Gate-Tunable Quantum Resonant Tunneling

Updated 19 August 2025

Gate-tunable quantum resonant tunneling is the control of discrete quantum states via external gates that adjust energy detuning and tunnel coupling, enabling coherent quantum transport.
Devices such as graphene double quantum dots and silicon MOS structures use multi-gate configurations to finely tune resonance conditions, observable as conductance peaks and honeycomb diagrams.
Experimental signatures, including peak splitting, FWHM analysis, and negative differential resistance, underpin advancements in quantum electronics and fast quantum operations.

Gate-tunable quantum resonant tunneling refers to the ability to control quantum tunneling rates and resonance conditions between discrete energy levels in mesoscopic or nanoscale systems using external gate voltages. This phenomenon is central to quantum electronics, enabling both precise mapping of charge and spin states and the realization of molecular regimes, quantum phase transitions, and fast, coherent quantum operations. The following sections provide a comprehensive review organized along the most significant experimental and theoretical dimensions.

1. Physical Principles and Device Archetypes

Gate-tunable quantum resonant tunneling emerges when discrete quantum states, such as those in quantum dots (QDs), quantum wells, or engineered artificial molecules, are coupled together such that transport is dominated by quantum coherence and resonance conditions. Electrostatic gates are used to modify two key parameters:

Energy detuning ( $\epsilon$ ): The energy difference between quantum states in adjacent sites (e.g., two dots).
Tunnel coupling ( $t$ ): The hybridization energy scale due to quantum mechanical overlap dependent on the tunnel barrier height and width.

In graphene double quantum dot (DQD) devices (Wang et al., 2010, Wei et al., 2013), multiple lateral gates (GL, GR, G1, GM, G2) are used to tune both the energy levels and the transparency of tunnel barriers. In Si/SiGe and silicon MOS structures, overlapping or stacked gate architectures define localized quantum islands and the alignment of these discrete levels with source/drain reservoirs (Spruijtenburg et al., 2013, Zajac et al., 2015). In oxide superlattices, the alignment of quantum well subbands by external bias can also be modulated by gate voltages (Choi et al., 2015).

The effect is observable as sharp peaks in conductance or transconductance when quantized energy levels align with each other or the chemical potential of the leads, leading to enhanced tunneling (resonance). The degree of control enabled by gating distinguishes these systems from traditional fixed-barrier resonant tunneling devices.

2. Gate Control of Tunneling Regimes

Gate voltages fundamentally determine whether a system is in the localized (atomic-like), hybridized (molecular-like), or interacting many-body regime:

Weak coupling regime ( $t \ll \epsilon$ ): Tunneling is suppressed; electrons are localized. The charge stability diagram exhibits isolated “honeycomb” cells, each corresponding to a unique electron occupation configuration $(N_L, N_R)$ . Resonant tunneling occurs only at triple points where chemical potentials align.
Strong coupling regime ( $t \sim \epsilon$ ): Tunneling is enhanced, resulting in coherent molecular orbitals extending over both dots/sites. The eigenstates are coherent superpositions:

$|{\Psi}_{B}\rangle = -\sin (\theta/2) e^{-i\phi/2}|{N_L+1, N_R}\rangle + \cos (\theta/2) e^{i\phi/2}|{N_L, N_R+1}\rangle,$

$|{\Psi}_{A}\rangle = \cos (\theta/2) e^{-i\phi/2}|{N_L+1, N_R}\rangle + \sin (\theta/2) e^{i\phi/2}|{N_L, N_R+1}\rangle,$

with mixing angle $\theta = \arctan(2t/\epsilon)$ .

The energy splitting is: $E_\Delta = U' + \sqrt{\epsilon^2 + (2t)^2},$ where $U'$ quantifies mutual capacitance.

Fully resonance-tuned:
- By tuning both detuning and tunnel coupling via gates, one can induce transitions between the two regimes (Wang et al., 2010, Wei et al., 2013).
- This process is evidenced by the transformation of the honeycomb diagram: triple points evolve from sharp vertices to broadened or merged features as tunnel coupling increases.

Devices with multi-gate configurations can independently modulate $\epsilon$ and $t$ :

In graphene DQDs, the middle gate (MG) specifically tunes $t$ exponentially, and plunger gates control $\epsilon$ (Wei et al., 2013).
In Si/SiGe architectures, dedicated barrier gates alter $t_c$ , and plunger gates control dot occupancy and local detuning (Zajac et al., 2015).

A summary of key formulas governing these regimes:

Parameter	Formula	Physical Meaning
Mixing angle	$\theta = \arctan \left(2t / \epsilon \right)$	Superposition weight control
Energy splitting	$E_\Delta = U' + \sqrt{\epsilon^2 + (2t)^2}$	Bonding-antibonding gap
Tunnel coupling	Extracted from linewidths/peak splitting	Quantifies resonant tunneling

3. Experimental Manifestations and Characterization

The experimental signatures of gate-tunable quantum resonant tunneling include:

Honeycomb charge stability diagrams in gate–gate parameter space, directly mapping out the fixed charge state regions and triple/triple-point resonances. The degree of splitting and blurring of vertices serves as a proxy for $t$ .
FWHM analysis of charge transition peaks: Peak widths in the detuning axis encode both electron temperature and the intrinsic tunnel coupling, with the relation,

$\mathrm{FWHM}(E) = \sqrt{(2t_c)^2 + (\alpha k_B T_e)^2}$

(Wei et al., 2013),

Excitation spectroscopy and Coulomb diamonds: The onset of current at the diamond boundaries sets the charging energy and indicates when discrete quantum levels align for tunneling, with further resonant structure revealing excited states (Spruijtenburg et al., 2013).
Current/voltage peak shifts under gate action: As gate voltages are swept, both peak position (reflecting $\epsilon$ ) and splitting/broadening (reflecting $t$ ) change continuously.
Negative differential resistance (NDR): Observed in vertical graphene heterostructures and oxide superlattices, where gate tuning of band alignment enhances or suppresses resonant tunneling (Fallahazad et al., 2014, Choi et al., 2015).

4. Theoretical Frameworks and Analytical Models

The theoretical description relies on an effective two-level Hamiltonian for double quantum dots or quantum wells: $\mathcal{H} = \frac{\epsilon}{2} \sigma_z + t \sigma_x$ with eigenenergies: $E_\pm = \pm \frac{1}{2} \sqrt{\epsilon^2 + (2t)^2}$ This model captures the essential mixing and energy hybridization between dots.

Extraction of key device parameters is performed via

Peak splitting measurements (directly yield $2t$)
Capacitance network analysis (relates gate voltages to detuning energies via lever arms and cross-capacitances)
Spectral response to temperature: Enables discrimination between thermally and tunnel-coupling–limited regimes.

The noise and intrinsic broadening analysis is crucial for establishing the lower bound on tunnel coupling and qubit operation speed, with thermal contributions subtracted via low- $t$ regime measurements (Wei et al., 2013).

5. Quantum Information and Device Implications

Gate-tunable quantum resonant tunneling is foundational for solid-state quantum engineering:

Fast gate operation: Large $t$ values achieved (e.g., $t \approx 727\,\mu$ eV in graphene DQDs (Wang et al., 2010)), enabling quantum gate times as short as $\tau \approx 50$ ps.
Long coherence times: Materials like graphene offer nuclear-spin-free environments, minimizing hyperfine decoherence channels and optimizing coherence for both charge and spin qubits.
Charge and spin manipulation: Electric gates provide localized, high-fidelity, and scalable control of both charge and spin states, essential for manipulating individual qubits in quantum circuits.
Molecular regime as qubit basis: The transition from localized to delocalized (molecular-like) states under precise gate control enables encoding of qubit states in spatial superpositions.

A comparison of fabrication and control approaches:

System	Gate Controls	Notable Features
Graphene DQD	Multi-lateral gates	Strong, fast-coupling, nuclear-spin-free
Si/SiGe, Si MOS QDs	Overlapping gates	Few-electron, charge sensing, valley splitting
Vertical heterostructures	Back, top gates	Band alignment, NDR, ultrathin barriers

6. Limitations and Scaling Considerations

Practical implementation requires:

Minimization of disorder: Edge and bulk disorder (especially in etched graphene) must be minimized to ensure reproducible tunability and low-variance tunnel couplings (Wei et al., 2013).
Precise gate cross-talk calibration: Capacity matrices must be well understood to relate voltages to energy detuning and lever arms for accurate extraction of $t$ and $\epsilon$ .
Suppression of charge noise: Low-frequency charge fluctuations can obscure quantum coherence. Careful design of the gating architecture is required.
Thermal management: As device scaling proceeds to larger arrays, heat dissipation and thermal broadening limit operation at higher temperatures unless materials and circuits are optimized.

7. Broader Context and Future Directions

Gate-tunable quantum resonant tunneling is actively being integrated into:

Quantum computing architectures: As the physical basis for fast and scalable charge/spin qubits, supporting essential operations such as the $\sqrt{\mathrm{SWAP}}$ and CNOT gates.
Quantum simulators: Enabling emulation of molecular physics (e.g., artificial molecules), Kondo and Luttinger liquid physics through tunable coupling and device engineering [see broader implications in (Mebrahtu et al., 2012, Liu et al., 2013, Karki, 7 Jul 2024)].
Next-generation nanoelectronics and quantum logic devices: Exploiting NDR and tunability for novel transistor concepts and logic elements (Fallahazad et al., 2014, Choi et al., 2015).
Integrated quantum-classical electronics: As realized by monolithic integration on advanced CMOS platforms for scalable quantum processor arrays (Bashir et al., 2021).

Gate-tunable systems allow for continuous and reversible in situ control over quantum coupling and resonance conditions, offering a powerful technological foundation for both fundamental quantum science and scalable quantum device applications.