The Multi-Regime Semiconductor Model

Table of Contents#

1. Purpose and Scope
2. Regime Definitions
3. The Classical Carrier Regime (CCR)
4. The Coherent Phase Regime (CPR)
5. The Transitional Regime (TR)
6. Material Parameters and Model Inputs
7. Regime Boundary Conditions
8. Derivation of Observable Parameters
9. SC Class as a Regime Indicator
10. Multi-Regime Interactions and Failure Modes
11. Process Effects on Regime Structure
12. Model Validation and Calibration
13. Design Implications
14. Glossary
15. Related Documents

1. Purpose and Scope#

1.1 Document Role#

This document establishes the physical theory underlying temporal manufacturing. The prior documents in the post-ASML era series — the Temporal Manufacturing Primer, the SCR Specification, the TGI Metrology Standard, the TCT Protocol, the TRS-Aware PDK Specification, and the Logic Folding Architecture Guide — treat empirical material parameters (Substrate Clarity SC, coherence length L_c, relaxation time τ_relax, coherence layer gradient CLG, and associated derived quantities) as externally measured inputs. This document defines the physical model from which those parameters arise.

The Multi-Regime Semiconductor Model (MRSM) provides:

• A conceptual and mathematical account of the three coexisting electronic regimes present in post-ASML substrates.
• Derivations connecting regime-level physics to each observable parameter used in the series.
• Physical explanations for inter-regime coupling, noise sources, and temporal address encoding.
• A framework for predicting how process variations and operating conditions perturb measured quantities.
• Validation criteria that allow experimental confirmation of model predictions.

This document is informative, not normative. It does not impose requirements on equipment, process, or design. Normative requirements appear in the documents listed in Section 15. Where this document makes claims about specific numerical values, those values are representative of the regime taxonomy and are not mandatory limits.

1.2 Scope#

The MRSM applies to:

• Post-ASML substrate materials classified as SC-I, SC-II, or SC-III under the TCT Protocol.
• The full z-extent of the substrate stack from Tier 1 (structural) through Tier 2 (temporal) layers.
• The TGI transition zone connecting these two tiers.
• Multi-zone SCR architectures where regime behavior must remain consistent across zone boundaries.

The MRSM does not extend to:

• Carrier transport in Tier 1 interconnect layers above the structural base (governed by classical electromigration and RC models).
• Packaging-level electromechanical interfaces.
• Equipment-level physics (TCU injection mechanisms, ARS readback transduction).

1.3 Relationship to Prior Documents#

The MRSM sits beneath all six prior documents in the causal hierarchy. Every empirical parameter that appears in normative documents is a projection of the regime model onto a measurement instrument's observables. The relationship is summarized as follows:

MRSM (physical model)
  └─ TCT Protocol        → measures SC, AER(Δτ), τ_relax, L_c
       └─ TGI Metrology  → measures CLG, IC, SC_eff, TAOE
            └─ PDK Spec  → derives Δτ_eff, Δτ_xtalk, SCLM, TDR rules
                 └─ Logic Folding  → consumes TSH, L_c, Δτ_eff, T_c
                      └─ SCR Spec  → uses T_c, SLF, zone sizing

Understanding this hierarchy is not required for production use of any individual document. It is, however, essential for process development, material qualification, and fault analysis that require connecting measurement anomalies to physical root causes.

2. Regime Definitions#

2.1 Overview#

Post-ASML substrates exhibit three distinct electronic regimes that can coexist simultaneously within a single die. The regimes are distinguished by the dominant mechanism governing the electronic state of the material at a given depth z and temperature T.

The three regimes are not discrete material phases in the thermodynamic sense. They are operational regimes: functional states of the electronic system under the boundary conditions imposed by the substrate stack structure, the applied process history, and the operating temperature and clock cycle.

2.2 Regime Assignment#

Regime assignment is a local property at position (x, y, z). For a given point:

• z < z_f (below the TGI floor): CCR dominant. CPR order parameter ψ(z) ≈ 0.
• z_f ≤ z ≤ z_c (within the TGI zone): TR active. ψ(z) interpolates continuously between 0 and ψ_bulk.
• z > z_c (above the TGI ceiling, within Tier 2): CPR dominant. ψ(z) ≈ ψ_bulk.

The boundaries z_f and z_c are the TGI floor and ceiling as defined in the TGI Metrology Standard. These boundaries are not sharp in the crystallographic sense; they are defined operationally as the depths at which the normalized phase order parameter crosses 0.05 and 0.95 respectively.

2.3 Regime Co-Presence in Device Operation#

In a functioning post-ASML device, all three regimes are simultaneously active:

• CCR governs signal routing, power delivery, and Tier 1 structural functions.
• TR (at the TGI) mediates the transfer of spatial patterning information into temporal address space during fabrication, and couples the two functional domains during operation.
• CPR in the Tier 2 bulk is the medium in which temporal addresses are committed, held, and read back.

A failure in any regime, or at any regime interface, propagates across all three. The MRSM provides the theoretical basis for diagnosing such failures by correlating observable anomalies with regime-level physical mechanisms.

3. The Classical Carrier Regime (CCR)#

3.1 Physical Mechanism#

In the CCR, the electronic state of the material is governed by band-structure charge transport. Mobile carriers — electrons in the conduction band and holes in the valence band — respond to local electric fields and chemical potential gradients according to the drift-diffusion equation:

J = q(n μ_e + p μ_h) E + q(D_e ∇n − D_h ∇p)

where:

• J = carrier current density [A m⁻²]
• q = elementary charge
• n,p = electron and hole carrier densities [m⁻³]
• μ_e, μ_h = electron and hole mobilities [m² V⁻¹ s⁻¹]
• E = local electric field [V m⁻¹]
• D_e, D_h = electron and hole diffusion coefficients [m² s⁻¹]

The CCR is the regime of classical CMOS operation. It supports transistor switching, voltage-domain logic, and capacitive signal storage. It does not support temporal address encoding, because temporal addressing requires a collective phase-coherent state that is absent when carriers are individually incoherent.

3.2 CCR Characteristic Parameters#

These parameters are standard inputs to SPICE-level models in Tier 1 design flows and are not redefined here. They are listed because they re-enter the MRSM at the regime boundary: the Debye screening length λ_D sets a lower bound on the CPR's spatial coherence at the TGI, as described in Section 7.

3.3 CCR Behavior Under Temporal Injection#

When TCU injection pulses are applied at depths within the CCR-dominant zone (z < z_f), the injection energy is absorbed by carrier scattering processes and dissipated as heat. No persistent phase state is established. The AER measured on a CCR-only coupon asymptotically approaches 0.5 at all Δτ spacings — the signature of a complete absence of discrimination. This behavior is used in calibration coupons during TCT instrument validation (TCT Protocol, Section 12).

A material whose Tier 2 layer remains in the CCR (due to insufficient phase-coherent doping, excessive thermal processing, or process damage) will fail SC classification entirely and cannot be qualified for temporal manufacturing.

4. The Coherent Phase Regime (CPR)#

4.1 Physical Mechanism and Order Parameter#

The CPR is characterized by the presence of a collective electronic phase order that spans macroscopic distances within the substrate. This order arises from the specific material composition of Tier 2 layers and is described by a complex scalar order parameter:

ψ(r, t) = |ψ(r, t)| exp(i φ(r, t))

where:

• |ψ(r, t)| = amplitude of the phase order at position r and time t
• φ(r, t) = phase angle of the order parameter [radians]

The order parameter ψ is a collective property of the electron system, not of individual carriers. Its amplitude |ψ| quantifies the degree of phase coherence at a given location. In a fully coherent bulk CPR material at operating temperature, |ψ| ≈ ψ_bulk, a material-specific constant.

4.2 Temporal Address Encoding#

The phase angle φ(r, t) is the physical carrier of temporal address information. When the TCU commits an operation at address τ, it imprints a specific target phase Φ_τ at the designated substrate site:

φ_committed(r, t_commit) = Φ_τ = 2π × (τ mod 1)

This mapping wraps the temporal address domain [0, 1) onto the full 2π phase circle. Addresses that differ by Δτ correspond to phases that differ by:

ΔΦ = 2π Δτ

Discrimination between two committed addresses therefore requires resolving two phase states separated by ΔΦ = 2π Δτ. The minimum discriminable phase difference is set by the phase noise of the CPR material (Section 4.4), which ultimately determines the Substrate Clarity rating.

4.3 Phase Stiffness and Coherence Length#

The key energetic parameter of the CPR is the phase stiffness ρ_s [J m⁻¹], which quantifies the energetic cost of spatial phase gradients:

F_phase = (ρ_s / 2) ∫ |∇φ(r)|² d³r

High phase stiffness suppresses spatial phase fluctuations and is associated with SC-I behavior. Low phase stiffness allows large spatial phase variations and is associated with SC-II behavior. Materials unable to sustain any meaningful stiffness at operating temperature fall into SC-III.

The coherence length L_c is derived from ρ_s and the thermal energy scale:

L_c = sqrt(ρ_s / (k_B T n_s))

where k_B is the Boltzmann constant, T is operating temperature, and n_s is the areal density of phase-active sites [m⁻²].

L_c is the distance over which a phase perturbation at one site influences the phase at a neighboring site. Two sites separated by r_pair experience inter-site phase coupling proportional to:

C_coupling(r_pair) ≈ exp(−r_pair / L_c)

This exponential decay is the physical basis for the inter-site interaction smearing mechanism in the TCT Protocol (§3.2). The reference interaction length L_ref used in the FSCP address spacing formula satisfies L_ref ≈ L_c × β_geo, where β_geo is a dimensionless geometric factor of order unity determined at instrument qualification.

4.4 Phase Noise Sources#

The total phase noise at a given substrate site has four contributions, corresponding directly to the four smearing mechanisms enumerated in the TCT Protocol (§3.2).

Thermal phase noise (σ_thermal):

σ_thermal² = k_B T / (ρ_s × a_site²)

where a_site is the effective site lattice constant. This is the irreducible noise floor set by thermodynamics. It decreases with lower temperature and increases with weaker phase stiffness.

Doping inhomogeneity noise (σ_doping):

σ_doping² = (∂Φ_τ/∂n_d)² × σ_n_d²

where n_d is the local dopant density and σ_n_d is the RMS spatial fluctuation of n_d across the Tier 2 layer. Doping variations shift the local phase reference, producing a position-dependent offset in Φ_τ that cannot be corrected by the global CCG distribution. This is the physical origin of TAOE_sys: a spatially varying systematic address overlay error that tracks the doping map.

Coherence-length smearing (σ_L_c):

When the address pair spacing d_pair < L_c, phase correlation between the two sites is significant and the addresses cannot be treated as independent:

σ_L_c² = (Δτ_eff)² × (1 − exp(−d_pair / L_c))² × f_coupling

where f_coupling is a dimensionless coefficient dependent on the spatial arrangement of the pair. At d_pair >> L_c, σ_L_c → 0. At d_pair ≈ L_c, this term contributes comparably to σ_thermal.

Inter-site interaction noise (σ_inter):

Phase-active sites are not isolated; the phase at each site is weakly pulled by the committed phases of its neighbors:

δφ_inter(r) = Σ_{r'≠r} J_coupling(|r − r'|) × sin(φ(r') − φ(r))

where J_coupling decays exponentially with distance. The temporal crosstalk model in the PDK Specification (TCRS, §8) is the design-time approximation of this physical coupling. The PDK formula:

Δτ_xtalk = K_xtalk × Δτ_eff × exp(−d_pair / L_ref)

follows directly from the linear approximation of δφ_inter for small inter-site phase differences, with K_xtalk being the density-tier multiplier tabulated in the PDK Specification (§8.2).

Combined phase noise:

The total normalized address-space noise σ_phase (in units of the full address range) is the quadrature sum of all four components:

σ_phase = sqrt(σ_thermal² + σ_doping² + σ_L_c² + σ_inter²) / (2π)

Each component is independently measurable through dedicated TCT sub-protocol variants described in Section 12.

4.5 Phase State Relaxation#

A committed phase state is metastable. The order parameter amplitude at the committed site relaxes toward the equilibrium (uncommitted) background state with characteristic time constant τ_relax:

|ψ(r, t)| = |ψ_background| + (|ψ_committed| − |ψ_background|) × exp(−(t − t_commit) / τ_relax)

The relaxation is governed by an activated process with Arrhenius temperature dependence:

τ_relax = τ_0 × exp(E_a / (k_B T))

where:

• τ_0 = pre-exponential attempt frequency factor [s]; typically 10⁻¹² to 10⁻¹³ s
• E_a = activation energy barrier for phase state decay [eV]

This equation has three important consequences for the measurement and design chain:

1. Temperature sensitivity of τ_relax. A 10 K increase in operating temperature reduces τ_relax by a factor of exp(−E_a / (10 k_B)). For E_a = 0.4 eV, this is approximately 0.85× at 300 K — a non-trivial reduction. This is why the CEC temperature control requirement in the TCT Protocol (R-CEC-01: ±0.05 K) is physically necessary, not merely a precision preference.

2. SC-class correlation with E_a. SC-I materials have larger E_a than SC-II materials. Higher phase stiffness corresponds to a deeper energy well for the committed state, hence a higher barrier to relaxation. This directly explains why SC-I materials exhibit longer Temporal State Hold counts (TSH ≥ 8 cycles typical) compared to SC-II materials at the same T_c.

3. Injection-to-readback constraint. The 10-minute ±30 s interval in the TCT Protocol (R-SEQ-02) ensures that relaxation-induced phase drift remains below the uncertainty budget threshold across all SC class levels. The correction formula:

   AER_corrected = AER_measured / exp(−Δt_elapsed / τ_relax)
   

   (TCT Protocol §7.4) is a first-order approximation valid for Δt_elapsed << τ_relax, which is guaranteed by R-SEQ-02 for all qualifying materials.

5. The Transitional Regime (TR)#

5.1 Nature of the Transitional Regime#

The Transitional Regime is the coexistence zone at the TGI where neither CCR nor CPR behavior is fully dominant. It arises because the phase-coherent order of the Tier 2 material does not switch on abruptly at a crystallographic boundary. Instead, the order parameter amplitude |ψ(z)| rises continuously from zero at z_f to its bulk value ψ_bulk at z_c.

Within the TR, the electronic system simultaneously sustains:

• Residual carrier transport from the Tier 1 CCR layer below (screening by mobile carriers).
• Partial phase order from the Tier 2 CPR layer above (growing coherence length).

The competition between these two behaviors is what makes the TGI the most metrology-intensive zone in the substrate stack.

5.2 The Phase Order Profile#

In the absence of process perturbations, the z-dependent order parameter amplitude follows a hyperbolic tangent profile:

|ψ(z)| / ψ_bulk = (1/2) × [1 + tanh((z − z_mid) / ξ_TGI)]

where:

• z_mid = (z_f + z_c) / 2 = midpoint of the TGI zone
• ξ_TGI = characteristic half-width of the order-parameter rise [nm]

The width ξ_TGI is not an independent parameter. It is related to the bulk coherence length L_c and the interface quality:

ξ_TGI ≈ L_c × sqrt(IC)

where IC is the Interface Continuity composite score from the TGI Metrology Standard (§4.2). A perfect interface (IC → 1.0) produces the sharpest possible transition (ξ_TGI → L_c). Interface damage or contamination (IC → 0) broadens the transition and reduces SC_eff.

5.3 The Coherence Layer Gradient#

The Coherence Layer Gradient measured by the TGI Metrology Standard is the observable projection of the order parameter gradient:

CLG(z) = d(|ψ(z)| / ψ_bulk) / dz = [1 / (2 ξ_TGI)] × sech²((z − z_mid) / ξ_TGI)

CLG is maximum at z = z_mid and falls to zero deep in either the CCR or CPR bulk. The peak value is:

CLG_max = 1 / (2 ξ_TGI) = 1 / (2 L_c sqrt(IC))

This formula interlocks with the TGI Metrology Standard's CLG penalty function (§4.6). The SC-I threshold of CLG_max = 0.010 and SC-II threshold of 0.020 correspond to minimum acceptable ξ_TGI values of:

ξ_TGI_min (SC-I)  = 1 / (2 × 0.010) = 50 nm
ξ_TGI_min (SC-II) = 1 / (2 × 0.020) = 25 nm

A material with L_c < 25 nm cannot form a compliant TGI under any IC condition; this sets a hard material qualification cutoff independent of SC bulk rating.

5.4 Carrier Screening at the TGI#

The CCR layer below the TGI supplies mobile carriers that screen the CPR's phase-coherent state via the electrostatic Debye effect. Within the TGI zone, the effective phase stiffness is reduced from its bulk value by a screening factor:

ρ_s_eff(z) = ρ_s_bulk × (1 − exp(−(z − z_f) / λ_D))

For typical Tier 1 materials, λ_D ≈ 5–15 nm. This screening is confined to z near z_f and has negligible effect at z > z_f + 3λ_D. The TGI proximity rule in the PDK (TPR category, §5.3) requiring a 5 nm ceiling clearance is physically grounded in this screening effect: it prevents temporal address sites from entering the carrier-screened sub-zone where effective phase stiffness is reduced.

5.5 Effective SC at the TGI#

The effective Substrate Clarity across the TGI zone is degraded relative to the Tier 2 bulk:

SC_eff(z) = SC_bulk × (|ψ(z)| / ψ_bulk) × f_screening(z)

f_screening(z) = 1 − exp(−(z − z_f) / λ_D)

Integrating SC_eff(z) across the TGI zone and weighting by the fraction of temporal operations nominally placed within that zone yields the effective SC reduction captured by the formula in the TGI Metrology Standard:

SC_eff = SC_bulk × IC × f(CLG_max)

The IC factor captures structural quality effects; f(CLG_max) captures the CLG-driven gradient penalty. Together they project the full SC_eff(z) integral onto two metrology-accessible scalar quantities.

6. Material Parameters and Model Inputs#

6.1 Parameter Taxonomy#

The MRSM requires a set of fundamental material parameters as inputs. These are properties of the bulk Tier 2 material and the TGI interface, determined by the material system and process conditions. They are anchored through the model-to-observable derivations in Section 8, not directly specified by the production metrology chain.

Primary bulk CPR parameters:

Primary TGI interface parameters:

CCR boundary parameters:

6.2 Temperature Dependence of Key Parameters#

The operating temperature T appears in three model formulas with distinct functional forms:

L_c(T)         = ξ_0 × (1 − T/T_c_mat)^(−ν)          [diverges as T → T_c_mat]
τ_relax(T)     = τ_0 × exp(E_a / (k_B T))              [Arrhenius; decreasing in T]
σ_thermal(T)   = sqrt(k_B T / (ρ_s × a_site²))         [increasing in T]

These dependencies are coupled. As T increases toward T_c_mat:

• L_c grows → inter-site coupling extends further → σ_L_c increases.
• τ_relax shrinks → committed states are shorter-lived → TSH decreases.
• σ_thermal grows → the discrimination noise floor rises.

The combined effect is SC degradation that accelerates nonlinearly as T approaches T_c_mat. The dimensionless reduced temperature:

t_red = T / T_c_mat

is a concise indicator of regime position. SC-I materials typically operate at t_red ≤ 0.25; SC-II at 0.25 < t_red ≤ 0.45. Materials with t_red > 0.55 at nominal operating temperature fall into SC-III.

6.3 Process-History Dependence#

MRSM parameters carry process history dependence; they are not purely intrinsic to the base material.

Section 11 provides a detailed treatment of how individual process steps shift these parameters.

7. Regime Boundary Conditions#

7.1 Continuity Requirements at the TGI#

The MRSM requires that certain physical quantities are continuous across the TGI boundary, while others are permitted to be discontinuous.

Continuous:

• Total electronic current density J (conservation of charge).
• Electrostatic potential V(z) (no macroscopic charge sheet at z_f).
• Phase order parameter amplitude |ψ(z)| (no first-order jump; tanh profile is smooth by construction).

Permitted to be discontinuous:

• Phase stiffness ρ_s (step from near-zero in CCR to ρ_s_bulk in CPR; this gives rise to CLG).
• Carrier mobility (steps from μ_CCR to effectively zero in the CPR bulk, where carriers are condensed into the phase-ordered state).
• Effective dielectric permittivity (different in Tier 1 vs. Tier 2 material).

A sharp discontinuity in ρ_s at z_f is energetically costly; the system responds by rounding the ρ_s(z) profile over a length scale ~ξ_TGI. For ideal interfaces (ξ_TGI → L_c), this rounding is minimal. For damaged interfaces (IC << 1), the rounding extends into the CPR bulk and degrades SC_eff.

7.2 Multi-Zone Boundary Conditions#

At a zone boundary within the SCR, the phase order must be maintained at a consistent reference across the boundary. This is enforced by the ZBI protocol (SCR Specification §7). At the physical level, this requires that the committed phase at a zone boundary site satisfies:

φ_committed(r_boundary, zone A) − φ_ref(zone A) = φ_committed(r_boundary, zone B) − φ_ref(zone B)

This condition is satisfiable if and only if the phase stiffness at the boundary site is sufficient to maintain the committed phase against the cross-zone coupling perturbation within one coherence time (~τ_relax). The SCR handoff latency L_handoff (SCR Specification, §7.3) must satisfy:

L_handoff × T_c < τ_relax × (1 − ε_handoff)

where ε_handoff is the maximum allowed fractional phase drift during handoff. For SC-I material, ε_handoff ≤ 0.05 is typically achievable; for SC-II, ε_handoff ≤ 0.15.

7.3 Thermal Boundary Conditions#

A zone-to-zone temperature gradient ΔT produces a mismatch in effective TSH:

ΔTSH / TSH ≈ (E_a / (k_B T²)) × ΔT

For E_a = 0.4 eV and T = 300 K, each 1 K of cross-zone gradient produces approximately 5% TSH mismatch. SCR zone design must account for this when specifying zone boundary placement relative to thermal management infrastructure.

8. Derivation of Observable Parameters#

8.1 AER Contrast Curve Derivation#

The Address Error Rate at a given Δτ spacing is the probability that a readback of a committed phase state is assigned to the wrong address. Assuming total phase noise is Gaussian with variance σ_phase² and the discrimination threshold is at the phase midpoint between two target addresses (ΔΦ/2 = π Δτ), the error probability is:

AER(Δτ) = (1/2) erfc(π Δτ / (σ_phase × sqrt(2)))

For Δτ >> σ_phase: AER → 0 (perfect discrimination). For Δτ << σ_phase: AER → 1/2 = AER_sat (chance-level discrimination).

The logistic sigmoid model used in TCT Protocol §8.2:

AER_model(Δτ) = AER_sat / (1 + exp(k(Δτ − Δτ_50)))

is the standard Fermi-function approximation to the complementary error function. The correspondence between sigmoid parameters and the physical model is:

Δτ_50  ≈ σ_phase × sqrt(2) × erfinv(0.5) / π   [≈ 0.477 σ_phase / π]
k      ≈ π / (σ_phase × sqrt(2) × ln(3))         [≈ π / (1.317 σ_phase)]
AER_sat ≈ 1/2                                    [exact in Gaussian model]

The R² ≥ 0.98 fit requirement (TCT Protocol R-AER-09) is met whenever actual phase noise is well-described by a single-component Gaussian. Bimodal or non-Gaussian distributions (arising from mixed-regime material or inhomogeneous doping) produce curve shapes that fail the R² criterion; this is the physical origin of the non-standard curve classification in TCT Protocol §8.3.

8.2 SC Rating Derivation#

Substituting Δτ_ref = 1/256 into the AER formula and normalizing by AER_sat:

SC = 1 − AER(Δτ_ref) / AER_sat
   = erf(π / (256 × σ_phase × sqrt(2)))

SC is a monotonically decreasing function of σ_phase. SC = 1 corresponds to σ_phase → 0 (ideal material). SC = 0 corresponds to complete loss of CPR order.

The SC class boundaries (0.92 for SC-I/SC-II; 0.75 for SC-II/SC-III) correspond to the following σ_phase thresholds:

σ_phase at SC = 0.92:  π / (256 × sqrt(2) × erfinv(0.92)) ≈ 0.00425
σ_phase at SC = 0.75:  π / (256 × sqrt(2) × erfinv(0.75)) ≈ 0.00801

Materials with σ_phase > 0.00801 are SC-III and cannot support temporal manufacturing.

8.3 Coherence Length Extraction from TCT Data#

The TCT Protocol generates an FSCP contrast curve across seven spacing levels S1–S7. At large Δτ spacings (S1, S2), inter-site interaction is negligible (d_pair >> L_c) and σ_L_c ≈ 0. At small spacings (S6, S7), inter-site coupling is significant and σ_L_c adds to the total. The difference in σ_phase² between large-Δτ and small-Δτ regimes isolates σ_L_c, from which L_c is extracted:

L_c = d_pair,ref / ln(σ_L_c,ref² / σ_L_c,target²)

where d_pair,ref is the pair separation at a reference spacing. This extraction is performed as part of full TCT data reduction and is reported as a material characterization output alongside the SC rating.

8.4 Coherence Layer Gradient Derivation#

The CLG measured by the ISP instrument is the depth-resolved gradient of the normalized SC:

CLG(z_i) = (SC_norm(z_{i+1}) − SC_norm(z_{i−1})) / (z_{i+1} − z_{i−1})

where SC_norm(z) = SC(z) / SC_bulk. In the MRSM, SC_norm(z) = |ψ(z)| / ψ_bulk. CLG peaks at z_mid with CLG_max = 1/(2ξ_TGI). The TGI Metrology Standard penalty function derives from:

SC_eff = SC_bulk × exp(−κ × CLG_max × Δz_TGI)

where κ is a dimensionless coupling coefficient and Δz_TGI = z_c − z_f. The penalty function parameters (SC-I: threshold 0.010, k = 15.0; SC-II: threshold 0.020, k = 8.0) are empirical fits to this exponential evaluated at representative Δz_TGI values for each SC class.

8.5 Temporal Density Limit Derivation#

The Temporal Density TD = N_ops / (A_substrate × C_cycle) is bounded at the physical level by the address spacing requirement. The minimum allowable address spacing Δτ_eff sets a maximum number of distinguishable addresses per coherence cycle:

N_addr_max = floor(1 / Δτ_eff)

Each substrate area element hosting one address is (3 L_c)² (from ASR-001: r_int = 3 L_c). Therefore:

TD_max = floor(1/Δτ_eff) / (9 L_c² × C_cycle)

This formula closes the loop between material parameters and the design-space limit. A material with larger L_c has fewer address sites per unit area, directly reducing TD_max despite potentially superior SC.

9. SC Class as a Regime Indicator#

9.1 SC Class and Regime Dominance#

The SC class is not merely a measurement outcome; it is a regime indicator. The three classes map to distinct physical regimes and operating envelopes:

SC-II materials are physically distinct from SC-I not merely by degree but by character. The 20% Δτ_margin requirement for SC-II (vs. 15% for SC-I) in the PDK Specification is calibrated to the higher σ_doping and σ_inter contributions typical of SC-II materials at elevated t_red, where the phase-order free-energy landscape is shallower and more susceptible to dopant-induced distortion.

9.2 Why SC-III Materials Cannot Support Temporal Manufacturing#

In SC-III materials, one or more of the following conditions holds:

1. T ≥ T_c_mat (t_red ≥ 1). The CPR order is completely suppressed by temperature. ψ_bulk = 0. No phase encoding is possible.
2. Severe doping inhomogeneity. σ_doping is so large that even at full CPR order, AER at Δτ_ref = 1/256 exceeds the SC-II floor. The material cannot be processed to SC-II even with perfect thermal control.
3. L_c < λ_D. The CPR coherence length is shorter than the CCR Debye screening length. Carrier screening from the Tier 1 layer penetrates the entire Tier 2 bulk, suppressing phase order everywhere. Increasing Tier 2 thickness does not recover temporal encoding capability; the screening is determined by λ_D of Tier 1, not by Tier 2 geometry.

Condition 3 is the regime-conflict condition: the CCR boundary condition wins over the CPR bulk order. It imposes a minimum L_c requirement on any candidate Tier 2 material as a function of the Tier 1 process node.

9.3 SC Lot-Level vs. Site-Level Properties#

The TCT Protocol assigns SC class at the lot level, with spatial uniformity characterized by the 5×5 subregion grid (SC_mean, SC_range, SC_σ). Since ρ_s, E_a, and n_s are bulk material properties determined by the growth process, they are expected to be spatially uniform within a lot. However, z_f and z_c (TGI boundary positions) are determined by the Tier 1/Tier 2 deposition process and can vary across the wafer due to step coverage and etch uniformity.

This is why SC_eff can be spatially non-uniform even within an SC-I lot, and why the TGI Metrology Standard requires die-level spatial characterization in addition to the bulk SC rating.

10. Multi-Regime Interactions and Failure Modes#

10.1 Regime Interaction Mechanisms#

The three regimes interact at their boundaries through three physical mechanisms:

Electrostatic coupling (CCR → TR). Mobile carriers in the CCR layer screen the TR region. Elevated carrier density (due to overimplant, junction leakage, or proximity to a charge trap layer) extends λ_D upward into the TGI zone, compressing the available CPR zone.

Thermal phonon coupling (CCR ↔ CPR). Phonons generated by switching activity in the CCR (Joule heating, capacitive discharge) propagate into the Tier 2 layer and elevate the local temperature. Since τ_relax decreases exponentially with T, localized Joule heating events cause temporary TSH reduction at nearby temporal address sites.

Phase boundary pinning (TR ↔ CPR). At the TGI ceiling z_c, the rapidly rising phase order can become pinned to interface defects (dangling bonds, contamination sites, crystallographic dislocations). A pinned site anchors the local phase to a fixed value regardless of the commit operation, producing a stuck-address failure mode that cannot be distinguished from a valid committed state by the ARS.

10.2 Regime Failure Mode Catalog#

RFM-01 through RFM-05 have direct counterparts in the SCR Specification failure mode table (§9). RFM-06 is a long-timescale effect managed through the TGI production monitoring tier structure (TGI Metrology Standard §11.3).

10.3 Failure Mode Interaction#

The most damaging failure scenarios involve coupled modes:

RFM-02 + RFM-04 cascade. A localized thermal event (RFM-02) that exceeds T_c_mat at a hot spot triggers RFM-04 within the heated zone. Recovery requires the zone to cool below T_c_mat, at which point phase order re-nucleates — but with a random initial phase angle. This permanently corrupts any address that was held through the collapse. This scenario motivates the SCR L3 severity classification (Coherence Collapse) and the requirement for full re-qualification following such an event.

RFM-01 + RFM-03 co-occurrence. CCR encroachment that reaches z_c simultaneously with surface defect pinning produces a phase-trapped zone where committed state is controlled by defect pinning rather than by the commit operation. Such zones cannot be corrected by PIMC because TAOE_sys correction assumes a smooth spatial function; a defect-pinned zone is stochastic in character.

11. Process Effects on Regime Structure#

11.1 Epitaxial Growth and ρ_s#

Phase stiffness ρ_s is primarily set during Tier 2 epitaxial growth.

Because ρ_s directly determines σ_thermal and (through L_c) σ_L_c, growth optimization is the primary lever for achieving SC-I classification. Post-growth process steps can only degrade ρ_s, not enhance it.

11.2 Activation Anneal and E_a#

The activation energy E_a is set by the dopant activation anneal following ion implantation. The anneal conditions control three competing effects:

1. Dopant site occupation. Dopants must occupy the specific substitutional or interstitial sites that provide CPR coupling. Under-annealed lots have partially activated dopants with lower E_a.
2. Defect annihilation. Implant damage (vacancies, interstitials) provides trapping sites that reduce E_a. Higher anneal temperature removes these defects, but at the expense of increased dopant redistribution risk.
3. Activation competition. High-temperature anneals can cause dopant diffusion into the TGI zone, shifting z_f upward and compressing the CPR bulk.

The MRSM predicts that optimal anneal temperature maximizes E_a subject to the constraint that dopant redistribution does not shift z_f by more than Δz_TGI/4.

11.3 TGI Formation: z_f and z_c Control#

z_f is controlled by:

• The endpoint of Tier 1 deposition.
• The Tier 1 top-surface preparation (oxide removal, cleaning protocol): contamination at this surface increases ξ_TGI and lowers IC.
• The first atomic layers of Tier 2 deposition: a nucleation layer with higher defect density than the bulk Tier 2 shifts z_f upward.

z_c is controlled by:

• The total Tier 2 layer thickness at deposition.
• Post-deposition planarization processes: over-polishing reduces Tier 2 thickness and may bring z_c inside the minimum CPR bulk zone requirement.

A common failure mode is symmetric shrinkage: contamination that simultaneously raises z_f and reduces Tier 2 thickness. Both effects compress the available CPR bulk, degrade IC, and increase ξ_TGI. The TGI Uniformity Index (TUI) measurement is specifically designed to catch this and its asymmetric variants.

11.4 Thermal Budget and τ_relax Stability#

Every thermal step in the process sequence following Tier 2 deposition perturbs τ_relax. The accumulated effect is tracked using a thermal budget integral:

ΔΘ_thermal = Σ_steps [t_step × exp(−E_a_relax / (k_B T_step))]

where E_a_relax is the effective activation energy for permanent τ_relax modification (distinct from E_a for state decay; typically higher). Thermal steps with ΔΘ_thermal above a process-node-specific threshold cause irreversible reduction of τ_relax. This constraint motivates the post-ASML trend toward low-thermal-budget back-end processes for any layer deposited above z_c.

12. Model Validation and Calibration#

12.1 Validation Strategy#

MRSM validation operates at three levels, each with increasing experimental effort and diagnostic specificity.

Level 1 — Internal consistency. Verify that TCT-derived L_c, τ_relax, and SC values are mutually consistent with model predictions given the measured T. If SC(measured) and σ_phase derived from L_c and T disagree by more than ±0.02, the single-Gaussian noise model is suspect and Level 2 decomposition is required.

Level 2 — Component decomposition. Separately measure each σ_phase component:

• σ_thermal: on high-symmetry samples at multiple temperatures; extrapolate to T → 0 to isolate non-thermal terms.
• σ_doping: on samples with intentionally varied dopant dose; extract slope vs. n_d.
• σ_L_c: from Δτ-dependent AER data as described in §8.3.
• σ_inter: by varying d_pair at fixed Δτ; extract the coupling decay constant vs. d_pair.

Level 3 — Cross-document consistency. Verify that parameters extracted from TCT data are consistent with TGI metrology observations:

12.2 Calibration Reference Hierarchy#

MRSM parameters are anchored through the calibration hierarchy established in the TCT Protocol:

National Standard (τ_n, Δτ_n)
    └─ RLSS (Reference Lot Standard Set)
          └─ FTRC (Fab Transfer Reference Coupon)
                └─ TAIS + ARS session calibration
                      └─ Per-lot TCT measurements
                            └─ MRSM parameter extraction

The critical calibration tie point is the FTRC, which establishes the absolute temporal address scale in units of T_c. All MRSM formulas involving the dimensional form of Δτ depend on this calibration tie being current. The FTRC recalibration interval (TCT Protocol §12.3) ensures this tie remains valid to the tolerance required for Level 1 consistency checks.

12.3 Model Update Triggers#

Model updates are logged in the MRSM revision history and trigger a PDK minor version increment when any parameter change propagates to a TDR rule or CBT value.

13. Design Implications#

13.1 Summary of Model-to-Design Parameter Paths#

13.2 SC-Class Selection as a Design Decision#

The MRSM makes clear that SC class selection is a physical commitment, not a performance grade. An SC-II material has a genuinely different physical regime structure from SC-I:

• The TR zone is relatively larger (shorter L_c → smaller ξ_TGI), making TGI placement constraints tighter in absolute terms.
• σ_phase is higher, requiring larger Δτ_margin and thus fewer usable addresses per cycle.
• τ_relax is shorter, requiring more aggressive TR refresh scheduling.
• Thermal sensitivity of all three quantities is greater (higher t_red → closer to T_c_mat).

Designs that target SC-I material must not be adapted for SC-II by simply applying more conservative Δτ_margins. The full set of changes — L_c-driven ASR rules, TSH recomputation, CBT rebalancing, and CLG penalty revalidation — must be applied as a coordinated PDK migration.

13.3 Multi-Zone Designs and Regime Uniformity#

In Zone-Distributed architectures (Logic Folding §4.4), the MRSM establishes that regime uniformity across zones is a physical requirement, not a design preference:

• All zones in a multi-zone architecture must use the same SC class. Mixing SC-I and SC-II zones creates a fundamental asymmetry in TSH and Δτ_eff that cannot be bridged by the ZBI protocol without either leaving SC-I margin on the table or violating SC-II refresh requirements.
• Zone boundary placement must respect the thermal boundary condition derived in §7.3: a 1 K temperature gradient produces ~5% TSH mismatch per zone. Zones should be placed to minimize cross-zone thermal gradients, or the TSH budget must be explicitly de-rated by the expected gradient.
• The cross-zone coherence budget fraction default of 0.20 (PDK Specification §6) corresponds, in the MRSM, to the fractional reduction in effective B_cycle caused by the L_handoff-induced dead time. For SC-II material at maximum L_handoff, this fraction may need to be increased to 0.25–0.30 to maintain functional timing closure.

14. Glossary#

15. Related Documents#

This document is INFORMATIVE. It does not impose normative requirements. All normative requirements are located in the documents listed above in §15. Where this document's formulas and the normative documents' rules appear to conflict, the normative documents take precedence and this document should be reviewed for revision.

Proposed revisions to the MRSM should be submitted via pull request with a linked issue. Because this document is the physical foundation for normative parameters in multiple downstream documents, any revision that changes a formula result by more than 5% at an SC class boundary requires a cross-impact analysis spanning the TCT Protocol, TGI Metrology Standard, PDK Specification, and Logic Folding Architecture Guide before the revision is merged. The cross-impact analysis must be included in the pull request body.