S-scheme Heterojunctions: Theory & Applications

• S-scheme heterojunctions are interfaces between distinct semiconductors that use staggered band alignments and engineered potentials to enhance efficient charge separation and transport.
• Mathematical frameworks leveraging the effective-mass Schrödinger equation and supersymmetric transformations enable smooth modeling across abrupt interface discontinuities.
• These heterojunctions underpin advanced devices in optoelectronics, photovoltaics, and photocatalysis, achieving breakthroughs in efficiency and scalable integration.

An S-scheme heterojunction is a type of interface between distinct semiconductors or materials that is engineered to facilitate efficient charge separation, transfer, and transport by exploiting selective band alignment, abrupt variations of effective mass, and tailored interface grading. In S-scheme configurations, either the band structure forms a staggered (step-like) profile across the junction or selective carrier recombination and retention mechanisms are enabled, often involving additional built-in electrostatic fields or engineered potential landscapes. These schemes have become central to high-efficiency optoelectronics, tunneling transistors, and photocatalytic architectures due to their capacity for superior charge extraction, suppressed recombination, and controllable electronic properties.

1. Fundamental Principles and Mathematical Framework

S-scheme heterojunctions generally involve joining two semiconductor regions with different band edges and effective masses, leading to spatial discontinuities in the electronic Hamiltonian. The effective-mass Schrödinger equation describing carrier dynamics across the heterojunction, particularly in the presence of abrupt band or mass changes, exhibits singularities at the interface: $$H_0(z)\psi_n(z) = \left[ -\frac{d}{dz} \left( \frac{\hbar^2}{2 m^*(z)} \frac{d}{dz} \right) + V_0(z) \right] \psi_n(z) = E_n \psi_n(z)$$ where $m^*(z)$ is the position-dependent effective mass with step-like grading implemented through Heaviside functions $G_j$, per region. A central technical challenge is that the kinetic energy operator becomes non-commutative with abrupt spatial variations in mass, producing delta-function singularities that physically correspond to rapid changes in the local band structure and effective mass—yielding nonphysical discontinuities in the wave function derivatives.

Supersymmetric quantum mechanics provides an avenue to circumvent these singularities by factorizing the Hamiltonian: $$H_0^{(j)} = A_j A_j^{\dagger} - \mathcal{E}, \qquad A_j = \frac{\hbar}{\sqrt{2m_j}} \frac{d}{dz} + W(z)$$ At the junction, a global ladder operator $A$ is constructed by patching together or averaging the local ladder operators $A_j$, yielding: $$A = \sum_j G_j A_j = \frac{\hbar}{\sqrt{2m^*(z)}} \frac{d}{dz} + W(z)$$ where $W(z)$ is a carefully chosen smooth superpotential. The partner Hamiltonian,

$\hat{H}_0(z) = A^{\dagger}A - \mathcal{E}$

is manifestly free of the problematic singular terms present in the original formulation, enabling physically sensible, smooth solutions across the interface.

2. Junction Matching and Analytical Potentials

The mapping between original and transformed wavefunctions,

$\psi(z) \propto A \tilde{\psi}(z)$

requires continuity at the interface. The matching conditions typically enforce that both the wavefunction and its derivative are continuous across the junction: $$\tilde{\psi}'(a_j+0) = \tilde{\psi}'(a_j-0) = \tilde{\psi}'(a_j)$$ These conditions guarantee regularity in the supersymmetrically transformed potential,

$$\tilde{V}_0(z) = W^2(z) - \left( \frac{\hbar}{\sqrt{2m^*(z)}} W(z) \right)' - \mathcal{E}$$

such that the delta-singular behavior is eliminated from $\hat{H}_0$.

Practically, in devices with graded intermediate layers (e.g., Ga$_{1-x}$Al$_x$As/GaAs), the mass grading can be modeled as an exponential: $m_2(z) = m_0 \beta^2 \exp(2\beta z/a_0)$ This grading scheme allows for the transformation of the original problem into a set of Morse-type potentials via appropriate choice of $W(z)$;

$$W(z) = \frac{\hbar}{a_0 \sqrt{2m_0}} \frac{1}{\mathcal{A} - \mathcal{B} e^{-\beta z/a_0}}$$

After transformation, the enveloping potential in each region is analytically tractable, facilitating exact solutions in terms of confluent hypergeometric functions.

3. S-scheme Band Alignment and Carrier Transport

In vertical tunneling field-effect transistors (VTFETs) and similar S-scheme architectures, the unique band alignment modulates the barrier height (rather than merely its width) at the critical junction region. Materials are selected to produce staggered conduction and valence bands, allowing for simultaneous onset of vertical and lateral tunneling at the pocket region, yielding exceptionally steep switching characteristics:

• Subthreshold swing as low as 16 mV/decade is achieved (Ganapathi et al., 2011).
• ON currents are enhanced via increased vertical tunneling area, even when the max current is reduced by the staggered alignment.

Quantum mechanical transport in these devices—including highly engineered band offsets and geometries—is rigorously modeled via NEGF approaches, often using a 4×4 k·p Hamiltonian to capture band structure and all relevant tunneling channels. Tuning alloy composition and geometry enables precise control of turn-on voltage, ON currents, and off-state leakage.

4. Interfacial Structure, Defects, and Local Percolation

Atomic-scale studies using conduction atomic force microscopy reveal that S-scheme heterojunctions, especially in solar cells with amorphous/crystalline contacts such as a-Si:H/c-Si, exhibit complex, highly non-uniform local behavior (Teferi et al., 2020). The:

• Presence of bandtail states in the amorphous region leads to current patches—nm-scale conduction pathways that dominate dark leakage current via trap-assisted tunneling.
• Local open circuit voltages can exceed the macroscopic bandgap due to strong potential fluctuations arising from randomly trapped charges.
• The device-level (macroscopic) behavior is an average over many non-uniform microscopic junctions, with random telegraph noise and spatial variations arising from defect-induced band edge shifts.

Mitigation requires careful engineering of the selective contact layer to minimize the density and energetic distribution of bandtail states where trap-assisted processes are active.

5. Photocatalysis and S-scheme Charge Transfer

S-scheme concepts have been extended to photocatalytic systems where charge carrier separation and selective recombination drive chemical transformations. Notably, vertically stacked SeWS/bilayer-SiC heterojunctions combine:

• Type-II direct band gap (1.67 eV), promoting efficient electron–hole separation (Yang et al., 23 Jun 2025).
• Large anisotropic hole mobility (9.58×10³ cm² V⁻¹ s⁻¹) and high visible-light absorption (10⁵ cm⁻¹).
• S-scheme mechanism: built-in electric field facilitates selective retention of high-energy electrons (SiC CBM) and holes (WSSe VBM), with low-energy carriers recombined at the junction for maximal redox potential.

External electric fields and biaxial strain enable further band gap modulation, with strong strain dependence of absorption coefficients. Hydrogen evolution efficiency achieves 22.15%, well beyond commercial viability thresholds.

6. Advanced Device Architectures and Scalability

S-scheme heterojunctions underpin numerous scalable device architectures:

• Self-aligned van der Waals heterojunctions offer sub-diffraction channel lengths (down to 135 nm) and dual-gate electrostatic control for precise modulation of depletion regions and anti-ambipolar characteristics (Sangwan et al., 2018).
• Vertical oxide p–n heterojunctions (e.g., SnO/β-Ga₂O₃) demonstrate large breakdown voltages and low on-resistance, enabled by carefully controlled type-I band alignment and thermionic emission transport mechanisms (Budde et al., 2020). Lowering substrate donor densities and optimizing contact geometry further improves breakdown performance and device isolation.
• Semiconductor grafting techniques with ultrathin oxygen-enriched interlayers (as in GaAs/Si tunnel diodes) allow high peak-to-valley ratios (PVCR = 36.38), robust negative differential resistance, and efficient quantum tunneling across lattice-mismatched interfaces (Zhou et al., 24 Sep 2024).

These approaches demonstrate the applicability of S-scheme concepts across multiple material platforms, ranging from organic-inorganic tandem solar cells to high voltage and high-frequency electronics.

7. Interface Engineering, Recombination, and Efficiency Optimization

Achieving high optoelectronic efficiency in S-scheme heterojunctions further relies on precise interface engineering:

• Optimization of TCO work function (e.g., 5.54 eV in a-Si/c-Si solar cells) is critical for favorable band alignment, built-in potential, and quantum efficiency, boosting cell efficiency up to 21.849% (AQing et al., 2013).
• Surface recombination velocity $S$ drastically affects electroluminescence (EL) and photoconversion: EL quantum efficiency is inversely proportional to $S$, while photoconversion efficiency exhibits only a weak logarithmic dependence. Temperature-dependent studies show injection-level-dependent radiative recombination, with tunnel currents dominating at low temperatures (Sachenko et al., 2017).
• Intense sulphurization and Ga-grading in chalcopyrite thin-film solar cells shift band edges to form a spike-like barrier, reduce p-doping, expand depletion width, and enhance selectivity, confirmed by DFT and device measurements (Cojocaru-Miredin et al., 2021).

Careful grading and interface treatments enable the design of S-scheme heterojunctions with minimal recombination, optimal carrier selectivity, and enhanced device efficiency.

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S-scheme heterojunctions embody a rigorous engineering paradigm for charge transport, separation, and extraction by leveraging advanced concepts in band alignment, supersymmetric transformations, quantum transport, and interface science. Their deployment has led to significant advancements in fields spanning nanoelectronics, photovoltaics, and photocatalysis, with ongoing research focused on further controlling local energetics, defect states, and scalable device integration.