Vertical Non-Local Switching

• Vertical non-local switching is a framework where a vertical control parameter induces state changes in physically or algebraically separated layers or regimes.
• It spans diverse applications such as excitonic relocation in quantum rings, remote spintronic switching, and cross-tier reassignment in heterogeneous radio networks.
• Quantitative studies demonstrate its utility with metrics like exciton dipole reversal, subthreshold swings in FETs, and power savings in network switching architectures.

Vertical non-local switching is a domain-dependent research term used for several mechanisms in which a vertically applied field, a vertically separated actuator, a vertical network tier, or a vertically indexed regime induces a state change that is expressed elsewhere in real space, configuration space, or regime space rather than at the point of actuation itself. The expression appears in semiconductor quantum rings, spintronic tunnel structures, vertical heterogeneous radio networks, non-Hermitian control of topological sectors, ferroelectric inversion-domain-wall motion, and systems of variational inequalities for optimal switching (McDonald et al., 2010, Chanthbouala et al., 2011, Salamatmoghadasi et al., 15 Jan 2026, Yeom et al., 2024, Naudin et al., 19 Jun 2026, Lundström et al., 2013).

1. Terminological scope and unifying structure

In the cited literature, the term does not denote a single canonical mechanism. Instead, it labels several architectures in which a “vertical” degree of freedom acts as the control channel, while the switched observable is remote, laterally displaced, algebraically coupled, or encoded in another layer or sector. The shared pattern is not material-specific; it is the decoupling between the control coordinate and the switched degree of freedom.

Context	Vertical element	Switched quantity
Quantum rings	Electric field $F_z$ along [001]	Lateral exciton localization
vHetNets	Cross-tier SBS $\rightarrow$ MBS/HAPS reassignment	Serving layer and SBS ON/OFF state
Optimal switching PDEs	Obstacle coupling across regimes	Active regime under Lévy dynamics
Spintronic devices	Vertical transport or torque delivery	Current branch, DW position, or free-layer state
Toric code / ferroelectrics	$W_y$ sector or vertical IDW	Topological parity or polarization state

A common misconception is to treat the phrase as fixed jargon with one physical definition. In practice, its meaning is field-specific. In semiconductor nanostructures it denotes a vertical electric field that repositions an excitonic complex laterally; in radio access networks it denotes cross-tier reassignment across terrestrial and aerial layers; in stochastic control it denotes coupling across the regime index; and in spintronics it denotes vertically separated torque generation and magnetic switching (McDonald et al., 2010, Salamatmoghadasi et al., 15 Jan 2026, Lundström et al., 2013, Biagi et al., 26 May 2026).

2. Excitonic switching in semiconductor quantum rings

In self-assembled InGaAs/GaAs quantum rings, vertical non-local switching refers to the lateral relocation of excitonic complexes induced by a vertical electric field through the ring’s intrinsic piezoelectric anisotropy (McDonald et al., 2010). The modeled structure is a 50% (In,Ga)As random alloy ring embedded in GaAs, grown on a thin 30% In wetting layer, with inner radius $\approx 5$ nm, outer radius $\approx 20$ nm, and maximum height $\approx 6$ nm. The crystallographic growth direction is [001], and the relevant in-plane diagonals are [110] and $[1\bar{1}0]$.

The key ingredient is the strain-induced piezoelectric potential with $C_{2v}$ symmetry. It alternates in sign every $90^\circ$, producing four lateral lobes around the ring and additional smaller lobes of opposite sign in the GaAs core directly above and below the ring. Those core lobes penetrate the electrically active region and directly shape the electronic ground states. A vertical field $F_z$ polarizes the exciton, creating a vertical dipole; the dipole then couples selectively to the sign of the piezoelectric core lobes. For one sign of $\rightarrow$0, the exciton localizes along [110]; reversing $\rightarrow$1 raises the energy along [110] and favors localization along $\rightarrow$2. The resulting rotation of the exciton probability density is roughly $\rightarrow$3 while maintaining binding.

The model is a two-band effective-mass Hamiltonian in the full three-dimensional confinement of the strained heterostructure,

$\rightarrow$4

Here $\rightarrow$5 is derived from an atomistic valence-force-field strain model, and $\rightarrow$6 is computed to second order in strain. The energetics follow the usual Stark expansion,

$\rightarrow$7

with $\rightarrow$8 and lateral polarizabilities $\rightarrow$9 and $W_y$0 probed by weak in-plane test fields.

The simulations use Path Integral Quantum Monte Carlo at $W_y$1 K, with full electron–hole Coulomb correlations in continuous 3D space and observables averaged over eight atomistic realizations of the same 50% random alloy ring after imposing average $W_y$2 symmetry. The complexes studied are the neutral exciton $W_y$3 and the biexciton $W_y$4. At zero field the exciton already prefers [110] with the hole lower in the ring, producing a permanent negative vertical dipole. Under applied fields up to $W_y$5 kV/cm, the predicted observables are a sign change of $W_y$6, a reversal of the in-plane polarizability anisotropy, and—for biexcitons—energy shifts beyond the normal quadratic Stark law. Under sufficiently strong negative $W_y$7, pair-correlation functions indicate partial biexciton dissociation, with a second peak near $W_y$8 nm at $W_y$9 kV/cm. No hysteresis is reported; the relocation follows the sign of $\approx 5$0 smoothly.

3. Spintronic and orbitronic realizations

In spintronic vertical heterostructures, vertical non-local switching denotes a separation between the local electrostatic or current-driving element and the degree of freedom that actually determines whether current or magnetization switching occurs. One realization uses spin-gapless semiconductors (SGSs) or spin-gapped metals (SGMs) as electrodes in a vertical 2D Schottky-barrier transistor (Şaşıoğlu et al., 2024). The simulated device is a VS$\approx 5$1/Ga$\approx 5$2O$\approx 5$3/VS$\approx 5$4 vertical stack with dual gating, transport along the stacking direction, and a small electron Schottky barrier $\approx 5$5 eV at both interfaces. In the antiparallel configuration, gate bias reduces $\approx 5$6 from $\approx 5$7 eV to $\approx 5$8 eV. In the parallel configuration, for $\approx 5$9 V, the drain lacks available states in the current-carrying spin channel, so vertical transmission is blocked despite local source-barrier modulation. The paper defines

$\approx 20$0

and reports $\approx 20$1 mV/dec at $\approx 20$2 nm and 300 K, $\approx 20$3 $\approx 20$4A/$\approx 20$5m for $\approx 20$6–$\approx 20$7 V, $\approx 20$8 for $\approx 20$9 nm, and 100% non-local GMR at 0 K. The transport framework is spin-resolved Landauer theory,

$\approx 6$0

A second realization is vertical current-driven domain-wall motion in MgO-based magnetic tunnel junctions (Chanthbouala et al., 2011). There, the write current is injected vertically through the MTJ, but the switched object is a domain wall in an extended free-layer half-ring electrode. The Landau–Lifshitz–Gilbert equation includes both in-plane and field-like spin-transfer torques,

$\approx 6$1

The experiments identify the out-of-plane, field-like torque as the dominant driver of steady domain-wall displacement. Reported current densities are $\approx 6$2 A$\approx 6$3cm$\approx 6$4, typically $\approx 6$5 A$\approx 6$6cm$\approx 6$7, about 100 times smaller than in in-plane current geometries. At $\approx 6$8 Oe, the resistance changes from 16.6 $\approx 6$9 to $[1\bar{1}0]$0 $[1\bar{1}0]$1 at $[1\bar{1}0]$2 mA and to $[1\bar{1}0]$3 $[1\bar{1}0]$4 at $[1\bar{1}0]$5 mA. The OOP torque reaches up to $[1\bar{1}0]$6 of the in-plane torque for $[1\bar{1}0]$7 mV, and $[1\bar{1}0]$8 Oe at $[1\bar{1}0]$9 A$C_{2v}$0cm$C_{2v}$1.

A third realization uses orbital transport in Ta/W-based three-terminal SOT-MTJs (Biagi et al., 26 May 2026). Here vertical non-local switching means that the source of angular momentum is physically separated from the free magnetic layer along the vertical direction. Ta acts as an orbital source, W as an orbital-to-spin converter, and the angular momentum is transported across a metallic spacer to the MTJ. The measured damping-like efficiencies follow

$C_{2v}$2

with $C_{2v}$3 nm and $C_{2v}$4 for W-only, $C_{2v}$5 nm and $C_{2v}$6 for Ta-only, and $C_{2v}$7 nm and $C_{2v}$8 for Ta/W, reduced only marginally to $C_{2v}$9 after 400$90^\circ$0C anneal. In proof-of-concept vertical non-local devices, Ta spacers of $90^\circ$1 nm and $90^\circ$2 nm still allow clear SOT switching with pulses down to 1 ns. COMSOL analysis yields $90^\circ$3–20.2 nm, whereas the experimental switching-current growth gives $90^\circ$4 nm, supporting an additional long-range contribution beyond pure current shunting.

4. Vertical non-local switching in vertical heterogeneous radio networks

In radio access networks, vertical non-local switching is a cross-tier cell-switching strategy in a vertical heterogeneous network comprising one HAPS acting as a super macro base station, one terrestrial MBS, and multiple SBSs (Salamatmoghadasi et al., 15 Jan 2026). In this usage, “vertical” refers to the layered terrestrial–aerial architecture, and “non-local” refers to traffic reassignment from a deactivated small cell to higher tiers rather than only to nearby terrestrial cells.

The studied network places one HAPS at altitude $90^\circ$5 km, one MBS at the center of a $90^\circ$6 km $90^\circ$7 $90^\circ$8 km dense-urban area, and $90^\circ$9 SBSs, with default $F_z$0 and small-cell radius $F_z$1 m. HAPS and MBS are always ON; SBSs can be ON or OFF. Each SBS $F_z$2 has a load factor $F_z$3, and when SBS $F_z$4 is turned OFF, its users are offloaded either to the MBS or the HAPS. The optimization variables are $F_z$5 for SBS state, $F_z$6 for offloading choice, and $F_z$7 for linearization of the product $F_z$8. Aggregate load constraints are enforced through

$F_z$9

$\rightarrow$00

with $\rightarrow$01 and $\rightarrow$02.

QoS is modeled by a deterministic received-power threshold $\rightarrow$03: if $\rightarrow$04, the selected serving tier must satisfy

$\rightarrow$05

The energy objective uses an EARTH-based linear load model, and the resulting MINLP is linearized into an MIP with McCormick constraints and solved with SCIP. Default capacities are $\rightarrow$06, $\rightarrow$07, and $\rightarrow$08.

With realistic 3GPP TR 38.901 and TR 38.811 channel models, the HAPS-enhanced cell-switching algorithm reduces total power consumption by up to 77% relative to All-ON at low load and about 40% at high load; the NoQoS variant reaches up to 90% and 47%, respectively. The method keeps total served traffic with QoS nearly equal to All-ON across loads. Sensitivity analysis shows that savings decrease as SBS density rises, with savings at $\rightarrow$09 dropping from $\rightarrow$10 for $\rightarrow$11 to $\rightarrow$12 for $\rightarrow$13. Runtime scales roughly linearly with $\rightarrow$14, with average runtime $\rightarrow$15 s for $\rightarrow$16 on the tested platform.

5. Topological-sector control and ferroelectric domain-wall motion

In non-Hermitian topological matter, vertical non-local switching refers to controlled transfer between the toric-code ground states distinguished by the vertical Wilson-loop sector $\rightarrow$17 (Yeom et al., 2024). The four ground states $\rightarrow$18 are labeled by the eigenvalues of the non-contractible Wilson loops $\rightarrow$19 and $\rightarrow$20. The vertical sector is the parity $\rightarrow$21 associated with $\rightarrow$22. To switch that sector, the symmetry-related perturbation

$\rightarrow$23

is introduced. In the ground-state subspace it becomes block diagonal and couples $\rightarrow$24 at fixed $\rightarrow$25, with eigenvalues

$\rightarrow$26

Exceptional points occur at $\rightarrow$27. Encircling an EP yields $\rightarrow$28 modulo 2. The work further reports orientation-dependent non-adiabatic transitions: clockwise encirclement can yield near-perfect switching in the quasi-adiabatic regime, whereas counterclockwise encirclement can suppress switching. For local non-Hermitian perturbations, EPs between ground states can be absent when both $\rightarrow$29 and $\rightarrow$30 are even because of symmetry constraints.

In wurtzite AlN-based ferroelectrics, vertical non-local switching denotes polarization reversal by lateral displacement of a vertical inversion domain wall parallel to [0001], rather than by local nucleation directly under the driving electrode (Naudin et al., 19 Jun 2026). The wall is electrically neutral, Ising-like, atomically sharp, and two atomic layers thick perpendicular to $\rightarrow$31. Its formation energy is $\rightarrow$32 meV \AA$\rightarrow$33. NEB calculations show that displacement by one atomic layer along $\rightarrow$34 proceeds through chain-by-chain reversal of [0001]-oriented atomic columns. For pristine AlN, the total chain-by-chain barrier is $\rightarrow$35 meV \AA$\rightarrow$36; with neutral V$\rightarrow$37 it becomes $\rightarrow$38 meV \AA$\rightarrow$39, while neutral V$\rightarrow$40 yields strong pinning with $\rightarrow$41 meV \AA$\rightarrow$42. In the coherent path, pristine AlN gives $\rightarrow$43 meV \AA$\rightarrow$44, V$\rightarrow$45 reduces it by $\rightarrow$46, Sc$\rightarrow$47 by $\rightarrow$48, B$\rightarrow$49 increases it by $\rightarrow$50, and O$\rightarrow$51 produces a single large barrier of $\rightarrow$52 meV \AA$\rightarrow$53. All studied defects except La$\rightarrow$54 are more stable at or near the wall, and the wall itself reduces the band gap by $\rightarrow$55 eV. The paper therefore links wall-mediated switching, defect segregation, coercive-field variation, and leakage pathways.

6. Mathematical and abstract formulations

A mathematically precise use of the phrase appears in optimal switching problems with non-local operators, where “vertical” refers to coupling across the regime index and “non-local” refers to the Lévy jump operator (Lundström et al., 2013). For $\rightarrow$56 regimes, value functions $\rightarrow$57 satisfy the coupled system

$\rightarrow$58

with terminal condition $\rightarrow$59. The obstacle term is the vertical coupling, while the non-local integral operator $\rightarrow$60 comes from the Lévy jump-diffusion dynamics. The paper establishes a comparison principle and then existence and uniqueness of a viscosity solution by Perron’s method under continuity, polynomial growth, terminal compatibility, and a no-loop condition on switching costs; a key contribution is that no sign restriction is imposed on the switching costs.

A related stochastic-game formulation appears in vertical competition with mixed impulse and regime-switching controls (Aïd et al., 2020). There, the upstream firm changes the commodity price through impulses, while the downstream firm changes the drift through regime switching. The state dynamics are written as

$\rightarrow$61

The resulting equilibrium problem couples a non-local impulse QVI for the upstream producer with a switching QVI for the downstream firm, and the analysis exhibits three equilibrium types: generic double-switch, transitory single-switch, and preemptive equilibria.

An even more abstract usage appears in Dynamic Switching Networks (Khalili, 2017). There, the network has eight outgoing directions per node, complete global code, and access to initial positions and motion equations of all particles. The law-to-switching conversion is written as minimizing the Euclidean distance between target positions $\rightarrow$62 and the positions generated by switching decisions $\rightarrow$63,

$\rightarrow$64

The framework is explicitly described as dynamic, non-local, and time independent. Vertical symmetry around the $\rightarrow$65-axis is implemented through the rules $\rightarrow$66 and $\rightarrow$67, and the network may use future events provided it does not create logical loops.

7. Recurring observables, limitations, and interpretive boundaries

Across these literatures, the switched quantity is typically not the variable directly driven by the control channel. In quantum rings, a field along [001] changes lateral localization and is expected to be observed through Stark shifts, dipole measurements, polarizability anisotropy, recombination-lifetime changes, and biexciton photoluminescence residuals after subtraction of a quadratic Stark background (McDonald et al., 2010). In SGS/SGM FETs, the vertical gate-controlled source barrier and the remote drain DOS are read out through $\rightarrow$68–$\rightarrow$69, $\rightarrow$70–$\rightarrow$71, subthreshold swing, and non-local GMR (Şaşıoğlu et al., 2024). In MgO MTJs and Ta/W SOT-MTJs, the observables are stepwise resistance states, switching-current asymmetries, torque-effective fields, and intrinsic $\rightarrow$72 extracted from pulse-width scaling (Chanthbouala et al., 2011, Biagi et al., 26 May 2026). In vHetNets, the corresponding observables are total power consumption, served-traffic-with-QoS, outage proxy compliance through $\rightarrow$73, and solver runtime (Salamatmoghadasi et al., 15 Jan 2026). In non-Hermitian toric-code control, the signatures are phase rigidity, eigenstate coalescence at exceptional points, and orientation-dependent switching probability under dynamic encirclement (Yeom et al., 2024). In AlN ferroelectrics, the relevant outputs are domain-wall migration barriers, defect segregation energies, gap narrowing, and leakage-related defect states (Naudin et al., 19 Jun 2026).

The main limitations are likewise domain-specific. In quantum rings, simulations are reported at 10 K and no hysteresis is found (McDonald et al., 2010). In vHetNets, the QoS model uses a received-power threshold $\rightarrow$74 as a proxy rather than explicit per-user rate or latency optimization (Salamatmoghadasi et al., 15 Jan 2026). In SGS-based proof-of-concept FETs, VS$\rightarrow$75 is noted to have a theoretically estimated $\rightarrow$76 K, motivating the search for higher-$\rightarrow$77 alternatives (Şaşıoğlu et al., 2024). In Ta/W MTJs, disentangling SHE and OHE channels remains intrinsically difficult, and the orbital diffusion length is still modest (Biagi et al., 26 May 2026). In toric-code switching, non-Hermiticity requires open-system or engineered gain/loss control, and even-by-even system sizes can forbid EPs between ground states (Yeom et al., 2024). In the AlN defect study, PBEsol is used, Berry-phase polarization is not explicitly computed, and charged-defect corrections are not applied (Naudin et al., 19 Jun 2026).

These differences matter because the phrase is operational rather than universal. This suggests that “vertical non-local switching” is best understood as a structural descriptor: a vertical control coordinate, vertical separation, or vertical regime index selects among states that are spatially displaced, topologically encoded, or algebraically coupled, with the concrete mechanism determined by the host field rather than by the phrase itself.