S-n relation is universal (−kB/e) thus independent of scattering and dimensionality, whereas the intercept retains dependence on the scattering parameter r and D through s (See Section 2.2 and Equation S43 of the Supplementary Information for detail). Consequently, for 1D, 2D, and 3D systems, alike, S/ln(n−1), the S-n curve therefore cannot discriminate dimensionality. Instead, the temperature dependence is decisive (see Supplementary Information 2.2 for detail):
S(T) ∝ D/2 * lnT (e.g. |S|≥200 μVK−1) (10)
A single S(T) sweep thus serves as a thermometer for D.

In the degenerate regime (η>0, practically η≥1), S acquires a strong carrier-density signature that does resolve dimensionality (see Supplementary Information 2.1 for detail):
S(n) ∝ n−2/D (e.g. |S|≤150 μVK−1) (11)
Traditional Pisarenko plots measured at a fixed temperature therefore reveal D directly from the power-law exponent. Conversely, in the degenerate regime the Seebeck coefficient S scales linearly with temperature T for every dimensionality, so an S-T plot offers no leverage for distinguishing D, but differ depending on the scattering.

In this work, we adopt a curve-wise m* assumption within the effective-mass framework32,33. For each S-n data set at fixed T (or S-T at fixed n), we fit with a single m* that captures the local band curvature and serves as a transport descriptor. When a single m* fails to reconcile the data with the D-dimensional Pisarenko relation, the systematic deviation is interpreted as evidence of non-parabolicity, band modification (e.g., convergence or inversion), or a change in the effective D. Importantly, our inference of D relies on the scaling exponent of S vs n, which is insensitive to the absolute magnitude of m* (i.e., m* sets the prefactor, not the exponent).

Carriers scattering mechanisms, such as acoustic phonon, ionized impurity, piezoelectric, or polar optical phonon scattering, impose unique dependencies on carrier energy and temperature31,34–36. Once the dominant mechanism is established, the associated transport exponents may become an independent fingerprint of dimensionality. For example, for ionized-impurity and polar-optical-phonon scattering, each effective DOS dimensionality D maps onto a unique s value (Table 1). Experimentally, s can be extracted by measuring the Seebeck coefficient S and electrical conductivity σ while systematically tuning the carrier concentration at a fixed temperature, then fitting the joint S-σ response. The resulting s can be matched to the theoretical s(D) catalogue (Table 1), providing a rapid, self-consistent check on the dimensionality already inferred from S(T) or Pisarenko S(n) analysis. (A detailed explanation can be found in the Introduction of Section 2 and in Section 2.1 of the Supplementary Information.).

Combining these measurements allows one to establish the effective DOS dimensionality in a range of low-dimensional or nanostructured systems.

Case Studies
Using our transport model framework, we next analyze representative low-dimensional systems to illustrate how their electronic dimensionality and dominant scattering mechanisms are revealed.

Dimensional crossover from 3D- to 2D-like transport in heavily doped SrTiO3. A cross-analysis of six independent data sets11,37–41 (Fig. 2) exposes a universal dimensional crossover in SrTiO3 that is triggered once the carrier density surpasses n ≈ 1 × 1020 cm−3. Below this threshold, the transport is well described by a conventional 3D DOS. The Seebeck coefficient varies with temperature as 3/2 lnT and the Pisarenko relation follow the canonical S ∝ n−2/3 dependence. As the carrier density rises beyond this threshold, both signatures evolve in concert toward the behaviour expected for a 2D DOS. Specifically, the logarithmic prefactor in S(T) collapses to unity, yielding S ∝ lnT, while the slope of the S-n curve steepens to S ∝ n−1 (the detailed calculation procedure is described in Supplementary Information 3.1 and illustrated in Figs. S1 to S3).

Taken together, these observations argue that heavily doped SrTiO3 develops a Fermi surface that is effectively 2D, in line with recent DFT predictions of flattened t2g sub bands under strong filling42. This quasi-2D behavior arises from orbital-selective hybridization: in-plane Ti-dxy and O-p coupling yields dispersive bands, whereas out-of-plane (z) hybridization is symmetry-suppressed, producing weak dispersion and cylindrical Fermi-surface sections. The result cautions that bulk SrTiO3 cannot always be treated within a simple 3D parabolic band picture once carrier densities approach 1020 cm−3, and it underscores how modest chemical doping can unlock low dimensional transport physics without artificial confinement.

Temperature-triggered transition from 1D-like to 3D-like transport in few-layer bismuth oxyselenide (Bi2O2Se). Bismuth Oxyselenide (Bi2O2Se) features a layered crystal structure consisting of alternating [Bi2O2]2n+ layers and [Se]2n- layers, held together by weak electrostatic interactions43. Recent studies by Yang et al. have investigated the carrier scattering mechanisms in Bi2O2Se flakes to be polar optical phonon/piezoelectric scattering mechanisms and their impact on TE performance44.

In this study, our model uncovers that the electron transport behavior undergoes dimensional evolution. As the T increases, transport behavior transitions from a 1D to higher-dimensional regime. This transition is discovered by analyzing the relationship between S and n (detailed calculation procedures can be found in Supplementary Information 3.2, along with Figs. S4 and S5). At 60 K, a 1D transport model fits the S-n data (Fig. 3b), whereas at 100 K, a 2D model is more appropriate (Fig. 3c). By 300 K, a 3D model best describes the S-n relationship (Fig. 3d). The change in transport exponent s further confirmed this transition. The S–σ relationship follows the s=1 relation at 60 K, s=1.5 at 100 K, and s=2 at 300 K. Assuming optical phonon/piezoelectric scattering mechanism thoughout, this observation indicates that the transport behaviour shifts from 1D to 3D as T increases (Table 1). This effect may originate from its layered crystal/electronic anisotropy: weak interlayer coupling along the out-of-plane direction produces a highly anisotropic Fermi surface and effective mass (nearly isotropic in-plane but elongated along kz)45. At low temperature, suppressed interlayer hopping confines carriers within the layers, yielding quasi−1D-like scaling; with increasing T, phonon-assisted interlayer transport progressively activates transverse conduction, driving a 1D→2D→3D evolution. These temperature-induced dimensional transitions expand our understanding of carrier dynamics in Bi2O2Se and highlight novel routes for property optimization.

Topological phase transition and emergent 1D-like transport in Pb1-xSnxTe alloy. Topological insulators (TIs) represent a unique class of quantum materials, characterized by metallic surface states coupled with insulating bulk electronic structures46,47. These properties offer significant promise for applications in next-generation electronic devices48. Remarkably, many TIs, particularly Bi2Te3-based alloys, also exhibit outstanding TE performance and are widely employed in commercial TE technologies49. Recent studies suggest that bulk electronic structures in TIs can induce novel TE behaviors. In particular, band inversion and the associated band warping in TIs can increase valley degeneracy and effective carrier pocket multiplicity50,51, enhancing the Seebeck coefficient S and power factor S2σ. Despite these promising insights, transport phenomena originating within the bulk states of TIs remain relatively understudied, especially in the context of unconventional TE effects driven by topological electronic structure transitions in the bulk electronic structure.

Here, we investigate the transport behavior across the topological phase transition in Pb1-xSnxTe, using our generalized transport model, supported by both previously reported data52–60 and systematic experiments performed in this work (further experimental details are provided in Method section). This alloy system spans from the normal insulator PbTe (trivial band ordering) to the topological crystalline insulator SnTe (inverted bands), providing an ideal platform to study how TE properties evolve with changes in band topology. As the alloy approaches a critical composition (xc ≈ 0.6), we observe a striking deviation from ordinary 3D transport toward a 1D-like transport regime (Fig. 4a). In compositions well away from xc, the S versus n follows the expected 3D trend S ∝ n−2/3 even as the effective mass (m*/me) varies with alloying (Fig. 4b). Near the critical point, however, the S-n relationship steepens to S ∝ n−2, closely resembling 1D transport behavior and clearly deviating from the 3D model (Fig. 4a). This motivates fitting with the 1D transport model, from which a self-consistent (curve-wise constant) 1D m* is extracted. This dimensional crossover is accompanied by a dramatic drop in the apparent effective mass: at x ≈ 0.6, m*/me is roughly three orders of magnitude lower than in the 3D regime (Fig. 4b) (further calculation details are provided in Supplementary Information 3.3 and Fig. S6). Such an abrupt reduction in m*/me is consistent with the emergence of massless or light Dirac-like carriers at the topological transition61. This behavior supports, rather than contradicts, the conclusion that transport channels exhibit 1D-like characteristics in the vicinity of the topological transition.

At the critical composition, the bulk band gap of Pb1-xSnxTe narrows to zero and band inversion occurs at the L-points of the Brillouin zone, resulting in the formation of linear Dirac cones (a hallmark of the topological phase transition)50,61. This inversion marks the point where the alloy’s electronic structure is reorganized from normal to topological. We hypothesize that the unusual 1D-like transport observed in the bulk Pb1-xSnxTe at this transition originates from topologically derived conducting channels that effectively reduce the dimensionality of charge transport. In other words, as the system’s bulk bands invert, new conducting states associated with the topological phase boundary can arise and carry current in a lower-dimensional manner than the 3D bulk.

One plausible scenario is the appearance of topologically protected boundary or edge states along internal interfaces or defects. For instance, microscopic regions of trivial and inverted band order may coexist near the transition (due to alloy fluctuations or strain), and the interfaces between these regions would host 2D topological surface states62. If these interfaces percolate or form networks, charge carriers might preferentially flow along them. In extreme cases, e.g. along line defects or along the intersection of facets, these percolating pathways could behave like quasi-1D conductors embedded in the 3D material63,64. This provides a reasonable explanation for the 1D-like S ∝ n−2 scaling: the transport is dominated by carriers confined to filamentary or stripe-like topological conductive paths, rather than the bulk 3D conduction band. Notably, analogous phenomena have been reported in other topological systems, lending credence to our interpretation. Bismuth antimony alloys (Bi1-xSbx) undergo a well-known topological band inversion; remarkably, theory and experiments have shown that in the inverted regime, crystal dislocations can host 1D helical modes that conduct along the dislocation line in the bulk crystal65. These 1D topological defect states (predicted in weak TIs) effectively act as internal wires with quantum-protected transport, and they have indeed been suggested to contribute to the electrical conduction in Bi1-xSbx crystals near the topological transition. Likewise, elemental bismuth, recently identified as a higher-order topological insulator, features conducting hinge states 1D channels running along the edges where crystal faces meet66. These topological hinge modes allow electrons in Bi to travel ballistically along the crystal edges, which helps explain Bi’s unusually long carrier mean free path and ability to conduct with very low carrier densities. The existence of such 1D conduction pathways in nominally 3D topological materials underscores that dimensionality of transport can be intrinsically altered by topology.

By analogy, we suggest that in Pb1-xSnxTe, the closing of the bulk gap and onset of band inversion create conditions for 1D-like transport channels. These may arise from topology-enabled line, hinge, or interface modes, or from percolating networks of locally inverted and trivial regions that can dominate conduction at the critical composition xc. Importantly, the 3D to 1D scaling appears not only in polycrystals studied but also in single-crystal datasets near xc (e.g., Orihashi et al.55), indicating that the effect does not rely on macroscopic crystallinity. Microstructure chiefly modulates the density and connectivity of these pathways, and thus the amplitude of the 1D-like signature, but is not required for their existence; high-quality single crystals may suppress extrinsic networks yet still exhibit intrinsic 1D-like scaling in the vicinity of xc.

Crucially, the emergence of these 1D-like pathways at xc offers a direct performance benefit: the effective transport dimensionality reduction leads to an enhanced Seebeck coefficient at a given carrier concentration (since S in a 1D-like system can achieve larger values than in 3D, for comparable n below 6×1019 cm−3) (Fig. 4a). In tandem with the ultralow effective mass of the Dirac-like bulk carriers, this drives a marked improvement in the TE quality factors. Indeed, compositions near x ≈ 0.6 are observed to exhibit significantly boosted TE performance52.

We propose that this enhancement originates from the topologically driven changes in the electronic structure; essentially, the material taps into high-mobility, low-dimensional transport channels at the transition.

This finding not only deepens our understanding of how topological phase transitions can influence bulk transport properties but also suggests a novel strategy for engineering superior TEs. By tuning a material to the brink of a topological phase change (through alloying, strain, or other means), one can invoke these protected, high-conductivity 1D-like pathways and achieve an optimal combination of high Seebeck coefficient and mobility.

In summary, the Pb1-xSnxTe alloy at its critical composition xc ≈ 0.6 demonstrates an intrinsic quasi-1D transport mode arising from a topological electronic transition. This discovery opens new avenues for exploiting topological transitions as a tool for designing TE materials, potentially guiding the development of next-generation TE materials with unprecedented performance through the controlled dimensionality of electronic transport.

Assessing electronic dimensionality in organic semiconductors: PBTTT as a model system. Despite the well-known practical difficulty of obtaining reliable Hall carrier densities in organic conductors67,68, we assessed the applicability of our framework using the extensively studied Poly[2,5-bis(3-dodecylthiophen-2-yl)thieno[3,2-b]thiophene] (PBTTT) as a model system. We analyzed a few literature datasets69–72 that report both S and n (Fig. 5). PBTTT shows a scattering evolution from s=3 to s=1 with increasing doping69–71 (Fig. 5a), implying a transition toward acoustic-phonon dominated transport22 and compatible with coherent, quasi-2D metallic behavior73,74 at high conductivity (σ≥200 Scm−1).

At the single-study level, some datasets (e.g., Chen69,70 (Fig. 5c, e) and Kang71 (Fig. 5d)) exhibit strong 2D-like trends, whereas others (e.g., Wang72 (Fig. 5f)) appear 3D-like. To mitigate dataset-specific Hall uncertainties, we pooled all points at 300 K (Fig. 5b). The aggregated data largely occupy the overlap region between our 3D- and 2D-discriminative bands, so we cannot claim a universal 2D assignment for PBTTT at high doping.

We infer that the apparent dimensionality in organics is highly sensitive to disorder, such as set by intrinsic microstructure, dopant distribution, and processing history (e.g., crosslinking), as highlighted in prior studies75–78. Establishing dimensionality robustly in organics will therefore benefit from independent carrier-density benchmarks (e.g., coulometry, IR sum-rules, or gate-capacitance areal densities) and cross-checks with microscopic probes where available.

Methods
Theoretical analysis
A simplified transport model was established to describe the dependence of the Seebeck coefficient on temperature S(T) or carrier concentration S(n) across different electronic dimensionalities. The model assumes a single dominant band and employs the relaxation-time approximation with power-law scattering. Full derivations and parameter details are provided in the Supplementary Information.

Materials synthesis
Samples with nominal compositions of Pb1-xSnxTe (x=0.1–0.9, Δx=0.1) were synthesized using high-purity Pb, Sn, and Te elements. Stoichiometric mixtures, each weighing approximately 8 g, were loaded into quartz tubes, evacuated to a residual pressure of 10−4 mbar, and sealed. The sealed tubes were then heated gradually to 1100 °C over 11 h, maintained at this temperature for 12 h to ensure complete melting and homogeneity, cooled to 900 °C over 2 h, held at this intermediate temperature for 6 h to facilitate phase stabilization, and subsequently furnace-cooled to room temperature.

The resulting ingots were mechanically ground into fine powders using ball milling. The powders were then densified via spark plasma sintering (SPS) at 500 °C for 5 min under an axial compressive pressure of 40 MPa in vacuum, producing high-density, polycrystalline bulk samples.

Transport measurements
Electrical conductivity and Hall coefficient measurements were performed using the standard 4-point Van der Pauw method under a 0.8 T magnetic field in a high-vacuum environment79. The Seebeck coefficients were measured by establishing a temperature gradient of approximately ±5 K across the sample, employing chromel-Nb thermocouples for accurate temperature detection80.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work introduces an innovative transport framework that utilizes thermoelectric measurements as an internal probe for characterizing electronic dimensionality. By establishing scaling relations for the Seebeck coefficient with carrier density and temperature, the method enables sensitive detection of dimensional crossovers without requiring sophisticated surface-sensitive techniques. It provides a rapid and scattering-independent approach to explore quantum confinement effects across various material systems, offering valuable insights for the development of thermoelectrics and other electronic applications. However, I recommend a major revision for this manuscript based on the following critical issues:\n1. In this manuscript, the model proposed by the authors is premised on the assumption of a constant effective mass. However, many thermoelectric materials exhibit an effective mass that varies with temperature or the position of the Fermi level. Therefore, the authors should explicitly state this assumption in the manuscript. Additionally, while analyzing the dimensional transition transport behavior in Pb₁₋ₓSnₓTe, the authors mention a three-order-of-magnitude decrease in the effective mass at x ~0.6. Does this significant change affect the conclusion that these samples exhibit 1D-like transport characteristics?\n2. The relationship between S and T provided by the authors under non-degenerate conditions is based on the assumption that n remains constant. This premise should be explicitly stated in the manuscript. In analyzing the dimensional transition transport behavior in SrTiO₃, did the authors consider the possible temperature-dependent variation of n? Furthermore, what values of the scattering parameter r were used in calculating η of SrTiO₃ (Figs. S1 and S2)? Was the possible temperature dependence of the scattering parameter taken into account? Additionally, η shown in Figs. S1 and S2 indicates that SrTiO₃ is in a non-degenerate state, however, the authors used a relationship between S and n derived under degenerate conditions to demonstrate the existence of dimensionally transition transport in SrTiO₃. This methodological inconsistency may undermine the validity of the conclusions. In summary, the argument regarding dimensionally transition transport phenomena in SrTiO₃ seems to lack rigorous logical consistency.\nAdditionally, the manuscript has several minor concerns:\n1. At line 143, the authors stated that the S-n relationship is independent of the scattering mechanism. This is not accurate.\n2. At line 179, could the authors clarify the motivation for mentioning the weighted mobility model? It appears somewhat abrupt.\n3. Could the authors explain the reason for the dimensional transition in electrical transport behavior observed in SrTiO3 and Bi2O2Se?\n4. The authors explained the reason for the transition from conventional 3D transport to a quasi-1D transport mode in Pb₁₋ₓSnₓTe polycrystalline materials, emphasizing the importance of grain boundaries and defects in enabling this transition. Would this imply that higher-quality crystalline materials would not exhibit this phenomenon? If so, experimental data validating this hypothesis would significantly strengthen the authors' conclusions.\n5. It would be helpful if the authors could expand the conclusion with a discussion on the general applicability, potential limitations, and future improvement directions of the proposed method.\n6. Some references lack DOI links, while others contain misassigned DOIs (e.g., [46], [52]). Additionally, Chinese characters appear in reference [28]. Please thoroughly review all citations.", "reviewer_2": "The authors develop a simple model for the relationship between the thermoelectric Seebeck coefficient and conductivity that takes into account carrier concentration or temperature and sample dimensionality. They conclude that the Pisarenko scaling law S ~ ln(n) works at low to moderate doping while at high doping and in degenerate materials, the scaling law becomes dependent on the dimensionality. The authors compare the simple model to numerous experimental data sets and demonstrate that the simple model can guide the analysis of experiments, including the prediction of unexpected reduced dimensionality in some cases. For this reason, I support the work for publication--simple models can be useful when they provide insight, which is the case in this manuscript.\nHowever, what is not clear to me from the manuscript is how the energy dependence of the scattering rate plays into the results. Dimensionality primarily affects the exponent of the energy dependence of the DOS, through which dimensionality makes its way into the energy dependence of the transport distribution functon(TDF). The TDF is then also dependent on the scattering rate and its exponent of energy dependence. The manuscript apparently shows the opposite---the S-n scaling in degenerate regime and the S-T scaling in the non-degenerate regime are independent of the scattering rate and only depend on dimensionality (in other words depend on D but not r). I think this is because S-sigma plots only show a dependence on s=r+D/2 in the degenerate (eta>>1) regime but the S vs n plot does not, and are independent of s at the low and medium doping. But I wish the manuscript would explain this a little better---how do we end up having S-sigma show a dependence on s (so depend on the r of the scattering rate) but the S-n plot does not depend on s through r, only on D. It seems that the r dependence of conductivity and mobility are canceling, leaving S vs n independent of r?\nThe other comment is that the S vs sigma going as ln(sigma) has been demonstrated for organic materials (see for example https://doi.org/10.1557/s43579-024-00628-2) and the Kang-Snyder model that is very analogous to the model in this paper has also been shown to work for organic materials. Would the conclusions given in this manuscript about dimensionality extend to organic materials as well? In organics, the Mott formula relates temperature dependence of conductivity to dimensionality, but it is only valid for a constant density of states, which is violated in virtually all organic materials.\nMinor error/typo: line 173 where it says Session 1.2 it should be Section 1.2.", "reviewer_3": "Zhang X.X., et al. introduced a generalized transport model that functions as an internal dimensionality meter. The method provided a scattering independent framework to design lower-dimensional transport materials. This work should be of widespread interest to researchers in the fields of materials and thermoelectrics.\nI am not an expert in this field, so I cannot verify the detailes. There doesn't seem to be any major concerns. I recommend to publish it in the current form."}, "author_response": "Dear Reviewers:\nWe sincerely thank the Reviewers for their careful reading of our manuscript and for providing insightful and constructive comments. We have carefully considered all suggestions and have revised the manuscript accordingly. Each comment has been addressed point by point in this response letter. In the revised manuscript, all changes are highlighted in yellow for clarity. We believe that these revisions have significantly improved the clarity and quality of our work.\nReviewer #1 ............................................................................................................................... 2\nReviewer #2 ............................................................................................................................. 21\nReviewer #3 ............................................................................................................................. 32\nReviewer #1\nThis work introduces an innovative transport framework that utilizes thermoelectric measurements as an internal probe for characterizing electronic dimensionality. By establishing scaling relations for the Seebeck coefficient with carrier density and temperature, the method enables sensitive detection of dimensional crossovers without requiring sophisticated surface-sensitive techniques. It provides a rapid and scattering-independent approach to explore quantum confinement effects across various material systems, offering valuable insights for the development of thermoelectrics and other electronic applications. However, I recommend a major revision for this manuscript based on the following critical issues:\nComment 1: In this manuscript, the model proposed by the authors is premised on the assumption of a constant effective mass. However, many thermoelectric materials exhibit an effective mass that varies with temperature or the position of the Fermi level. Therefore, the authors should explicitly state this assumption in the manuscript. Additionally, while analyzing the dimensional transition transport behavior in Pb₁₋ₓSnₓTe, the authors mention a three-order-of-magnitude decrease in the effective mass at x ~0.6. Does this significant change affect the conclusion that these samples exhibit 1D-like transport characteristics?\nResponse: We thank the reviewer for highlighting the role of an energy- and temperature-dependent effective mass. We agree that in many thermoelectric semiconductors the apparent effective mass can vary with temperature or Fermi-level position. In our analysis, the constant assumption is applied curve-wise: for each isothermal (or isodoping) data set we use a single, self-consistent m* as a transport descriptor of the local band curvature. This is standard within the effective mass framework and provides a clear diagnostic: if a single m* cannot reconcile a data set with the D-dimensional Pisarenko relation, then the deviation signals non-parabolicity, band evolution (e.g., convergence/inversion), or a change in the effective transport dimensionality (Adv. Funct. Mater. (2022) 32, 2112772; arXiv (2017) 1710.06896).\nCrucially, our identification of dimensionality relies on the scaling exponent of S with n (the Pisarenko slope), which is robust to the absolute magnitude of m*. The value of m* and scattering parameter (s) sets the prefactor, whereas the exponent is set by D (detailed in Section 2.1 of Supplemental Material).\nThus, even when m* varies modestly with composition or temperature, the inference of D from the S-n slope remains valid.\nFor Pb1-xSnxTe, near the topological critical composition (x = 0.6), the S-n relation steepens from the 3D trend to approximately S ∝ n^-2/3, which is the hallmark of 1D-like transport. When we fit the data with the 1D model, the extracted (curve-wise constant) m* is small, physically consistent with the emergence of light/Dirac-like carriers as the band gap closes at band inversion (J. Phys. Chem. Solids (2023) 183, 111635. J. Appl. Phys. (2008) 103, 053710. Mol. Syst. Des. Eng. (2019) 4, 407–420. J. Am. Chem. Soc. (2018) 140, 13097–13102; Innovation (2025) 6, 100782). Therefore, the pronounced change in m* does not undermine the dimensionality conclusion; rather, it is consistent with the physics of the topological transition and supports the 1D-like transport behavior inferred from scaling.\nWe have revised the manuscript to (i) state explicitly that our model adopts a curve-wise constant assumption, (ii) clarify how deviations from this assumption serve as a diagnostic of band or dimensionality changes, and (iii) explain why the small m* near x = 0.6 in Pb1-xSnxTe is expected and does not affect the conclusion of 1D-like transport.\nRevision made:\nIn “Experimental strategy for identifying electronic dimensionality” Session\nTraditional Pisarenko plots measured at a fixed temperature therefore reveal D directly from the power-law exponent. Conversely, in the degenerate regime the Seebeck coefficient scales linearly with temperature for every dimensionality, so an S-T plot offers no leverage for distinguishing D, but differ depending on the scattering.\nAssumption and diagnostic. In this work we adopt a curve-wise assumption within the effective-mass framework (Adv. Funct. Mater. (2022) 32, 2112772; arXiv (2017) 1710.06896). For each data set at fixed T (or at fixed n), we fit with a single m* that captures the local band curvature and serves as a transport descriptor. When a single m* fails to reconcile the data with the D-dimensional Pisarenko relation, the systematic deviation is interpreted as evidence of non-parabolicity, band modification (e.g., convergence or inversion), or a change in the effective D. Importantly, our inference of D relies on the scaling exponent of S vs n, which is insensitive to the absolute magnitude of m* (i.e., m* sets the prefactor, not the exponent).\nIn “Topological phase transition and emergent 1D-like transport in Pb1-xSnxTe alloy” Session\nIn compositions well away from xc, the S versus n follows the expected 3D trend even as the effective mass (m*/me) varies with alloying [Fig. 4(b)]. Near the critical point, however, the relationship steepens to S ∝ n^-2, closely resembling 1D transport behavior and clearly deviating from the 3D model [Fig. 4(a)]. This motivates fitting with the 1D transport model, from which a self-consistent (curve-wise constant) 1D m* is extracted. This dimensional crossover is accompanied by a dramatic drop in the apparent effective mass: at x ≈ 0.6, m*/me is roughly three orders of magnitude lower than in the 3D regime [Fig. 4(b)] (further calculation details are provided in Supplemental Material 3.3.1 and Fig. S5.). Such an abrupt reduction in m*/me is consistent with the emergence of massless or light Dirac-like carriers at the topological transition (J. Am. Chem. Soc. (2018) 140, 13097–13102). This behavior supports, rather than contradicts the conclusion that transport channels exhibit 1D-like characteristics in the vicinity of the topological transition.\nComment 2: The relationship between S and T provided by the authors under non-degenerate conditions is based on the assumption that n remains constant. This premise should be explicitly stated in the manuscript. In analyzing the dimensional transition transport behavior in SrTiO₃, did the authors consider the possible temperature-dependent variation of n? Furthermore, what values of the scattering parameter r were used in calculating η of SrTiO₃ (Figs. S1 and S2)? Was the possible temperature dependence of the scattering parameter taken into account? Additionally, η shown in Figs. S1 and S2 indicates that SrTiO₃ is in a non-degenerate state, however, the authors used a relationship between S and n derived under degenerate conditions to demonstrate the existence of dimensionally transition transport in SrTiO₃. This methodological inconsistency may undermine the validity of the conclusions. In summary, the argument regarding dimensionally transition transport phenomena in SrTiO₃ seems to lack rigorous logical consistency.\nResponse: We appreciate the reviewer’s careful examination and for pointing out the assumptions and methodological consistency in our treatment of SrTiO3.\n1) Assumption behind non-degenerate S-T.\nYes. In the non-degenerate analysis we treat the carrier concentration as temperature-independent along each trace, and we now state this explicitly. This assumption is appropriate for La/Nb-doped and oxygen-deficient SrTiO3 over 100−1000K, where Hall data show n is essentially flat once donors are ionized (J. Appl. Phys. (2005) 97, 034106; Appl. Phys. Express (2008) 1, 015007; EcoEnergy (2025) 3, 459). Thus, variations in S(T) primarily reflect scattering and dimensionality rather than changes in n(T). Operationally, for each sample in Fig. 2(a) we fix n to its 300K Hall value when fitting S(T). For datasets that exhibit pronounced freeze-out or thermally activated n(T), we refrain from the constant n diagnostic and instead use the degenerate analysis or report the case as ambiguous.\n2) How we treated the scattering parameter r.\nTo computing the reduced chemical potential η from the measured S, we use the full Fermi–Dirac–integral expressions with a relaxation-time model (detailed in Section 1.2 and Table S1 of the Supplemental Material). We parameterize scattering by r via τ(E) ∝ E^r, so that deformation-potential acoustic-phonon scattering corresponds to s = r+D/2 (Phys. Rev. B (2018) 97, 235201), yielding s=1 in 3D and s=1.5 in 2D. We therefore take s=1 as the baseline for all curves (now stated in the Fig. S1).\nTo assess robustness, we allow a mild temperature dependence consistent with prior reports (J. Appl. Phys. (2005) 97, 034106), sweeping over a conservative envelope r(T) ∈ [r0, r0+0.5]. This maps to s(T) ∈ [1, 1.5]. Recomputing η across these bounds shifts η(T) modestly and does not alter the regime assignment (degenerate (η > 0) for S(n) vs non-degenerate (η < 0) for S(T)) or any dimensionality conclusions (Fig. R1). Dimensionality is ultimately inferred from the appropriate slope of S(T) in the non-degenerate regime and S(n) in the degenerate regime, so small variations of s (namely, r) affect prefactors more than the slope-based diagnostic.\nFig R1: Thermoelectric transport fingerprints of n‑type SrTiO3. Data are compiled from La doped, Nb doped and oxygen deficient crystals and films in Refs (Nat. Mater. (2007) 6, 129-134; Appl. Phys. Lett. (2010) 97, 042106; Phys. Rev. B (2001) 63, 113104; Phys. Rev. (1964) 134, A442; J. Alloys Compd. (2005) 392, 306-309; J. Appl. Phys. (2005) 97, 034106).\n(a) Temperature dependence of S for different n. Low-n traces follow the behavior expected for a 3D DOS, whereas high-n traces approach the form S ∝ 3/2ln(T) characteristic of a 2D DOS, indicating a dimensional crossover near n ~ 10^20 cm-3. Middle and bottom subpanels: reduced chemical potential computed from Eq. S33 using s=1 (baseline, deformation-potential) and s=1.5. For all, η(T) remains negative (typically η ≤ -1) from 100–1000 K for all samples, validating use of the non-degenerate diagnostic. With s=1.5, η(T) shifts upward but stays predominantly negative, leaving the crossover conclusion unchanged. (b) Room-temperature S versus carrier concentration n. Data follow the 3D Pisarenko trend up to ~7×10^20 cm-3, then deviate toward the 2D prediction; for n > 7×10^20 cm-3 the data align with the 2D curve (solid red) rather than the 3D curve (blue dashed), consistent with a completed 3D to 2D crossover. Middle and bottom subpanels: η(n) at 300 K computed with s=1 and s=1.5, respectively. The light-red band denotes the boundary between non-degenerate and degenerate regimes; the vertical guideline marks η=0. Only in the degenerate regime (η > 0) does the Pisarenko slope S(n) ∝ n^-D/2 resolve dimensionality. Increasing s to 1.5 shifts the crossover slightly to lower n but does not change the overall conclusion.\n3) Degeneracy limits and diagnostics.\nWe correct the definitions and use them throughout: degenerate when η > 0 (practically η ≥ 1) non-degenerate when η < 0 (practically η ≤ -1); -1 ≤ η ≤ +1 is an intermediate window. The model expressions for S(n) and S(T) are defined for all η, but their ability to discriminate dimensionality depends on the regime.\nS(T), (fix n).\n• non-degenerate (η < 0): for 1D/2D/3D alike, S ∝ ln(T), the curve cannot distinguish D.\n• degenerate (η > 0): acquires a dimension-dependent slope, S(T) ∝ T, so Pisarenko plots at fixed n reveal D from the power-law exponent. For robustness we recommend focusing on data with η ≥ 1.\nS(n), (fix T).\n• non-degenerate (η < 0): the temperature dependence is decisive under acoustic deformation-potential scattering, S(n) ∝ 2lnT, so a single sweep serves as a “thermometer” for D. Signals are clearest for η ≤ -1.\n• degenerate (η > 0): for all D (with scattering-dependent prefactors), S(n) ∝ n^-D/2, so does not discriminate dimensionality here.\nRegime-gated workflow (what we actually use).\n(a) Compute η for each datum.\n(b) If η < 0: infer D from the log-slope of S(T) at fixed n (S(T) ∝ 2lnT).\n(c) If η > 0: infer D from the slope of S(n) at fixed T (S(n) ∝ n^-D/2).\n(d) If -1 ≤ η ≤ +1: treat as ambiguous; we avoid S-n and S-T assignments and rely on trends only when η is clearly negative.\nWe have revised the manuscript to reflect these points and to state explicitly that all dimensionality claims are gated by η and use only S(T) in the non-degenerate regime and only S(n) in the degenerate regime.\nRevision made:\nIn “3.1 Dimensional crossover from 3D- to 2D-like transport in heavily doped SrTiO3” Session\nA cross analysis of six independent data sets (Nat. Mater. (2007) 6, 129-134; Appl. Phys. Lett. (2010) 97, 042106; Phys. Rev. B (2001) 63, 113104; Phys. Rev. (1964) 134, A442; J. Alloys Compd. (2005) 392, 306-309; J. Appl. Phys. (2005) 97, 034106) [Fig. 2] exposes a universal dimensional crossover in SrTiO3 that is triggered once the carrier density surpasses n ≈ 1 × 10^20 cm-3. Below this threshold, the transport is well described by a conventional 3D DOS. The Seebeck coefficient varies with temperature as S ∝ 3/2ln(T) and the Pisarenko relation follows the canonical S ∝ n^-2/3 dependence. As the carrier density rises beyond this threshold, both signatures evolve in concert toward the behaviour expected for a 2D DOS. Specifically, the logarithmic prefactor in S(T) collapses to unity, yielding S ∝ ln(T), while the slope of the S-n curve steepens to S ∝ n^-1 (the detailed calculation procedure is described in Supplemental Material 3.1 and illustrated in Figs. S1 to S3).\nIn “3.1 Dimensional crossover from 3D- to 2D-like transport in heavily doped SrTiO3” Session in Supplemental Material\nThe temperature dependence of the Seebeck coefficient is plotted in Fig. 2(a) for samples with different carrier concentrations. In analyzing these relations under non-degenerate conditions, we assume the carrier concentration is constant over 100–1000K, consistent with Hall measurements on SrTiO3-based samples (J. Appl. Phys. (2005) 97, 034106; Appl. Phys. Express (2008) 1, 015007; EcoEnergy (2025) 3, 459). The reduced chemical potential η corresponding to each measured Seebeck coefficient was obtained using Equation S33 with s=1 (deformation-potential acoustic-phonon scattering), where s is determined from the S-σ trends (Fig. S1). As shown in Fig. S2, η stays negative (typically η ≤ -1) across the full temperature range for all samples in Fig. 2(a), validating the non-degenerate treatment. As discussed in Section 2.2, examining the temperature dependence of the Seebeck coefficient at fixed carrier concentration [Fig. 2(a)] has proven to be an effective approach for identifying the dimensionality of the samples. Samples with n = 7.5 × 10^19 cm-3 and n = 9 × 10^19 cm-3 follow a S ∝ 2ln(T) dependence, indicating 3D-like transport. Samples with n = 1.6 × 10^20 cm-3 and n = 3.3 × 10^20 cm-3 exhibit S ∝ ln(T) behavior in S(T), consistent with 2D-like transport, thereby evidencing a dimensional crossover near n = 10^20 cm-3.\nRoom-temperature S versus n is plotted in Fig. 2(b). Using Eq. S33 with s=1, we compute η at each doping level; Fig. S3 shows that at 300 K, for n ≤ 7×10^20 cm-3 (non-degenerate), where η < 0, S-n cannot resolve dimensionality, and above this range (degenerate), where η > 0, the Pisarenko slope S(n) ∝ n^-D/2 does discriminate dimensionality. Consistently, the high-n data in Fig. 2(b) steepen toward the 2D trend, evidencing a 3D to 2D evolution with increasing n.\nSummary, S(T) identifies the onset of dimensional crossover near n = 10^20 cm-3, while S(n) at 300K confirms 2D-like behavior once the system is degenerate (n ≥ 7×10^20 cm-3). Together, these diagnostics provide a consistent picture of the 3D to 2D transition in heavily doped SrTiO3.\nFigure S1: Room temperature Seebeck coefficient (S) versus electrical conductivity (σ) for La doped and oxygen deficient SrTiO3 single crystals and thin films. Dashed curves are illustrative transport fits using s=1 (deformation-potential scattering). The shaded band marks the range where the dimensional crossover becomes apparent.\nFigure S2: Reduced chemical potential η for the data in Fig. 2(a), computed from Eq. S33 with s=1. For all four samples, η stays negative (typically η ≤ -1) over 100−1000 K, confirming the non-degenerate regime in which S(T) distinguishes dimensionality.\nFigure S3: Reduced chemical potential η at 300 K versus carrier concentration n for the data in Fig. 2(b). Vertical guideline at n ≈ 7×10^20 cm-3 separates the non-degenerate (η < 0) from the degenerate (η > 0) region; only in the latter does the Pisarenko slope S(n) ∝ n^-D/2 resolve dimensionality.\nComment 3: At line 143, the authors stated that the S-n relationship is independent of the scattering mechanism. This is not accurate.\nResponse: Thank you for pointing this out. Our original sentence was imprecise. In the non-degenerate regime (η < 0), the Pisarenko relation can be expressed as S = (kB/e)(s+1-η), n ∝ e^η with s = r+D/2. Eliminating η gives S = -(kB/e)lnn + (kB/e)(r+D/2+1+const.). Thus, the slope is universal (independent of scattering and dimensionality) only in the non-degenerate regime; the intercept depends on scattering and dimensionality via s. Outside the non-degenerate limit, the slope regains dependence. We have revised the text to state explicitly that it is the slope in the non-degenerate limit that is universal, not the entire S-n relation.\nRevision made:\nIn “Experimental strategy for identifying electronic dimensionality” Session\nIn the non-degenerate regime (η < 0, practically η ≤ -1), the Pisarenko slope of S-n relation is universal (S ∝ -lnn) thus independent of scattering and dimensionality, whereas the intercept retains dependence on the scattering parameter r and dimensionality D via s (See Section 2.2 and Equation S43 of the Supplemental Material for detail). Consequently, for 1D, 2D, and 3D systems alike, S ∝ -lnn, the curve therefore cannot discriminate dimensionality. Instead, the temperature dependence is decisive (see Supplemental Material 2.2 for detail): (e.g. S(T) ∝ 2lnT) (10) |S| ≥ 200 μV/K A single sweep thus serves as a “thermometer” for D.\nIn “2.2 Non-degenerate limit” Session in Supplemental Material\n2.2 Non-degenerate limit\nFor the non-degenerate limit (η < 0, practically η ≤ -1), the Maxwell–Boltzmann approximation holds, making: F_i(η) → Γ(i+1) ∙ e^η (S41) where the exponential factor captures the low-occupancy states above the band edge. Eq. S33 simplifies to: S = (kB/e)(s+1-η) (S42) Meanwhile, Eq. S17 reduce to n ∝ e^η, with s = r+D/2. eliminating η gives: S = -(kB/e)lnn + (kB/e)(r+D/2+1+const.) (S43) The slope is universal (independent of scattering and dimensionality), the intercept depends on scattering and dimensionality via s.\nComment 4: At line 179, could the authors clarify the motivation for mentioning the weighted mobility model? It appears somewhat abrupt.\nResponse: We thank the reviewer for this helpful observation. The reference to the weighted mobility model (Adv Mater (2020) 32, e2001537) was meant to emphasize the conceptual foundation of our framework, as briefly introduced in the Introduction. That model was originally developed for analyzing bulk 3D systems; in our work we extend its logic to low-dimensional systems, where dimensionality and scattering mechanisms need to be disentangled. To avoid the impression of an abrupt transition, we have revised the text accordingly.\nRevision made:\nIn “Case Studies” Session\nUsing our transport model framework, we next analyze representative low-dimensional systems to illustrate how their electronic dimensionality and dominant scattering mechanisms are revealed.\nComment 5: Could the authors explain the reason for the dimensional transition in electrical transport behavior observed in SrTiO3 and Bi2O2Se?\nResponse: We thank the reviewer for raising this important point. The observed dimensional transition in electrical transport for SrTiO3 and Bi2O2Se can be understood from their underlying electronic structures.\nIn SrTiO3, The quasi-2D transport originates from the orbital-selective hybridization between Ti-d and O-p orbitals. Specifically, Ti-d_xy orbitals strongly hybridize with O-p orbitals within the x-y plane, leading to a strong π-interactions that impart dispersive character to the bands. By contrast, along the out-of-plane (z) direction, such hybridization is symmetry-forbidden, leaving only weak δ-interactions. This weak coupling produces nearly non-dispersive character. The coexistence of strong in-plane and week out-of-plane interactions results in quasi-2D band dispersion and cylindrical Fermi surfaces, which ultimately manifest as 2D-like transport behavior. More detailed discussions can be found in Refs. (Angew. Chem. Int. Ed. (2019) 58, 5503–5512).\nIn Bi2O2Se, the low-dimensional transport behavior primarily stems from its anisotropic crystal and electronic structure. Weak interlayer electrostatic interactions along the out-of-plane direction leads to a highly anisotropic Fermi surface and effective mass: nearly isotropic in-plane but elongated along the kz direction (Matter. (2022) 5, 4274-4314). This anisotropy confines carriers predominantly within the layers, giving rise to quasi-2D transport characteristics. At low temperatures, this anisotropy is even more pronounced and can result in quasi-1D transport features. With increasing temperature, additional scattering and thermal activation gradually weaken the confinement, and the system exhibits a crossover from 1D- to 2D-like transport, eventually recovering 3D transport at sufficiently high temperatures.\nRevision made:\nIn “3.1 Dimensional crossover from 3D- to 2D-like transport in heavily doped SrTiO3” Session\nTaken together, these observations argue that heavily doped SrTiO3 develops a Fermi surface that is effectively 2D, in line with recent DFT predictions of flattened t2g sub bands under strong filling (Angew. Chem. Int. Ed. (2019) 58, 5503–5512). This quasi-2D behavior arises from orbital-selective hybridization: in-plane Ti-d_xy and O-p coupling yields dispersive bands, whereas out-of-plane (z) hybridization is symmetry-suppressed, producing weak dispersion and cylindrical Fermi-surface sections. The result cautions that bulk SrTiO3 cannot always be treated within a simple 3D parabolic band picture once carrier densities approach 10^20 cm-3, and it underscores how modest chemical doping can unlock low dimensional transport physics without artificial confinement.\nIn “3.2 Temperature-triggered transition from 1D-like to 3D-like transport in few-layer bismuth oxyselenide (Bi2O2Se)” Session\nIn this study, our model uncovers that the electron transport behavior undergoes dimensional evolution. As the T increases, transport behavior transitions from a 1D to higher dimensional regime. This transition is discovered by analyzing the relationship between S and n (detailed calculation procedures can be found in Supplemental Material 3.2, along with Figs. S3 and S4). At 60 K, a 1D transport model fits the data [Fig. 3(b)], whereas at 100 K, a 2D model is more appropriate [Fig. 3(c)]. By 300 K, a 3D model best describes the S-n relationship [Fig. 3(d)]. The change in transport exponent s further confirmed this transition. The S-σ relationship follows the s=1 relation at 60K, s=1.5 at 100K, and s=2 at 300K. Assuming optical phonon/ piezoelectric scattering mechanism thoughout, this observation indicates that the transport behaviour shifts from 1D to 3D as T increases (Table 1). This effect may originates from its layered crystal/electronic anisotropy: weak interlayer coupling along the out-of-plane direction produces a highly anisotropic Fermi surface and effective mass (nearly isotropic in-plane but elongated along kz (Matter. (2022) 5, 4274–4314)). At low temperature, suppressed interlayer hopping confines carriers within the layers, yielding quasi-1D-like scaling; with increasing T, phonon-assisted interlayer transport progressively activates transverse conduction, driving a 1D → 2D → 3D evolution. These temperature-induced dimensional transitions expand our understanding of carrier dynamics in Bi2O2Se and highlight novel routes for property optimization.\nComment 6: The authors explained the reason for the transition from conventional 3D transport to a quasi-1D transport mode in Pb₁₋ₓSnₓTe polycrystalline materials, emphasizing the importance of grain boundaries and defects in enabling this transition. Would this imply that higher-quality crystalline materials would not exhibit this phenomenon? If so, experimental data validating this hypothesis would significantly strengthen the authors' conclusions.\nResponse: Thank you for the thoughtful question. While our initial discussion proposed grain-boundary/defect–assisted pathways as one possible route to quasi-1D transport in polycrystalline Pb1-xSnxTe, our analysis indicates that the 3D to 1D crossover also appears in single-crystal datasets near the critical composition (see Ref. (J. Phys. Chem. Solids (2000) 61, 919–923)). This supports the view that the dimensional evolution is not contingent on polycrystallinity but is instead an intrinsic consequence of the topological transition in the bulk electronic structure.\nMechanistically, two levels can operate, often concurrently:\n(a) Intrinsic topological channels (microstructure-agnostic). Near xc, the bulk gap closes and bands invert at the L-points, creating light/Dirac-like carriers and enabling topological line/hinge/interface modes (e.g., along dislocation lines, domain boundaries between locally inverted and trivial regions, or crystal hinges). Such modes can exist within single crystals and can dominate transport when they percolate, yielding the observed S ∝ n^-2 scaling.\n(b) Microstructural facilitation (polycrystals). Grain boundaries and extended defects can increase the density and connectivity of these conducting paths, amplifying the 1D-like signature, but they are not required for the effect to occur.\nAccordingly, higher-quality crystals may reduce the amount of such pathways but need not eliminate the 1D-like scaling near xc. This reconciles the agreement between polycrystalline and single-crystal data and supports our conclusion that the dimensional crossover is intrinsic to Pb1-xSnxTe near the topological transition.\nTo strengthen this point, we have (i) tempered the language that could be read as “grain-boundary–exclusive,” and (ii) added a short paragraph noting that single-crystal data exhibit the same scaling trend.\nRevision made:\nIn “3.3 Topological phase transition and emergent 1D-like transport in Pb1-xSnxTe alloy” Session\nBy analogy, we suggest that in Pb1-xSnxTe, the closing of the bulk gap and onset of band inversion create conditions for 1D-like transport channels. These may arise from topology enabled line, hinge, or interface modes, or from percolating networks of locally inverted and trivial regions that can dominate conduction at the critical composition xc. Importantly, the 3D to 1D scaling appears not only in polycrystals studied but also in single-crystal datasets near xc (e.g., J. Phys. Chem. Solids (2000) 61, 919–923), indicating that the effect does not rely on macroscopic crystallinity. Microstructure chiefly modulates the density and connectivity of these pathways, and thus the amplitude of the 1D-like signature, but is not required for their existence; high-quality single crystals may suppress extrinsic networks yet still exhibit intrinsic 1D-like scaling in the vicinity of xc.\nComment 7: It would be helpful if the authors could expand the conclusion with a discussion on the general applicability, potential limitations, and future improvement directions of the proposed method.\nResponse: We appreciate the suggestion. We have expanded the Conclusion to explicitly cover (i) general applicability, (ii) limitations, and (iii) directions for improvement. Briefly, the framework is applicable to large public datasets (e.g., StarryData2) and can be combined with ARPES, quantum oscillations, and first-principles calculations for cross-validation. Its main limitations stem from standard simplifying assumptions (parabolic bands, relaxation-time approximation, single-band, curve-wise constant m*) and the requirement for reliable S, σ, n(T) data across regimes. We outline concrete extensions (non-parabolic and multi-band treatments, anisotropy, energy-dependent scattering, and Hall-factor corrections, and data-driven screening) to broaden scope and robustness.\nRevision made:\nIn “Conclusion” Session\nWe introduced a simple yet powerful transport framework for diagnosing electronic dimensionality in semiconductors from standard measurements (S, σ, n). Applied to SrTiO3, Bi2O2Se and Pb1-xSnxTe, the analysis captures 1D/2D-like to 3D crossovers driven by doping, temperature, and alloying, linking macroscopic transport to low-dimensional electronic structure.\nApplicability. The method can operate directly on published datasets and open repositories (e.g., StarryData2), enabling broad screening for dimensionality signatures. It is complementary to microscopic probes, such as ARPES, quantum oscillations, and first-principles calculations, which can validate the inferred D and scattering. The framework is agnostic to chemistry and extends naturally to other semiconductors and layered materials.\nLimitations. To remain tractable, we adopt standard assumptions: relaxation-time approximation with power-law scattering, single dominant band, and curve-wise constant m*. These can be strained in narrow-gap or strongly anisotropic systems, in multi-band regimes, or where strong correlations/disorder dominate.\nOutlook. Future work may (i) incorporate multi-band formulations; (ii) treat anisotropic effective-mass tensors and beyond-power-law energy-dependent scattering; and (iii) integrate high-throughput computation and machine-learning over large databases to identify materials with unconventional dimensional transport. Beyond inorganic systems, the same logic can be adapted to organic thermoelectrics, providing a unified perspective on dimensional transport across diverse material classes.\nOverall, the framework deepens understanding of carrier dynamics and offers practical guidance for designing next-generation thermoelectrics and electronic materials, where controlling dimensionality can unlock enhanced performance.\nComment 8: Some references lack DOI links, while others contain misassigned DOIs (e.g., [46], [52]). Additionally, Chinese characters appear in reference [28]. Please thoroughly review all citations.\nResponse: Thank you for flagging this. We conducted a full audit of the bibliography and in-text citations.\nReviewer #2\nThe authors develop a simple model for the relationship between the thermoelectric Seebeck coefficient and conductivity that takes into account carrier concentration or temperature and sample dimensionality. They conclude that the Pisarenko scaling law S ~ ln(n) works at low to moderate doping while at high doping and in degenerate materials, the scaling law becomes dependent on the dimensionality. The authors compare the simple model to numerous experimental data sets and demonstrate that the simple model can guide the analysis of experiments, including the prediction of unexpected reduced dimensionality in some cases. For this reason, I support the work for publication--simple models can be useful when they provide insight, which is the case in this manuscript.\nComment 1: However, what is not clear to me from the manuscript is how the energy dependence of the scattering rate plays into the results. Dimensionality primarily affects the exponent of the energy dependence of the DOS, through which dimensionality makes its way into the energy dependence of the transport distribution functon(TDF). The TDF is then also dependent on the scattering rate and its exponent of energy dependence. The manuscript apparently shows the opposite---the S-n scaling in degenerate regime and the S-T scaling in the non-degenerate regime are independent of the scattering rate and only depend on dimensionality (in other words depend on D but not r). I think this is because S-sigma plots only show a dependence on s=r+D/2 in the degenerate (eta>>1) regime but the S vs n plot does not, and are independent of s at the low and medium doping. But I wish the manuscript would explain this a little better---how do we end up having S-sigma show a dependence on s (so depend on the r of the scattering rate) but the S-n plot does not depend on s through r, only on D. It seems that the r dependence of conductivity and mobility are canceling, leaving S vs n independent of r?\nResponse: We thank the reviewer for this insightful question. As correctly pointed out, dimensionality modifies the energy dependence of the density of states (DOS), whereas the scattering mechanism enters through the relaxation time τ(E) ∝ E^r. Below, we clarify why the S-σ relation is sensitive to the scattering exponent r, while the S-n relation is largely governed by dimensionality D.\nThe carrier concentration is obtained from the weighted integral of the DOS with the Fermi-Dirac distribution function, n = ∫ 0^∞ g(E)f(E) dE, Which depends only on the DOS and Fermi level, and is independent of the carrier scattering mechanism. Explicitly, for different dimensionalities, n = a ∙ (2m*DkB T)^3/2 / (2π^2 ℏ^3) ∙ F1/2(η), for 3D; n = a ∙ (m*DkB T) / (πℏ^2) ∙ F0(η), for 2D; n = a ∙ (2m*DkB T)^1/2 / (πℏ) ∙ F-1/2(η), for 1D. Which in the degenerate regime (η > 0, practically η ≥ 1) reduce to: n = a ∙ (2m*DkB T)^3/2 / (3π^2 ℏ^3) ∙ η^3/2, for 3D; n = a ∙ (m*DkB T) / (πℏ^2) ∙ η, for 2D; n = a ∙ (2m*DkB T)^1/2 / (πℏ) ∙ η^1/2, for 1D.\nBy contrast, the electrical conductivity involves the transport distribution function: σ = ∫ e^2 𝜏(E) 𝑣x^2(E) g(E) (-𝜕f/𝜕E) dE, Where the scattering explicitly enters through σ(E) = e^2 𝜏(E) 𝑣x^2(E) g(E). The Seebeck coefficient is written in a similar form, S = (1/eT) ∫ (E-EF) σ(E) (-𝜕f/𝜕E) dE / ∫ σ(E) (-𝜕f/𝜕E) dE. In the degenerate limit, both σ and S reduce to expressions that depend on the combined parameter s = r+D/2: (Nat. Mater. (2017) 16, 252-257) σ = σE0 ∙ η^s, S = (kB/e) ∙ (π^2/3) ∙ η^-1. So, we obtain: S = (kB/e) ∙ (π^2/3) ∙ (σ/σE0)^-1/s. This explains why S-σ plots show scaling with s, reflecting both dimensionality and scattering.\nWhen S is expressed in terms of carrier concentration, we obtain: S ∝ s∙n^-2/3, for 3D; S ∝ s∙n^-1, for 2D; S ∝ s∙n^-2, for 1D. Here, the dependence on s appears only as a linear prefactor. Since the typical value of s (e.g., APS: 1, POP: 2) are relatively close, this weak prefactor is overwhelmed by the strong dimensional scaling n^-2/D. Consequently, the S-n relation primarily reflects dimensionality, and its sensitivity to the scattering exponent is very low.\nIn contrast, in the S-σ relation, as shown in Eq. (7), S exhibits a power-law dependence on σ with the exponent of -1/s. Different scattering mechanisms directly affect the slope of the curve (especially in a logarithmic plot), rather than merely scaling the magnitude. Consequently, the S-σ relation can clearly reveal the scattering exponent s, allowing distinct mechanisms to be distinguished from the curve slope.\nIn summary, the key distinction arises because n is determined solely by the DOS, while both σ and S carry the same scattering dependence. As a result, in the S-n relation the scattering exponent enters only as a weak prefactor, leaving dimensionality as the dominant factor, whereas in the S-σ relation the scattering exponent governs the slope and can be directly resolved.\nRevision made:\nIn “Experimental strategy for identifying electronic dimensionality” Session\nCarrier scattering mechanisms, such as acoustic phonon, ionized impurity, piezoelectric, or polar optical phonon scattering, impose unique dependencies on carrier energy and temperature (Meas. Sci. Technol. (2002) 13, 230-230; Heavily Doped Semiconductors (1969) 3, 77-205; Phys. Rev. (1950) 80, 72; Phys. Rev. (1956) 101, 944). Once the dominant mechanism is established, the associated transport exponent s may become an independent fingerprint of dimensionality. For example, for ionized-impurity and polar-optical-phonon scattering, each effective DOS dimensionality D maps onto a unique s value (Table 1). Experimentally, s can be extracted by measuring the Seebeck coefficient and electrical conductivity while systematically tuning the carrier concentration at a fixed temperature, then fit the joint S-σ response. The resulting s can be matched to the theoretical catalogue (Table 1), providing a rapid, self-consistent check on the dimensionality already inferred from S(T) or Pisarenko analysis (A detailed explanation can be found in the Introduction of Section 2 and in Section 2.1 of the Supplemental Material.).\nIn “2 Analysis of S(n) and S(T) dependency” Session in Supplemental Material\nSince the carrier concentration is obtained by the weighted integration of the density of states g(E) with the Fermi-Dirac distribution function f, it remains independent of carrier scattering mechanisms. Consequently, within the same dimensionality, S(n) follows a consistent dependence across different scattering mechanisms, and the same holds for S(T) (as shown in Table 1). In contrast, electrical conductivity explicitly contains the scattering time τ(E) ∝ E^r, which enters the transport function. As a result, S-σ plots reflect both dimensionality and scattering through the combined exponent s = r+D/2, while S-n plots remain determined solely by dimensionality D.\nIn “2.1 Degenerate limit” Session in Supplemental Material\n2.1 Degenerate limit\nη > 0\nIn the degenerate limit (η > 0, practically η ≥ 1), one can employ the Sommerfeld expansion or Mott relation. In this regime, the Fermi-Dirac integral simplifies through a power-series approximation (arXiv (2008) 0811.0116): Fi(η) → η^(i+1) / Γ(i+2) (S34) By employing the Sommerfeld expansion, the conductivity and Seebeck coefficient become (Nat. Mater. (2017) 16, 252-257): σ = σE0 ∙ η^s (S35) S = (kB/e) ∙ (π^2/3) ∙ η^-1 (S36) So, we obtain: S = (kB/e) ∙ (π^2/3) ∙ (σ/σE0)^-1/s (S37) This explains why S-σ plots show scaling with s, reflecting both dimensionality and scattering.\nWhile in the degenerate regime, Eq. S17 reduce to: n = a ∙ (2m*DkB T)^3/2 / (3π^2 ℏ^3) ∙ η^3/2, for 3D; n = a ∙ (m*DkB T) / (πℏ^2) ∙ η, for 2D; n = a ∙ (2m*DkB T)^1/2 / (πℏ) ∙ η^1/2, for 1D. When S is expressed in terms of carrier concentration, we obtain: S ∝ s∙n^-2/3, for 3D; S ∝ s∙n^-1, for 2D; S ∝ s∙n^-2, for 1D. Here, the exact pre-factors in the expressions for S(n) depend on effective mass, scattering parameter and other fundamental constants. However, since the typical value of s (e.g., Acoustic phonon scattering: 1, Polar optical phonon scattering: 2) are relatively close, this weak pre-factor is overwhelmed by the strong dimensional scaling n^-2/D, so that the power-law exponents in S(n) remain robust to leading order. Physically, lower dimensionality enhances the power-law dependence of S on n. On the other hand, at fixed T, we find S ∝ η^-1, implying that S(T) has essentially the same functional form of η (since η ∝ T) across all dimensions under degenerate conditions.\nComment 2: The other comment is that the S vs sigma going as ln(sigma) has been demonstrated for organic materials (see for example https://doi.org/10.1557/s43579-024-00628-2) and the Kang-Snyder model that is very analogous to the model in this paper has also been shown to work for organic materials. Would the conclusions given in this manuscript about dimensionality extend to organic materials as well? In organics, the Mott formula relates temperature dependence of conductivity to dimensionality, but it is only valid for a constant density of states, which is violated in virtually all organic materials.\nResponse: We appreciate the insightful comment. The observed behavior in organics and the Kang–Snyder framework are indeed closely aligned with our transport formalism. Our model does not require a constant density of states; however, whether our dimensionality inference (via slope diagnostics of S(T) in the non-degenerate regime and S(n) in the degenerate regime) extends to organic materials depends on meeting several conditions that are often challenging in organics:\n(a) Reliable carrier density n. Dimensionality from S(n) requires trustworthy volumetric n. In many organics, Hall measurements can be unreliable due to hopping, inhomogeneity, and Hall-factor uncertainties (Nature Electronics (2024) 7, 510–512; Phys. Rev. Materials (2023) 7, 034603).\n(b) Transport regime. Our D-diagnostics are most robust in the degenerate, extended-state regime with a dominant scattering mechanism. Many organic conductors at moderate doping remain disorder/hopping dominated (violating Mott’s constant-DOS assumption).\n(c) Sample-to-sample comparability. Microstructure, dopant distribution, and processing (e.g., crosslinking) strongly tune disorder and coherence, which can shift the apparent dimensional fingerprints.\nTo probe applicability, we examined PBTTT as a model system using literature sets that report both S and n. Within individual high-doping datasets [Fig. R1(c), (d), (e)] (e.g., Chen (Adv. Mater. (2025) 37, 2417954; Adv. Energy Mater. (2023) 13, 2202797) and Kang (Mater. Today Phys. (2019) 8, 112)), the trends often lean 2D-like, consistent with prior reports of coherent, metallic transport at high conductivity (Nat. Mater. (2016) 15, 896, Sci Adv (2020) 6, aay8065). However, when aggregating multiple datasets at 300 K, the points largely fall in the overlap between 3D- and 2D-discriminative regions, so we cannot claim a universal 2D assignment for PBTTT across studies. We therefore conclude: our dimensional analysis can be applicable to organics in the highly doped, coherent regime with reliable n, but it does not yet generalize across organic systems where disorder and hopping remain significant.\nFIG R2: Thermoelectric properties of PBTTT at 300K. Data compiled from Chen, Ma et al. Chen et al., Wang et al., and Kang et al. (see refs: Adv. Mater. (2025) 37, 2417954; Adv. Energy Mater. (2023) 13, 2202797; Mater. Today Phys. (2019), 8, 112; Nature (2024) 632, 528). (a) S-σ relation at 300K. The slopes indicates an evolution of the scattering descriptor from s=3 at low conductivity toward s=1 at high conductivity. (b) S-n relation at 300K for the aggregated dataset. Reference curves show the degenerate Pisarenko scalings for 3D (S ∝ n^-2/3) and 2D (S ∝ n^-1); the shaded band marks the region where S-n discriminates dimensionality (red for 3D, and blue for 2D). When combined, most points fall in the 3D/2D overlap, so a universal 2D assignment for PBTTT at 300 K cannot be made. Differences likely reflect variations in microstructure/disorder, dopant distribution, and processing. (c–f) S-n plotted separately for Chen et al., Kang et al., Chen, Ma et al., and Wang et al., using the same reference curves. Some individual datasets lean 2D-like at high doping, whereas others appear more 3D-like.\nRevision made:\nIn “3.4 Assessing electronic dimensionality in organic semiconductors: PBTTT as a model system” Session\n3.4 Assessing electronic dimensionality in organic semiconductors: PBTTT as a model system\nDespite the well-known practical difficulty of obtaining reliable Hall carrier densities in organic conductors (Nature Electronics (2024) 7, 510–512; Phys. Rev. Materials (2023) 7, 034603), we assessed the applicability of our framework using the extensively studied Poly[2,5-bis(3-dodecylthiophen-2-yl)thieno[3,2-b]thiophene] (PBTTT) as a model system. We analyzed a few literature datasets (Adv. Mater. (2025) 37, 2417954; Adv. Energy Mater. (2023) 13, 2202797; Mater. Today Phys. (2019) 8, 112; Nature (2024) 632, 528) that report both S and n [Fig. 5]. PBTTT shows a scattering evolution from s=3 to s=1 with increasing doping (Adv. Mater. (2025) 37, 2417954; Adv. Energy Mater. (2023) 13, 2202797; Mater. Today Phys. (2019) 8, 112) [Fig. 5(a)], implying a transition toward acoustic-phonon dominated transport (Nat. Mater. (2017) 16, 252) and compatible with coherent, quasi-2D metallic behavior (Nat. Mater. (2016) 15, 896, Sci Adv (2020) 6, aay8065) at high conductivity (σ ≥ 200 S cm^-1).\nAt the single-study level, some datasets (e.g., Chen [Fig. 5(c), (e)]: Adv. Energy Mater. (2023) 13, 2202797; Adv. Mater. (2025) 37, 2417954; and Kang [Fig. 5(d)]: Mater. Today Phys. (2019) 8, 112) exhibit strong 2D-like trends, whereas others (e.g., Wang [Fig. 5(f)]: Nature (2024) 632, 528–535) appear 3D-like. To mitigate dataset-specific Hall uncertainties, we pooled all points at 300 K [Fig. 5(b)]. The aggregated data largely occupy the overlap region between our 3D- and 2D-discriminative bands, so we cannot claim a universal 2D assignment for PBTTT at high doping.\nWe infer that the apparent dimensionality in organics is highly sensitive to disorder, such as set by intrinsic microstructure, dopant distribution, and processing history (e.g., crosslinking), as highlighted in prior studies (Nature (2014) 515, 384; JACS (2022) 144, 3005; MRS Communications (2024) 14, 1054; ACS Mater. Lett. (2024) 6, 4351). Establishing dimensionality robustly in organics will therefore benefit from independent carrier-density benchmarks (e.g., coulometry, IR sum-rules, or gate-capacitance areal densities) and cross-checks with microscopic probes where available.\nFIG 5: Thermoelectric properties of PBTTT at 300K. Data compiled from Chen, Ma et al., Chen et al., Wang et al., and Kang et al. (see refs: Adv. Mater. (2025) 37, 2417954; Adv. Energy Mater. (2023) 13, 2202797; Mater. Today Phys. (2019), 8, 112; Nature (2024) 632, 528). (a) S-σ relation at 300K. The slopes indicates an evolution of the scattering descriptor from s=3 at low conductivity toward s=1 at high conductivity. (b) S-n relation at 300K for the aggregated dataset. Reference curves show the degenerate Pisarenko scalings for 3D (S ∝ n^-2/3) and 2D (S ∝ n^-1); the shaded band marks the region where S-n discriminates dimensionality (red for 3D, and blue for 2D). Most points fall in the 3D/2D overlap, so a universal 2D assignment for PBTTT at 300 K cannot be made. Differences likely reflect variations in microstructure/disorder, dopant distribution, and processing. (c–f) S-n plotted separately for Chen et al., Kang et al., Chen, Ma et al., and Wang et al., using the same reference curves. Some individual datasets lean 2D-like at high doping, whereas others appear more 3D-like.\nComment 3: Minor error/typo: line 173 where it says Session 1.2 it should be Section 1.2.\nResponse: We thank the reviewer for catching this typographical error. The manuscript has been corrected: “Session 1.2” on line 173 has been changed to “Section 1.2.” We have also proof-read the manuscript for similar typographical slips.\nRevision made:\nIn “Table 1: Note” Session\nNote: the scattering exponent (commonly denoted r) encodes the underlying scattering mechanism (See Section 1.2 of the Supplemental Material for detail). Of particular relevance is the relationship s = r+D/2, where D is dictated by dimensionality, taking values of 1 for 1D systems, 2 for 2D systems, and 3 for 3D systems (see Section 1.3 of the Supplemental Material for a detailed derivation). Consequently, changing r directly shifts the transport exponent s.\nReviewer #3\nZhang X.X., et al. introduced a generalized transport model that functions as an internal dimensionality meter. The method provided a scattering independent framework to design lower-dimensional transport materials. This work should be of widespread interest to researchers in the fields of materials and thermoelectrics.\nI am not an expert in this field, so I cannot verify the details. There doesn't seem to be any major concerns. I recommend to publish it in the current form.\nResponse: We sincerely thank the reviewer for the positive comments and for recognizing the significance of our work."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I appreciate the authors' responses to all the comments and the additional data provided. All my concerns have been properly addressed, and the manuscript has been significantly improved. The revised manuscript meets the standards of Nature Communications and should be accepted.", "reviewer_2": "The authors have revised their manuscript in accordance with reviewer comments. They have done a commendable job in doing so. I recommend the manuscript for publication in its present form."}, "author_response": ""}]

Decision letter