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Threshold voltage

Threshold voltage, commonly denoted as V_{th} or V_T, is the minimum gate-to-source voltage (V_{GS}) required in a metal-oxide-semiconductor field-effect transistor (MOSFET) to induce strong inversion at the semiconductor surface, forming a conductive channel between the source and drain terminals and allowing significant drain current to flow.^[1] This parameter marks the boundary between the subthreshold region, where the transistor conducts weakly via diffusion, and the strong inversion region, where channel conduction dominates via drift, making it essential for MOSFET operation in switching and amplification applications.^[1] In n-channel MOSFETs, V_{th} is typically positive (around 0.2–1 V for modern devices), while for p-channel devices it is negative, reflecting the polarity needed to attract opposite charge carriers to the channel.^[1] The threshold voltage arises from the physics of the MOS capacitor structure underlying the MOSFET, where the gate voltage must overcome the flat-band voltage (V_{FB}), twice the bulk Fermi potential ($2\phi_B), and the voltage drop across the depletion layer to achieve inversion.^[2] The classical expression for V_{th} in an n-channel MOSFET is given by
V_{th} = V_{FB} + 2\phi_B + \frac{\sqrt{4\epsilon_s q N_A \phi_B}}{C_{ox}},
where V_{FB} accounts for work function differences and oxide charges, \phi_B is the bulk potential (approximately (kT/q) \ln(N_A / n_i)), \epsilon_s is the semiconductor permittivity, q is the electron charge, N_A is the substrate doping concentration, and C_{ox} = \epsilon_{ox} / t_{ox} is the oxide capacitance per unit area with t_{ox} as oxide thickness.^[2] Key factors influencing V_{th} include substrate doping (higher N_A increases V_{th}), oxide thickness (thinner oxides reduce V_{th}), temperature ( V_{th} decreases with rising temperature due to \phi_B variation), and short-channel effects in scaled devices, which can lower V_{th} through charge sharing and drain-induced barrier lowering (DIBL).^[2]^[1] Measurement of V_{th} is critical for device characterization and process control, with common methods including constant-current extraction (where V_{GS} is defined at a specific drain current, e.g., 100 μA × W/L), linear extrapolation from the I_D-V_{GS} curve in the linear regime, and second-derivative techniques for precision in subthreshold analysis.^[1] Variations in V_{th} due to body bias, known as the body effect, further modulate the parameter as V_{th} = V_{th0} + \gamma (\sqrt{2\phi_B + V_{SB}} - \sqrt{2\phi_B}), where \gamma is the body-effect coefficient and V_{SB} is the source-to-body voltage, impacting circuit performance in applications like CMOS logic and power electronics.^[1]

Fundamentals

Definition and significance

The threshold voltage, denoted as V_{th}, in a metal-oxide-semiconductor field-effect transistor (MOSFET) is defined as the minimum gate-to-source voltage (V_{GS}) required to induce a conductive inversion channel between the source and drain terminals, thereby transitioning the device from the cutoff (subthreshold) region to strong inversion where significant drain current flows.^[1] This parameter marks the onset of channel formation at the semiconductor-oxide interface, enabling the transistor to act as a switch or amplifier.^[1] The significance of threshold voltage lies in its direct influence on key performance metrics of MOSFET-based integrated circuits, including switching speed, power consumption, and logic level thresholds. It determines the overdrive voltage (V_{GS} - V_{th}), which governs the drive current and thus the transistor's on-state conductance; lower V_{th} enhances speed and efficiency but increases leakage current in the off-state, necessitating careful optimization for low-power designs.^[1]^[3] In digital circuits, V_{th} affects noise margins and propagation delays, with higher values improving immunity to noise at the cost of slower operation.^[3] Historically, the threshold voltage concept emerged with the invention of the MOSFET by Mohamed M. Atalla and Dawon Kahng at Bell Laboratories, who demonstrated the first working silicon-based device in 1959 using a thermally grown silicon dioxide gate insulator to overcome surface state issues, with the seminal report published in 1960.^[4] This breakthrough enabled scalable transistor fabrication, laying the foundation for modern microelectronics. MOSFETs exhibit varying V_{th} polarities: positive for enhancement-mode n-channel (NMOS) devices, which require V_{GS} > V_{th} to form a channel, and negative for enhancement-mode p-channel (PMOS) devices; depletion-mode variants, conversely, possess an inherent channel at zero gate bias, with V_{th} negative for NMOS and positive for PMOS to deplete it.^[5] In contemporary semiconductor scaling, threshold voltage remains pivotal for enabling low-power operation in advanced nodes, yet it presents challenges below 10 nm, where maintaining precise control becomes difficult due to quantum effects and variability, potentially increasing off-state leakage and complicating gate stack materials for tunable work functions.^[6]

Mathematical formulation

The threshold voltage V_{th} of a long-channel metal-oxide-semiconductor field-effect transistor (MOSFET) represents the gate-source voltage at which strong inversion occurs at the semiconductor surface, enabling significant channel conduction. This baseline model is derived from the MOS capacitor structure by solving Poisson's equation in the semiconductor under the depletion approximation, assuming no mobile carriers in the depletion region and a uniform electric field across the oxide. The derivation considers the electrostatics of the gate oxide and the p-type substrate for an n-channel MOSFET (NMOS), where the gate voltage must counteract the work function difference, achieve the required surface potential for inversion, and support the depletion charge.^[7] The flat-band voltage V_{FB} is the first component, defined as the gate voltage needed to produce zero band bending at the semiconductor surface in the absence of charges. It is expressed as V_{FB} = \phi_{ms} - \frac{Q_{ox}}{C_{ox}}, where \phi_{ms} is the work function difference between the gate material and the semiconductor, Q_{ox} is the fixed oxide charge per unit area (typically positive and located near the silicon interface), and C_{ox} = \frac{\epsilon_{ox}}{t_{ox}} is the oxide capacitance per unit area with \epsilon_{ox} as the oxide permittivity and t_{ox} as the oxide thickness. This term accounts for inherent material asymmetries and any trapped charges in the oxide.^[7] The second component is the surface potential at the onset of strong inversion, which is $2\phi_f, where \phi_f is the Fermi potential given by \phi_f = \frac{kT}{q} \ln \left( \frac{N_A}{n_i} \right) for a p-type substrate. Here, k is Boltzmann's constant, T is the absolute temperature, q is the elementary charge, N_A is the uniform acceptor doping concentration, and n_i is the intrinsic carrier concentration of silicon. The factor of 2 arises because strong inversion occurs when the minority carrier (electron) concentration at the surface equals the majority carrier (hole) concentration in the bulk, requiring the surface potential to be twice the Fermi potential relative to the bulk.^[7] The third component compensates for the voltage drop across the depletion region due to the space charge. The depletion charge density is Q_d = -\sqrt{2 [q](/page/Q) \epsilon_s N_A (2 \phi_f)}, obtained by integrating Poisson's equation \frac{d^2 \psi}{dx^2} = \frac{[q](/page/Q) N_A}{\epsilon_s} (with \psi as the electrostatic potential and \epsilon_s as the semiconductor permittivity) under the depletion approximation from the surface to the depletion edge. The corresponding voltage contribution is \frac{|Q_d|}{C_{ox}} = \frac{\sqrt{2 [q](/page/Q) \epsilon_s N_A (2 \phi_f)}}{C_{ox}}, which can also be written using the body-effect coefficient \gamma = \frac{\sqrt{2 [q](/page/Q) \epsilon_s N_A}}{C_{ox}} as \gamma \sqrt{2 \phi_f}. This term increases with higher substrate doping or thinner oxide, reflecting greater depletion charge that the gate must induce.^[7] Combining these contributions yields the full threshold voltage equation for a long-channel NMOSFET under zero substrate bias:

V_{th} = V_{FB} + 2 \phi_f + \frac{\sqrt{2 q \epsilon_s N_A (2 \phi_f)}}{C_{ox}}

This expression assumes an ideal long-channel device with uniform doping, no short-channel effects, negligible interface traps, and operation at room temperature where quantum-mechanical effects on the threshold are minor. The model relies on the gradual-channel approximation, where the vertical field dominates over the lateral field along the channel.^[7] For a p-channel MOSFET (PMOS) with an n-type substrate (donor concentration N_D), the counterpart equation has opposite polarity due to the inversion of carrier types and substrate doping: V_{th} is negative, \phi_f = -\frac{kT}{q} \ln \left( \frac{N_D}{n_i} \right) (or taken as positive with sign adjustments), and the depletion term changes sign to -\frac{\sqrt{2 q \epsilon_s N_D (-2 \phi_f)}}{C_{ox}}, resulting in V_{th} = V_{FB} + 2 \phi_f - \gamma \sqrt{-2 \phi_f}. This ensures the gate voltage must be sufficiently negative to attract holes to the surface.^[7]

Physical dependencies

Oxide thickness dependence

The threshold voltage V_{th} in MOSFETs depends on the gate oxide capacitance per unit area C_{ox}, defined as C_{ox} = \epsilon_{ox}/t_{ox}, where \epsilon_{ox} is the oxide permittivity and t_{ox} is the physical oxide thickness. The depletion charge contribution to V_{th} is inversely proportional to C_{ox}, so decreasing t_{ox} enhances the gate-to-channel capacitive coupling, thereby reducing V_{th}. This relationship arises because a thinner oxide allows stronger electrostatic control over the channel potential, shifting the onset of strong inversion to lower gate biases.^[8] Historical scaling of t_{ox} has directly lowered V_{th} to enable higher performance in integrated circuits, but it has also introduced trade-offs. In the 1970s, t_{ox} typically ranged from 50–100 nm for gate lengths around 7.5 µm, but aggressive scaling reduced it to 1.1–1.6 nm by the 45 nm node in the mid-2000s, with modern sub-5 nm nodes achieving equivalent thicknesses below 1 nm. While this trend decreases V_{th} and boosts drive current, it exacerbates direct tunneling leakage through the oxide, limiting further reductions without alternative materials.^[9]^[10] For t_{ox} < 2 nm, quantum mechanical effects counteract the classical reduction in V_{th}. Carrier quantization in the inversion layer causes the electron charge centroid to shift away from the Si/SiO₂ interface and induces effective bandgap widening, increasing the effective V_{th} by 0.1–0.2 V depending on the surface electric field and doping. These effects, prominent in ultrathin oxides, necessitate quantum-corrected models for accurate device simulation.^[11] To sustain thin effective thicknesses without excessive leakage, high-κ dielectrics like HfO₂ were adopted at the 45 nm node in 2007, replacing SiO₂. HfO₂, with a dielectric constant of ~25, enables physical thicknesses of several nanometers while maintaining an equivalent oxide thickness (EOT) comparable to 1 nm SiO₂, thereby stabilizing V_{th} and minimizing tunneling current. This transition preserved the capacitive benefits of scaling for V_{th} control.^[12] Contemporary V_{th} predictions incorporate EOT in the C_{ox} term, where EOT = t_{ox} \cdot (\epsilon_{SiO_2}/\epsilon_{high-κ}) + t_{IL} (with t_{IL} as any interfacial layer thickness), ensuring the formulation accounts for hybrid stacks and accurately reflects the effective capacitive coupling.^[13]

Substrate bias (body effect)

In MOSFETs, the substrate bias effect, commonly known as the body effect, arises when a reverse bias voltage (V_{BS} < 0 for NMOS devices) is applied between the source and the substrate, modifying the threshold voltage V_{TH}. This bias widens the depletion region beneath the channel, increasing the concentration of fixed charges that the gate must overcome to induce strong inversion at the silicon surface. As a result, a higher gate-to-source voltage V_{GS} is required to form the inversion layer, thereby elevating V_{TH}.^[2] The modified threshold voltage under substrate bias is given by:

V_{TH} = V_{TH0} + \gamma \left( \sqrt{|2\phi_F - V_{BS}|} - \sqrt{|2\phi_F|} \right)

where V_{TH0} is the zero-bias threshold voltage, \gamma is the body effect coefficient, and \phi_F is the Fermi potential. This equation accounts for the bias-induced shift, with the square root terms reflecting the change in surface potential and depletion charge.^[2]^[14] The derivation stems from the increased depletion charge density under reverse bias. The bulk charge per unit area Q_B in the depletion region is Q_B = -\sqrt{2 \epsilon_{Si} q N_A |2\phi_F - V_{BS}|}, where \epsilon_{Si} is the permittivity of silicon, q is the elementary charge, and N_A is the substrate acceptor doping concentration. This charge contributes to the gate voltage as -Q_B / C_{ox}, where C_{ox} is the oxide capacitance per unit area, leading to the \gamma term defined as \gamma = \sqrt{2 q \epsilon_{Si} N_A} / C_{ox}. The body effect thus directly ties to the enhanced electrostatic control needed to counter the widened depletion width.^[2] In bulk CMOS technology, the body effect coefficient \gamma typically ranges from 0.3 to 0.5 V^{1/2}, depending on oxide thickness and doping levels, allowing predictable V_{TH} adjustments for circuit design. This effect is leveraged in stacked transistor configurations, where reverse biasing the body of off-transistors in a series stack raises their V_{TH}, significantly suppressing subthreshold leakage currents—often by factors of 10 or more per stacked device—while maintaining performance in active paths.^[14]^[15] In silicon-on-insulator (SOI) devices, the body effect is altered due to the isolated body region, which reduces the magnitude of \gamma compared to bulk MOSFETs because of the thinner silicon film and limited depletion charge modulation; however, floating body dynamics can introduce additional variability in effective bias response.^[16]

Temperature dependence

The temperature dependence of the threshold voltage in MOSFETs arises primarily from the variation of the Fermi potential, \phi_f, with temperature. The Fermi potential is expressed as \phi_f = \frac{kT}{q} \ln\left(\frac{N_a}{n_i}\right), where N_a is the acceptor doping concentration, n_i is the intrinsic carrier concentration, k is Boltzmann's constant, T is the absolute temperature, and q is the elementary charge. The intrinsic carrier concentration n_i increases strongly with temperature, following n_i \propto T^{3/2} \exp\left(-\frac{E_g}{2kT}\right), where E_g is the bandgap energy. This exponential growth in n_i dominates, causing \phi_f to decrease with increasing T, which in turn reduces the threshold voltage V_{th} by approximately 1-2 mV/K in typical silicon devices.^[17] A comprehensive model for the temperature coefficient incorporates both the Fermi potential shift and the depletion charge term:

\frac{dV_{th}}{dT} \approx -\frac{d(2\phi_f)}{dT} - \frac{1}{2} \frac{\sqrt{2\epsilon_s q N_a (2\phi_f)}}{2\phi_f} \cdot \frac{d(2\phi_f)}{dT},

where \epsilon_s is the semiconductor permittivity. The first term reflects the direct decrease in surface potential, while the second accounts for the temperature sensitivity of the space-charge contribution. For silicon MOSFETs, this yields a negative coefficient dV_{th}/dT typically ranging from -1.5 to -4 mV/°C, with the exact value depending on doping and oxide thickness. This inherent negative dependence means V_{th} decreases as temperature rises, shifting device characteristics across operating conditions.^[17]^[18] Operationally, the reduced V_{th} at elevated temperatures increases subthreshold leakage current exponentially, posing challenges for power efficiency in integrated circuits, particularly in standby modes. Conversely, the on-state drive current is diminished because carrier mobility \mu degrades with temperature, following \mu \propto T^{-1.5} due to intensified phonon scattering, which partially offsets the leakage increase but limits performance at high T. To mitigate these effects and maintain stability over automotive and industrial ranges (-40°C to 125°C), compensation techniques include dual V_{th} processes, which assign low-V_{th} transistors to speed-critical paths and high-V_{th} ones to leakage-sensitive areas, and adaptive body biasing, which dynamically adjusts substrate voltage to tune V_{th} against thermal drifts.^[19]^[20]^[21] In advanced technology nodes, FinFET architectures exhibit weaker temperature dependence of V_{th} compared to planar MOSFETs, attributed to enhanced gate electrostatics that better confine carriers and reduce thermal perturbations in the channel potential. This results in more stable V_{th} variation, typically under 2 mV/°C, aiding reliability in scaled devices for high-performance computing.^[22]^[23]

Doping concentration effects

The threshold voltage V_{th} of a MOSFET is significantly influenced by the substrate doping concentration N_a, which primarily affects the depletion charge term in the device. In the standard long-channel model, the contribution from the bulk charge is given by \frac{\sqrt{2 \epsilon_s q N_a (2 \phi_f)}}{C_{ox}}, where \epsilon_s is the permittivity of silicon, q is the elementary charge, \phi_f is the Fermi potential, and C_{ox} is the oxide capacitance per unit area. Higher N_a increases this term, thereby raising V_{th} because a greater gate voltage is required to overcome the increased positive space charge in the p-type substrate for n-channel devices. Typical substrate doping levels in modern CMOS processes range from $10^{16} to $10^{18} cm^{-3} , balancing short-channel control with mobility degradation.^[24]^[25] Channel doping, often implemented through halo or pocket implants, further modulates V_{th} to counteract short-channel effects while introducing specific trade-offs. Halo implants, typically using boron for nMOS, create non-uniform doping profiles with higher concentrations near the source and drain junctions, effectively increasing the average channel doping and elevating V_{th} by enhancing the depletion barrier. However, these implants can lead to increased junction leakage currents due to band-to-band tunneling at the higher doped regions. In advanced nodes, such as 90 nm and below, halo doping doses are optimized around $10^{13} cm^{-2} to maintain V_{th} stability without excessive parasitic effects.^[26]^[27] The gate work function difference \phi_{ms} between the gate material and the semiconductor also plays a key role in setting V_{th}, as it directly shifts the flat-band voltage V_{fb}. Traditional n^+ polysilicon gates have a work function of approximately 4.05 eV, while metal gates in high-k/metal-gate stacks can be tuned from 4.1 to 5.2 eV, resulting in V_{th} shifts of 0.5 to 1 V compared to polysilicon. For CMOS integration, dual work function metals—such as titanium aluminide for nMOS (low work function) and titanium nitride for pMOS (high work function)—enable balanced V_{th} values around ±0.3 to 0.4 V without heavy channel doping. This approach reduces reliance on implants and improves performance in sub-45 nm technologies.^[28]^[29] Threshold voltage adjust implants provide post-fabrication fine-tuning of V_{th}. For nMOS, boron ions are implanted into the channel to increase p-type doping and raise V_{th}, with doses typically in the $10^{12} to $10^{13} cm^{-2} range to achieve adjustments of 50-200 mV. In pMOS, BF_2 implants are preferred over pure boron to minimize penetration through the gate oxide while similarly shifting V_{th} negatively, avoiding excessive fluorine-related degradation. These implants are performed after gate definition but before spacer formation, ensuring precise control in dual-threshold CMOS processes.^[30] Doping concentration choices involve inherent trade-offs in device performance and variability. Higher doping levels reduce threshold voltage variation by minimizing depletion width fluctuations but exacerbate short-channel effects like drain-induced barrier lowering due to increased lateral electric fields. Conversely, lower doping supports low-power operation by enabling near-ideal subthreshold slopes and higher carrier mobility but amplifies random dopant fluctuation impacts, leading to greater device-to-device V_{th} spread in scaled technologies. Optimal profiles, often combining uniform substrate and localized channel doping, are determined through process simulation to balance these factors for specific applications like low-standby-power logic.^[31]^[24]

Variations and fluctuations

Random dopant fluctuation

Random dopant fluctuation (RDF) arises from the statistical variation in the number and position of dopant atoms within the channel depletion region of MOSFETs, leading to device-to-device mismatches in threshold voltage (Vth). This phenomenon is governed by Poisson statistics, where the variance in the number of dopants scales with the mean number, resulting in fluctuations that become increasingly significant as device dimensions shrink. The standard deviation of Vth due to RDF can be approximated as \sigma_{V_{th}} \approx \frac{q t_{ox}}{\epsilon_{ox}} \sqrt{\frac{N_A W_{dep}}{3 W L}}, where q is the elementary charge, t_{ox} is the oxide thickness, \epsilon_{ox} is the oxide permittivity, N_A is the channel doping concentration, W_{dep} is the depletion width, W is the channel width, and L is the channel length. This formula highlights how higher doping and smaller channel areas amplify the variability, as fewer dopant atoms lead to larger relative fluctuations in the electrostatic potential barrier.^[32] Pelgrom's law empirically describes the scaling of these mismatches, stating that \sigma_{V_{th}} \propto 1/\sqrt{W L}, indicating that threshold voltage matching improves with larger transistor areas due to averaging over more dopant atoms. This relationship, derived from measurements across various CMOS technologies, underscores RDF as a key contributor to mismatch in analog circuits and static random-access memory (SRAM) cells, where precise Vth control is essential for functionality and yield.^[33] RDF becomes the dominant source of Vth variability below 65 nm technology nodes, where the number of dopant atoms in the channel drops to tens or fewer, exacerbating statistical noise. In 28 nm nodes, typical \sigma_{V_{th}} values range from 20-50 mV for minimum-sized transistors, significantly limiting manufacturing yield and necessitating design margins in high-density circuits like SRAM arrays.^[34] To mitigate RDF, strategies focus on reducing channel doping concentration, such as employing lightly doped or undoped channels in conjunction with high-k/metal gate stacks to maintain Vth control via work function tuning. Advanced structures like undoped FinFETs further suppress RDF by minimizing the depletion region's dopant dependence, achieving \sigma_{V_{th}} below 10 mV in sub-20 nm nodes and improving overall variability.^[35] Monte Carlo simulations are widely used to model RDF by randomly placing dopant atoms according to Poisson distributions and solving the 3D Poisson equation to compute Vth distributions, enabling statistical analysis of variability without exhaustive fabrication testing. These methods reveal non-Gaussian tails in Vth histograms and correlations with other parameters, guiding process optimization.^[36]

Short-channel effects

In short-channel MOSFETs, where the channel length L becomes comparable to the depletion region widths of the source and drain junctions (typically L < 100 nm), electrostatic control by the gate diminishes, leading to deterministic degradations in threshold voltage V_{th}. These short-channel effects (SCEs) primarily manifest as V_{th} roll-off and drain-induced barrier lowering (DIBL), which reduce the gate's ability to modulate the channel potential uniformly and increase off-state leakage. Charge sharing between the source/drain depletion regions and the channel depletion region under the gate reduces the effective depletion charge, thereby lowering V_{th}. This roll-off becomes prominent as L scales down, with an approximate shift given by \Delta V_{th} \approx -\left( \frac{\epsilon_{si} t_{ox}}{\epsilon_{ox} L} \right) V_{ds}, where \epsilon_{si} and \epsilon_{ox} are the permittivities of silicon and oxide, t_{ox} is the oxide thickness, and V_{ds} is the drain-to-source voltage; this expression captures the two-dimensional field penetration from the drain.^[37] DIBL further exacerbates V_{th} reduction by allowing the high drain potential to lower the source-channel potential barrier, facilitating easier carrier injection into the channel even at low gate voltages. The DIBL coefficient \eta, defined as \eta = \Delta V_{th} / \Delta V_{ds}, quantifies this effect and typically ranges from 50-100 mV/V in bulk silicon MOSFETs, indicating significant barrier lowering with increasing V_{ds}. Multi-gate architectures, such as FinFETs, suppress DIBL to below 50 mV/V by enhancing gate control through multiple sides of the channel.^[38] These SCEs became evident in commercial processes around the 0.25 \mum technology node in the late 1990s, where V_{th} roll-off exceeded 100 mV and DIBL contributed to subthreshold leakage currents rising by over an order of magnitude compared to longer-channel devices. The introduction of FinFETs in the early 2000s provided superior three-dimensional electrostatic control, reducing SCEs by factors of 2-5 relative to planar bulk MOSFETs at equivalent nodes, enabling scaling to 22 nm. Gate-all-around (GAA) structures, emerging in the 2010s, further mitigate SCEs by fully encircling the channel, achieving \eta < 30 mV/V in sub-10 nm devices. By 2025, GAA nanosheet transistors are in production at 3 nm and 2 nm nodes, achieving \eta < 20 mV/V through enhanced gate control and undoped channels.^[39]^[40] The onset of SCEs is governed by the natural length \lambda = \sqrt{\frac{\epsilon_{si} t_{ox} W_{dep}}{\epsilon_{ox}}}, where W_{dep} is the channel depletion width; significant degradations occur when L \lesssim 5\lambda, as this allows substantial sub-surface leakage paths. To counteract SCEs, halo (pocket) doping introduces higher dopant concentrations near the source/drain junctions to steepen the potential barrier and restore gate dominance, while raised source/drain structures shallow the junctions to minimize charge sharing—though both techniques trade off with increased random dopant fluctuation variability.

Measurement and modeling

Extraction methods

The extraction of threshold voltage (Vth) in MOSFETs is typically performed from experimental current-voltage (I-V) characteristics, as the precise definition of Vth can vary depending on the operational regime and device scaling. Common methods focus on identifying the gate-source voltage (Vgs) at which the device transitions from weak to strong inversion, often using drain current (Id) versus Vgs curves measured in either the linear or saturation regions. These techniques address ambiguities arising from non-ideal behaviors, such as mobility degradation and subthreshold leakage, but require careful selection to ensure consistency across devices and conditions.^[41] One widely adopted approach is the constant current method, which defines Vth as the Vgs value where Id reaches a predefined criterion, typically Id = 0.1 μA × (W/L), with W and L being the channel width and length, respectively. This method is straightforward and applicable to both linear and saturation regions, making it suitable for automated extraction in circuit simulations and production testing. However, it is sensitive to carrier mobility variations, as changes in mobility can shift the Id level without altering the underlying inversion threshold, leading to inaccuracies in devices with non-uniform doping or interface scattering. An adjusted variant normalizes the current criterion to account for substrate bias, improving reliability for short-channel devices.^[41]^[42]^[43] Another established technique is the linear extrapolation method, which relies on the square-law model of MOSFET operation in saturation. Here, the square root of Id (√Id) is plotted against Vgs, and Vth is determined by extrapolating the linear portion of this curve to the Vgs-axis intercept. This approach assumes ideal strong inversion behavior where Id ∝ (Vgs - Vth)^2, providing a physically grounded estimate that mitigates some subthreshold contributions. It is particularly effective for long-channel devices but can be distorted in nanoscale MOSFETs due to velocity saturation and series resistance, necessitating corrections for accurate results.^[44]^[45] The transconductance change method offers a derivative-based alternative, identifying Vth at the gate voltage where the transconductance (gm = dId/dVgs) reaches its maximum or where the second derivative d²Id/dVgs² equals zero. This technique captures the onset of strong inversion by highlighting the inflection point in the Id-Vgs curve, often using low drain voltage to emphasize channel conductance changes. It is less dependent on absolute current levels than the constant current method and provides insights into interface quality, though it requires high-resolution measurements to resolve the peak accurately, especially in noisy data. Variants based on the transconductance-to-current ratio (gm/Id) further refine extraction by minimizing drain voltage dependencies, making it robust for analog design applications.^[46]^[47]^[48] Challenges in Vth extraction arise particularly in the subthreshold regime, where the current follows an exponential form Id = I0 exp((Vgs - Vth)/(n kT/q)), with n as the subthreshold swing factor, k as Boltzmann's constant, T as temperature, and q as the elementary charge; Vth is thus defined at a specific low Id level, but the choice of this level introduces ambiguity. In nanoscale devices, short-channel effects (SCE) exacerbate this by blurring the weak-to-strong inversion transition through drain-induced barrier lowering, leading to extracted Vth values that vary by up to 100-200 mV across methods and requiring bias-dependent adjustments.^[49]^[50] For advanced characterization, capacitance-based methods like split-CV provide an alternative by measuring gate-to-channel and channel-to-substrate capacitances separately as functions of Vgs, from which Vth is inferred at the point of minimum capacitance or flat-band shift. This technique is valuable for isolating oxide and interface contributions without relying on current transport, offering high precision in ultra-thin body devices, though it demands specialized probing to decouple parasitic effects. Additionally, cryogenic measurements enable temperature-dependent studies, revealing Vth shifts due to freeze-out and trap effects down to 4.2 K, with extraction often combining I-V and C-V data to validate models for quantum computing applications.^[51]^[52]^[53]

Modeling in devices

In compact models for circuit simulation, the threshold voltage V_{th} serves as a fundamental parameter that governs the transition between weak and strong inversion regimes in MOSFETs. These models integrate V_{th} dependencies on physical factors such as temperature, substrate bias, and channel dimensions to predict device behavior accurately across operating conditions. Widely adopted industry-standard models like BSIM and EKV exemplify this approach, enabling efficient simulation in tools such as SPICE for analog, digital, and mixed-signal circuits.^[54]^[55] The BSIM family, developed by the Berkeley Short-channel IGFET Model group, treats V_{th} as a core parameter with explicit dependencies. In BSIM4, a threshold-voltage-based model for bulk MOSFETs, the long-channel V_{th} at nominal conditions, denoted V_{TH0}, is adjusted for temperature via

V_{th}(T) = V_{TH0} + KT1 \left( \frac{T}{T_{NOM}} - 1 \right) + KT2 (V_{bseff} + 2 V_{TH0}) \left( \frac{T}{T_{NOM}} - 1 \right),

where KT1 (default -0.11 V) captures the primary temperature coefficient, KT2 (default 0.022 V) accounts for body-bias interaction with temperature, T is the operating temperature, T_{NOM} is the nominal temperature, and V_{bseff} is the effective source-to-body voltage. The body effect is incorporated through coefficients K1 (default 0.5 V^{1/2}) and K2 (default 0), yielding terms like K1 (\sqrt{|V_{bs} + 2\phi_F|} - \sqrt{2\phi_F}) - K2 V_{bs}, where \phi_F is the Fermi potential. Short-channel and narrow-width adjustments, such as drain-induced barrier lowering (DIBL) via \Delta V_{th,DIBL} = \theta_{th}(L_{eff}) (V_{ds} - \phi_s - V_{bi}), ensure scalability with effective channel length L_{eff} and width W_{eff}. BSIM6 extends this to a charge-based framework, maintaining similar temperature and body dependencies while emphasizing symmetry for analog/RF applications, with V_{th} derived from pinch-off potential calculations that include non-uniform doping effects.^[54]^[56] The EKV model, a charge-based formulation continuous across inversion levels, defines V_{th} implicitly through the pinch-off voltage V_P, which marks the boundary where the inversion coefficient i_c = I_D / I_S = 1, transitioning from weak (i_c < 0.1) to strong inversion (i_c > 10). Here, V_P = V_{G'} - n V_T \ln(1 + \exp((V_{G'} - 2\phi_F - V_{ch})/ (n V_T))), with effective gate overdrive V_{G'} = V_{GS} - V_{FB} - \phi_s, slope factor n, and thermal voltage V_T; the long-channel V_{th} (VTO, default 0.5 V) shifts with temperature as V_{TO}(T) = V_{TO} - TCV (T - T_{nom}). This avoids abrupt discontinuities, making EKV suitable for low-voltage designs where subthreshold operation is prevalent.^[55] Parameter extraction for these models typically involves fitting to measured current-voltage (I-V) characteristics. Threshold voltage V_{th}, body-effect coefficient \gamma (or [K1](/page/K1)), and mobility \mu are optimized using nonlinear least-squares methods on transfer curves (I_D vs. V_{GS}) at various V_{BS} and V_{DS}, often starting with linear extrapolation or constant-current methods for initial V_{th} estimates, followed by global optimization to match saturation and linear regimes. For statistical variations, random dopant fluctuation (RDF) is incorporated via Pelgrom's matching law, where the standard deviation \sigma(\Delta V_{th}) = A_{VT} / \sqrt{W L}, with A_{VT} (typically 1-5 mV·μm) derived from Monte Carlo simulations calibrated to measured mismatch data; this extends compact models for variability-aware design in scaled technologies.^[57] Modern extensions address non-planar devices, such as FinFETs, where fin width W_{fin} influences V_{th} due to quantum confinement and corner effects. The BSIM-CMG model for multi-gate FinFETs augments the core V_{th} with fin geometry terms, like \Delta V_{th,fin} \propto 1/W_{fin} from volume inversion, calibrated via 3D technology computer-aided design (TCAD) simulations that solve Poisson's equation in full 3D structures to capture electrostatics and dopant distributions. TCAD tools, such as Synopsys Sentaurus, enable predictive modeling by simulating V_{th} roll-off and DIBL in fin arrays before compact model tuning.^[58]^[59] Reliability effects, including hot-carrier injection (HCI) in nMOSFETs and negative bias temperature instability (NBTI) in pMOSFETs, induce V_{th} shifts that degrade circuit performance over time. HCI generates interface traps under high V_{DS}, causing a \Delta V_{th} decrease of 50-100 mV over a 10-year lifetime at accelerated stress, while NBTI creates oxide traps under negative V_{GS} and elevated temperature, increasing |\Delta V_{th}| by similar magnitudes in pMOSFETs. These are modeled as power-law time dependencies, \Delta V_{th} \propto t^n, with n \approx 1/4 for NBTI (reaction-diffusion framework) and n \approx 1/2 for HCI (lucky-electron model), integrated into compact models via bias- and temperature-dependent lookup tables or empirical functions for lifetime prediction in circuits.^[60]^[61]

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