Beyond Wave-Nature Signatures: h-Independent Transport in Strongly-Scattering Quasi-2D Quantum Channels

Abstract

The Landauer-Büttiker formalism provides a fundamental framework for mesoscopic transport, typically expressing conductance in units of the quantum of conductance, e²/h. Here, we present a theoretical study of electron transport in a quasi two-dimensional (2D) quantum wire. This system features a wide transverse confinement and a longitudinal, high-energy, narrow potential barrier. The derivation, performed within the Landauer framework, yields an analytical expression for the total conductance that is explicitly independent of Planck’s constant (h). Instead, the conductance is found to depend solely on the Fermi energy, the electron effective mass, the wire width, and the effective barrier strength. We interpret this as an emergent phenomenon where the explicit signature of the electron’s wave-like nature, commonly manifest through Planck’s constant (h) in the overall scaling of conductance, is effectively absorbed within the energy- and geometry-dependent sum of transmission probabilities. This allows the conductance to be primarily governed by the Fermi energy, representing a ‘state-counting’ quantum parameter rather than more wave-like characteristic.

1. Introduction

The Landauer-Büttiker [1,2] has served as a cornerstone of mesoscopic physics for decades, providing an elegant and powerful framework for describing electrical conduction in quantum systems. A central tenet of this approach is the quantization of conductance, universally expressed as an integer or fractional multiple of the quantum of conductance, e²/h. Hence, in the case of an orifice with N transverse mode the conductance and resistance are

$$G_0=2{\displaystyle \frac{e^2}{h}}N,\mathrm{and}{\Re }_0={\displaystyle \frac{h}{2N{e}^2}}$$

(1)

respectively. This fundamental constant, derived from the core principles of quantum mechanics, unequivocally establishes Planck’s constant (h) as an indispensable parameter in characterizing quantum transport phenomena [3,4]. Its explicit appearance reflects the wave-particle duality and the inherent quantization of energy and momentum, underpinning phenomena ranging from ballistic transport in semiconductor nanowires [5,6] to the precise plateaus observed in the integer and fractional quantum Hall effects [7,8].

Here, we investigate the longitudinal conductance of a quasi two-dimensional (2D) quantum wire subjected to a highly localized potential barrier. Such systems are central to modern quantum electronic devices, forming the basis for quantum point contacts, tunable field-effect transistors, and components in quantum computing architectures [3,9]. Our theoretical model considers a wide quasi 2D wire (width W) where $w{k}_F>>1$, allowing for a multitude of propagating transverse modes. The model is valid as long as the transverse confinement in the third dimension is strong enough to ensure that only the lowest subband is occupied. Crucially, the wire incorporates a strong and narrow potential barrier characterized by height $V_D\left(x\right)$ and width $2a$, satisfying $\max V_D>>E_F$ and $a{k}_F<<1$. These conditions place the system squarely in a tunneling regime, where electrons interact strongly with the barrier, yet its narrowness permits an accurate approximation as a short-range scatterer, a limit widely explored in quantum scattering theory [10].

Contrary to the ubiquitous presence of Planck’s constant in quantum transport laws, the derivation of the conductance for this specific system yields an analytical expression that is explicitly independent of Planck’s constant (h) and the electron density, depending solely on the Fermi energy ($E_F$), the electron effective mass ($m^{*}$), the wire’s width (w), and the barrier’s integrated strength (${\displaystyle \int dx{V}_D\left(x\right)}$). This finding is particularly striking because the system operates in a ‘heavy quantum regime’ where tunneling through a classically impenetrable barrier is the dominant transport mechanism. While the Fermi energy itself is unequivocally a quantum mechanical concept, originating from the Pauli exclusion principle and the filling of discrete quantum states [11], its definition does not, in itself, necessitate an explicit reliance on the electron’s wave-like properties in the same manner as interference or typical tunneling probabilities do. The quantum of conductance, e²/h, on the other hand, is a direct consequence of the wave nature of charge carriers propagating through quantum channels [3].

The study of quantum transport in low-dimensional systems has been a vibrant area of research for decades. A foundational body of work, comprehensively reviewed in Ref. [12], has focused on quasi-1D systems, such as atomic-sized conductors and metallic quantum point contacts. This research has revealed fascinating phenomena, including quantized conductance and the stability of atomic chains, where transport is governed by a small number of transverse modes. Our work complements these studies by investigating a different physical regime: a quasi-2D wide wire in a semiconductor heterostructure, where a large number of transverse modes are available. We explore a unique transport phenomenon driven by a specific electromagnetic barrier configuration, leading to findings that are distinct from those observed in the highly confined quasi-1D systems.

The explicit cancellation of h and n in our final conductance formula therefore points to a profound emergent behavior. It suggests that in this specific parameter regime—a wide wire with a high, narrow barrier—the ‘wave-like’ quantum characteristics, normally encoded in the explicit appearance of h, are effectively suppressed or averaged out in the global transport observable. This leaves a conductance value primarily dictated by the Fermi energy, which appears to function as a more fundamental ‘state-counting’ quantum parameter. This unique manifestation of quantum transport, where the ‘residual’ signature of the wave nature of the electron seemingly vanishes from the conductance, is puzzling and invites a re-examination of how and when different facets of quantum mechanics (state quantization vs. wave phenomena) explicitly shape macroscopic observables. Our work highlights a subtle yet significant departure from typical quantum conductance scaling, potentially offering new insights into the interplay of fundamental constants in mesoscopic systems and hinting at novel regimes for electronic device design.

The remainder of this paper is organized as follows. Section 2 details the theoretical framework for a generic analytical approximate expression for the resistance of a quasi-2D quantum wire. Section 3 provides an exact numerical derivation for a rectangular barrier case, and illustrate the validity of the generic expression. Section 4 presents a possible experimental realization. Finally Section 5 concludes with a summary of our findings and outlines potential avenues for future theoretical and experimental investigations.

2. Generic Approximate Solution

The system, illustrated in Figure 1, comprises a wide, planar quantum wire (extending in the y-direction) featuring a narrow, high barrier of width 2a placed along the electron transport direction (x-axis). This system is governed by the stationary Schrödinger equation:

$$\left[-{\displaystyle \frac{{\hslash }^2}{2m^{*}}}\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)+V\left(x,y\right)\right]\mathsf{\psi }\left(x,y\right)=E\mathsf{\psi }\left(x,y\right)$$

(2)

where the potential consists of two parts:

$$V\left(x,y\right)=U_W\left(y\right)+V_D\left(x\right)$$

(3)

The wire’s boundaries

$$U_W\left(y\right)=\left\{\begin{array}{cc}0& 0<y<w\\ \infty & else\end{array}\right.$$

(4)

and the narrow membrane (barrier)

$$V_D\left(x\right)=\left\{\begin{array}{cc}V\left(x\right)& \left|x\right|<a\\ 0& else\end{array}\right.$$

(5)

It should be emphasized that these two requirements can be less restrictive, i.e., the wire’s boundaries potential do not have to be infinite, and the barrier does not have to vanish beyond the [−a, a] domain. It just needs to be localized there.

Due to the Cartesian symmetry of the system, the propagating eigenstates can be written simply as

$${\mathsf{\psi }}_n\left(x,y\right)=\sin \left(n\mathsf{\pi }{\displaystyle \frac{y}{w}}\right){\mathsf{\chi }}_n\left(x\right)$$

(6)

In case where the barrier width is considerably narrower than the electron’s de-Broglie wavelength, i.e., $a{k}_F<<1$, or $a\sqrt{2m^{*}{E}_F}/\hslash <<1$, where the subscript “F” represents the Fermi wavenumber and Fermi-energy respectively, the solution for the longitudinal component can be written using the one-dimensional (1D) Green Function

$$K_n\left(\left|x\right|\right)=i{\displaystyle \frac{2m}{{\hslash }^2}}{\displaystyle \frac{\exp \left(i{k}_n\left|x\right|\right)}{2k_n}}$$

(7)

where $k_n=k_F\sqrt{1-{\left({\displaystyle \frac{n\mathsf{\pi }}{w{k}_F}}\right)}^2}$ The Green Function (7) solves the differential equation

$$-{\displaystyle \frac{{\hslash }^2}{2m^{*}}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}{K}_n\left(\left|x\right|\right)-{\displaystyle \frac{{\left(\hslash k_n\right)}^2}{2m}}{K}_n\left(\left|x\right|\right)=\mathsf{\delta }\left(x\right)$$

(8)

Therefore, in the narrow barrier regime, the solution reads

$${\mathsf{\chi }}_n\left(x\right)=\exp \left(i{k}_{n}x\right)-{\displaystyle \frac{{\displaystyle \int dx{V}_D\left(x\right)\exp \left(i{k}_{n}x\right)}}{1+{\displaystyle \int dx{V}_D\left(x\right)K_n\left(\left|x\right|\right)}}}{K}_n\left(\left|x\right|\right)$$

(9)

After substituting the Green function $K_n\left(\left|x\right|\right)$ in (9), the stationary solution reads

$${\mathsf{\chi }}_n\left(x\right)=\exp \left(i{k}_{n}x\right)-{\displaystyle \frac{i{D}_n}{1+i{D}_n}}\exp \left(i{k}_n\left|x\right|\right)$$

(10)

where

$$D_n\equiv {\displaystyle \frac{2m^{*}}{{\hslash }^2}}{\displaystyle \frac{1}{2k_n}}{\displaystyle \int dx{V}_D\left(x\right)}$$

(11)

Therefore, beyond the barrier,

$${\mathsf{\chi }}_n\left(x>a\right)={\displaystyle \frac{1}{1+i{D}_n}}\exp \left(i{k}_{n}x\right)$$

(12)

and the transmission coefficient of the nth transverse propagating mode is

$$T_n={\displaystyle \frac{1}{1+{\left(\frac{m^{*}}{{\hslash }^2}\frac{1}{k_n}{\displaystyle \int dx{V}_D\left(x\right)}\right)}^2}}$$

(13)

When the barrier has a rectangular profile, i.e., $V_D\left(x\right)=V$ then

$$T_n={\displaystyle \frac{1}{1+{\left(\frac{m^{*}}{{\hslash }^2}\frac{2aV}{k_n}\right)}^2}}$$

(14)

Equations (13) and (14) are consistent with the transmission of a delta function potential barrier, i.e., $V_D\left(x\right)=\mathsf{\delta }\left(x\right){\displaystyle \int d{x}^{\prime }{V}_D\left(x^{\prime }\right)}$.

In a 1D wire, the transmission function T(k) can often exhibit sharp resonances due to wave interference within the barrier. However, a key assumption of our model is that the barrier is considerably narrower than the electron’s de Broglie wavelength. In this specific regime, a 1D transmission coefficient is a smooth, non-resonant function of energy, with a functional form such as T(k)∼1/(1 + C/k²). Our quasi-2D case is a direct extension of this principle to each transverse mode. The primary analytical difference is the replacement of the single wavenumber k with a mode-dependent wavenumber k_n in Equation (14).

Note, that the only difference between the 1D transmission coefficient and the quasi-2D one (14) is that $k$ is replaced by $k_n$, which depends both on the Fermi energy and the transverse mode number n.

The conductance is reached by substituting (13) in [3]

$$G=2{\displaystyle \frac{e^2}{h}}{\displaystyle \sum _{n=1}^N{T}_n}$$

(15)

where $N=⌊w{k}_F/\mathsf{\pi }⌋=⌊2w\sqrt{2m^{*}{E}_F}/h⌋$ is the number of propagating modes.

In the regime, where the wire’s width is much wider than the electron’s Fermi wavelength, i.e., $w{k}_F>>1$, and therefore the number of modes is very large $N>>1$ the summation in the conductance formula can be replaces with a corresponding integral

$$G=2{\displaystyle \frac{e^2}{h}}{\displaystyle \frac{wk}{\mathsf{\pi }}}{\displaystyle \underset{0}{\overset{1}{\int }}dx\frac{1}{1+D/\left(1-x^2\right)}}$$

(16)

where

$$D\equiv {\left({\displaystyle \frac{m^{*}}{{\hslash }^2}}{\displaystyle \frac{1}{k_F}}{\displaystyle \int dx{V}_D\left(x\right)}\right)}^2$$

(17)

This integral has an exact analytical solution

$$G=2{\displaystyle \frac{e^2}{h}}{\displaystyle \frac{w{k}_F}{\mathsf{\pi }}}\left(1-D{\displaystyle \frac{{\tanh }^{-1}\left(1/\sqrt{1+D}\right)}{\sqrt{1+D}}}\right)$$

(18)

This generic solution depends on the Planck constant both in the universal conductance $2e^2/\hslash$ and in D, however, in case $D>>1$

$$G\cong 2{\displaystyle \frac{e^2}{h}}{\displaystyle \frac{w{k}_F}{\mathsf{\pi }}}{\displaystyle \frac{1}{6/5+\left(3/2\right)D}}$$

(19)

Therefore, the resistance can be written

$$\Re \cong {\Re }_0+\Delta \Re ={\displaystyle \frac{h}{2e^2}}{\displaystyle \frac{\mathsf{\pi }}{w{k}_F}}\left(1+{\displaystyle \frac{1}{5}}+{\displaystyle \frac{3}{2}}D\right)$$

(20)

The first term in (20) correspond to the resistance of a free wire ${\Re }_0=h/\left(2e^{2}N\right)$. However, the second term in (20), which represents the resistance of the barrier, is independent of the Planck constant:

$$\Delta \Re \cong {\displaystyle \frac{3}{2}}\left({\displaystyle \frac{{\mathsf{\pi }}^2}{e^{2}w}}{\displaystyle \frac{\sqrt{2m^{*}}}{4E_F^{3/2}}}\right){\left({\displaystyle \int dx{V}_D\left(x\right)}\right)}^2$$

(21)

It should be stressed again that our findings are not dependent on the specific shape of the potential barrier. Equations (13) and (21) are completely generic and apply to any barrier profile, provided its width is much smaller than the particle’s de Broglie wavelength. This demonstrates that the emergent ℏ-independence is a general characteristic of this class of systems and is not an artifact of assuming a specific barrier shape like the rectangular potential.

In case the barrier has rectangular profile

$$\Delta \Re \cong {\displaystyle \frac{3}{2}}\left({\displaystyle \frac{{\mathsf{\pi }}^2}{e^{2}w}}{\displaystyle \frac{\sqrt{2m^{*}}}{E_F^{3/2}}}\right){\left(aV\right)}^2$$

(22)

While Equations (21) and (22) show a dependence on the Fermi energy, a quantum mechanical property, no residue of its wave-like nature is evident.

This unique behavior, where the conductance becomes ℏ-independent, only appears when the wire is wide and the number of transverse modes is large. When the number of modes is small, and particularly in the case of a single-atom wire [13,14,15], the effect vanishes. In this regime, even with the same large scattering strength ($D>>1$), the barrier’s resistance remains ℏ-dependent:

$$\Delta \Re \cong {\displaystyle \frac{h}{2e^2}}{\displaystyle \frac{3}{2}}D={\displaystyle \frac{3}{2e^{2}h}}{\displaystyle \frac{m^{*}}{E_F}}{\left(\mathsf{\pi }{\displaystyle \int dx{V}_D\left(x\right)}\right)}^2$$

This expression clearly shows that the explicit dependence on Planck’s constant persists. Therefore, a wide wire is a prerequisite for our mechanism, required to eliminate this last remnant of the quantum constant.

3. Exact Numerical Solution

Even when the number of modes is finite and the barrier has a finite width (unlike a delta function potential), Equations (20)–(22) remain valid. To show this, we numerically solve for the conductance of a finite-width wire with a finite-width barrier. The rectangular barrier case, i.e., $V_D\left(x\right)=V$, can be solved analytically (see, for example, Ref. [16]). In this case the longitudinal wavefunction obeys

$${\mathsf{\chi }}_n\left(x\right)=\left\{\begin{array}{cc}\exp \left(i{k}_{n}x\right)+r_n\exp \left(-i{k}_{n}x\right)& x<-a\\ a_n\exp \left(-{\alpha }_{n}x\right)+b_n\exp \left(+{\alpha }_{n}x\right)& \left|x\right|\le a\\ t_n\exp \left(i{k}_{n}x\right)& x>a\end{array}\right.$$

(23)

where

$${\alpha }_n=k_F\sqrt{{\displaystyle \frac{V}{E_F}}-\left(1-{\left({\displaystyle \frac{n\mathsf{\pi }}{w{k}_F}}\right)}^2\right)}$$

The coefficients $a_n,b_n,r_n$ and $t_n$ can be derived from the boundary conditions on the wavefunction and its derivative (for details, see [16]). The final transmission coefficients $T_n={\left|t_n\right|}^2$ can be substituted in (15) to obtain

$$G=2{\displaystyle \frac{e^2}{\hslash }}{\displaystyle \sum _{n=1}^N{\left[1+{\left(\frac{mV}{{\hslash }^2{k}_n{\mathsf{\alpha }}_n}\right)}^2{\sinh }^2\left(2{\mathsf{\alpha }}_{n}a\right)\right]}^{-1}}$$

(24)

From (24), the exact barrier resistance can be derived $\Delta \Re =1/G-{\Re }_0$. Figure 2 displays the plot against the barrier’s normalized width. It can be observed that the approximate expression is consistent with the accurate one over at least an order of magnitude.

4. Experimental Realization

To observe this effect in GaAs/AlGaAs 2DEG we suggest choosing realistic parameters: electron density n_s = 1 × 10¹¹ cm⁻² and low temperature ($<1 \mathrm{K}$) giving Fermi energy $E_F~10 \mathrm{meV}$. Since the electron mass is $m^{*}=0.067m_e=6.1\times {10}^{-32}\mathrm{Kg}$ then the Fermi wavelength is ${\lambda }_F\cong 48 \mathrm{nm}$ [17]. Choosing wire width $w\cong 1 \mathsf{\mu }\mathrm{m}$ correspond to $N>40$ modes. Modern MBE and e-beam lithography produce ~2 nm barriers via AlGaAs insertion or split-gate depletion [18,19,20,21,22,23,24]. Accordingly, we take barrier width $a=2\mathrm{nm}$ and height $V=200\mathrm{meV}=2\times {10}^{-19}\mathrm{J}$, i.e., $V/E_F\cong 20$. For these parameters ${\Re }_0=h/2e^{2}N\cong 315\Omega$ while $\Delta \Re \cong 3\mathrm{M}\Omega$.

These values can eaily be fabricated and measured. In the presence of disorder, finite temperature, or contact resistance fluctuations, extracting $\Delta \Re$ may require careful subtraction of background contributions [5,19]. Nonetheless, the predicted $\Delta \Re \cong 3\mathrm{M}\Omega$ is large enough to appear above such noise floors.

While our model operates under the zero-temperature assumption, it is important to consider the effect of finite temperatures. For our system, the Fermi energy, E_F ∼ 10 meV, is significantly larger than the operating temperature, T < 1K, where kT ≈ 0.086 meV. In this regime, the thermal smearing of the Fermi-Dirac distribution is small (kT ≪ E_F ), and the temperature-dependent corrections to the conductance are expected to be on the order of (kT/E_F )². Given that our zero-temperature result for conductance is independent of ℏ, these small thermal corrections would not reintroduce a dominant ℏ-dependence. Therefore, the emergent ℏ-independence of the conductance remains robust and is not fundamentally altered by finite temperature effects.

The functional dependence of the “classical” resistance on the system parameters can be measured by changing the voltage of a split-gate to control the barrier height and by modifying the voltage of a back gate to control the Fermi energy.

The resistance measurements would typically be performed using a four-terminal setup, where current is sourced through two outer leads and the voltage is measured across two inner leads. This configuration effectively eliminates the influence of contact resistance, as the voltmeter’s high input impedance ensures a negligible voltage drop across the contacts. While a two-terminal measurement could be employed, its validity depends on the sample resistance being much larger than the contact resistance. For the high-resistance regime of our barrier, this condition is satisfied, making a two-terminal approach a reasonable approximation. Furthermore, to address the challenge of a low signal-to-noise ratio in the case of a very opaque barrier, a lock-in amplifier would be used. This technique, which involves modulating the current and detecting the voltage signal phase-sensitively over an extended period, effectively averages out random noise and allows for the precise measurement of even very small currents.

Furthermore, while our theoretical model assumes a ballistic transport regime, it is important to consider the effects of interface roughness or impurity scattering in actual samples. The presence of such scattering mechanisms can indeed modify the resistance of the quantum wire. However, in our system, the engineered potential barrier is designed to be highly opaque, which causes it to dominate the total resistance. As long as the barrier’s resistance is significantly larger than the resistance of the wire itself, the influence of impurity scattering becomes a negligible perturbation.

This effect can be experimentally verified. By gradually increasing the barrier height (for example, by tuning the voltage on a split-gate), the barrier’s resistance can be made to increase dramatically. The point at which the measured total resistance becomes dominated by the barrier can be identified by first measuring the wire’s resistance in the absence of the barrier. Once the barrier’s contribution is much greater than this baseline resistance, the effects of impurities and interface scattering become a minor correction, and the ℏ-independent transport characteristics predicted by our model can be reliably measured.

5. Summary

In this work, we have presented a theoretical investigation of electron transport through a quasi 2D quantum wire containing a narrow but high potential barrier. The system was analyzed under specific conditions: a wide wire ($w{k}_F>>1$) ensuring multiple open transverse modes, and a barrier characterized by extreme parameters ($\max V_D>>E_F$ and $a{k}_F<<1$), allowing for its treatment as a short-range scatterer dominating the longitudinal transport.

The central finding is the derivation of an analytical expression for the wire’s conductance, performed within the Landauer-Büttiker formalism, that exhibits an unexpected independence from both Planck’s constant (h) and the electron density ($n_s$). Instead, the conductance is found to depend solely on the Fermi energy ($E_F$), the electron effective mass ($m^{*}$), the wire width (w), and the effective strength of the potential barrier (${\displaystyle \int dx{V}_D\left(x\right)}$). This result, given by $\Delta \Re \cong {\displaystyle \frac{3}{2}}\left({\displaystyle \frac{{\mathsf{\pi }}^2}{e^{2}w}}{\displaystyle \frac{\sqrt{2m^{*}}}{4E_F^{3/2}}}\right){\left({\displaystyle \int dx{V}_D\left(x\right)}\right)}^2$, underscores a unique limit of quantum transport.

The h-independence of the overall conductance is particularly intriguing because the system operates in a profoundly quantum regime, where transport occurs via tunneling through a classically impenetrable barrier. We suggest that while the Fermi energy is undeniably a quantum mechanical concept rooted in the Pauli exclusion principle and state counting, its definition does not, by itself, explicitly rely on the wave-like properties of the electron that are intrinsically tied to h. Thus, the absence of h in our final conductance formula indicates that for this specific set of physical conditions, the ‘wave-like’ signatures of quantum mechanics, usually manifest through h in the explicit scaling of conductance, appear to be effectively absorbed or cancelled within the energy- and geometry-dependent sum of transmission probabilities. This leads to an emergent transport behavior primarily dictated by the system’s energy scale ($E_F$) and scattering strength, rather than the explicit quantum unit of conductance or carrier concentration.

The concept of ‘classical-like’ behavior in this context warrants further clarification, particularly in comparison to the semiclassical limit where h→0. In the standard semiclassical approach, a system’s behavior approaches classical mechanics as quantum fluctuations become negligible. This often leads to a gradual disappearance of wave-like phenomena like interference and tunneling. However, our system operates in a regime where tunneling—a profoundly quantum wave-like effect—is still dominant. The ‘classical-like’ nature we observe here is therefore fundamentally different. It arises not from the vanishing of quantum effects, but from the collective behavior of an ensemble of quantum states. Specifically, while the individual transmission probability for each mode remains a quantum wave-like phenomenon, the sum over all transverse modes averages out the explicit dependence on the wave-like unit, h. The resulting macroscopic conductance is then primarily dictated by the density of available states at the Fermi energy (a state-counting property) and the scattering strength, with the explicit quantum wave scale effectively canceled out. This suggests an alternative pathway to classical emergence, where the system’s collective behavior is governed by state-counting principles, even as individual particle dynamics remain quantum wave-like.

This work sheds new light on an unconventional manifestation within the Landauer framework, demonstrating how the intricate interplay of system geometry and potential characteristics can lead to unexpected forms of scaling in macroscopic observables. Our findings offer a unique perspective on how different facets of quantum mechanics (e.g., state quantization versus wave phenomena) contribute to observable transport properties.

Looking forward, several avenues for future research emerge from these results. It would be valuable to explore whether similar h-independent transport regimes can be found in other low-dimensional systems or with different types of scatterers.

The fundamental conditions for the emergence of this h-independent transport—namely, a quasi-2D system with wide transverse confinement and a narrow, opaque barrier—are not specific to the GaAs/AlGaAs platform. In principle, this effect should also be observable in other quasi-2D conducting systems, such as graphene nanoribbons [25] or topological insulator edges [26], provided these systems can be engineered to meet the necessary geometric and scattering conditions.

This ‘classical’ phenomenon is not a mathematical artifact but a genuinely observable effect. Our analysis has shown that a similar ‘classical’ expression can be derived for any narrow barrier profile (Section 2), demonstrating that the cancellation of ℏ is a general characteristic of this class of systems, not a coincidence of a specific potential shape. Furthermore, this ℏ-independent behavior is not limited to a single point but holds as a valid approximation across a wide range of energies (Section 3 and Figure 2). The manuscript’s experimental realization section (Section 4) details how a practical experiment using a split-gate and back gate can be designed to achieve the specific physical conditions of our model, allowing for a direct measurement of this predicted phenomenon and its distinct transport characteristics.

Investigating the robustness of this behavior against small deviations from the ideal conditions (e.g., finite barrier width effects, intermediate barrier heights) would also be crucial. Furthermore, our findings could stimulate experimental efforts to detect this intriguing quantum phenomenon, providing empirical validation and potentially opening new pathways for the design of quantum electronic devices with tailored transport characteristics.