Conservation Laws at Physical Origins of Universal Mobility in MOSFET Inversion Layers in Consilience with River Flow in a Gravitational Field

— The salient properties of charge flow (or current) along the MOSFET’s inversion layer are shown to be analogous to a river’s flow in a gravitational potential field, insofar as both are fundamentally governed by energy conservation principles, and their laminar and turbulent conditions determined by friction losses at shallow depths. We formulate a simple, yet accurate, model for a non–uniform mobile charge density ρ(z) giving rise to a mean potential ψ* across an inversion layer of finite extent , which we measure by means of a novel, sensitive, experimental method involving nulls of harmonic distortion components (D 2 ≈ D 3 ≈ 0) of the drain current under sinusoidal excitation below saturation. We thus establish analytically that the low–field, “universal” effective, mobility µ eff , long reported to vary as ∼ ( E* for transversal fields manifestation and consequence of both energy and momentum conservation under laminar flow conditions and quantum mechanical effects, in which case the inversion layer’s mean thickness <z> also varies as ∼ ( E* T ) –1/3 up to a maximum value E* T ≈ 0.35 MV/cm at 300K, determined only by fundamental while 𝜼 varies between 1/2 and 1/3. We reconcile our model and observations with those of , establish, analytically and experimentally, that the higher order, derivative, parameter θ Τ quantifying µ eff ’s E* T ∼ ( E* corroborating the foregoing effects and interpretations thereof.


I. INTRODUCTION
The MOSFET 1 has been a critical component of modern CMOS 2 integrated circuits since their inception in the 1960's. The latter' evolution towards miniaturization continues unabated, whose main purpose is to increase the density and speed of ubiquitous digital and analog circuits. The physical mechanisms underlying the operation of MOSFETs properly continue to receive much attention in the technical literature because they are key to ongoing developments of accurate and efficient models and simulations for devices and circuits, especially as the latter are downscaled. Indeed the technological and commercial import of these mechanisms was recognized from the outset, as attested by Stern and Howard's landmark analysis of the inversion layer in 1967 [5], by Schwarz and Russek's empirical model of the same in 1983 [3], and by the indispensable experimental data of Takagi et al, in 1994 [4], whose scattering models have not been revised, ostensibly in the interim, in any substantial manner. While the pertinent literature is otherwise considerable, these prominent forerunners circumscribe the essential features of the subject at hand, thereby providing both context and reference for our analyses and observations.
The characteristics of the inversion layer, or current-carrying channel, which underlie the MOSFET's current conduction, include its "effective" mobility µ eff , conductance, transconductance, various associated capacitances, and transit/switching speeds, all of which are key to the performance of digital and analog circuit functions. In modern, downscaled, devices, whose gate oxides are near to, and below, tox ≈ 20Å, µ eff is observed to fall more steeply with increasing E*T than expected from its "universal" behavior, whereby µ eff varies as ∼ (E*T) -1/3 , which devices with thicker oxides and inversion layers have displayed over many prior technological generations. This mobility degradation is widely attributed to the growing, and increasingly dominant, influence of irregularities and/or "scattering" effects occurring at the Si/Gate-insulator interface, as E*T exceeds 0.5 MV/cm approximately. These trends occur in tandem with higher bulk doping concentrations, higher gate insulating dielectric constants, and thinner gate oxides, among other evolving physical parameters, as entailed by downscaling.
Consequently, accurate models of inversion layers of extents comparable to a single silicon inter-atomic distance (≈ ± 5Å) are required towards further developments, not only because their deleterious effects increasingly encroach upon critical MOSFET performance, but because the latter' complex interactions are likely further to complicate the task of disentangling their respective influences as well.
In section II, we establish the similar, general, relationships pertaining to a river's flow under gravitation, and to the MOSFET's current flow under low longitudinal field conditions, which relate the depths z of the respective "fluids" to the relevant parameters of these systems. These considerations readily lead to the origins of the behavior of "universal mobility" 3 , µ eff ∼ (E*T) -1/3 , as a consequence of laminar flow, and energy and momentum conservation under quantum-mechanical uncertainty relationships, as elaborated in II.C, along with a maximum value for E*T beyond which laminar flow ceases.
Typical models implemented in standard simulation programs achieve computational efficiencies by modeling µ eff and its modulation coefficient θΤ by means of "curve fitting" parameters (such as "UA", "UB", "UC", etc., in the industry standard BSIM4), which must be optimized for a given process, hence repeated (and/or verified) for each process generation, so as to meet prescribed levels of accuracy. Moreover, a key premise underlying MOSFET conduction models in archetypal circuit simulation programs is the assumption of a "negligibly thin" inversion layer, or a charge sheet. While this assumption simplifies drain current computations, it forsakes consequential information about the charge distribution ρ(z) within, and mean potential ψ* across, the inversion layer, both of which are critical to the conceptual modeling of channel conductivity, and of the modulation thereof by transverse electric fields E*T. The limitations and drawbacks entailed by these considerations are discussed in section III, along with those resulting from constant mobility models.
Absent efficient models derived from first principles, this quasi-Procrustean approach has entailed ever more delicate refinements and compromises as novel effects have gained prominence with the advent of downscaling. While some of these effects are inconsequential to certain circuit functions, they are detrimental to those, analog especially, wherein such performance parameters as distortion components are important, among other high order effects (e.g. noise), yet rendered inaccurately by charge sheet models [11]. Our results indicate that (i) the inversion layer's detailed characteristics remain consequential even as this layer shrinks with process downscaling and, thus, (ii) the conventional assumption of a "negligibly" thin inversion layer may not warrant its outright neglect, absent contrary and/or contextual justification.
In section IV, we apply an experimental method originally conceived to suppress harmonic distortion components in analog integrated filters [10,11], to measure the mean potential ψ* across the inversion layer, a novel element in this context, which yields the corresponding mean transverse E*T, (including ), as well as the ratio of mean to surface potentials (ψ*/ψs) from which we deduce ρ(z), and subsequently model analytically in section V. The accuracy of this model is confirmed, in section VI, by further, sensitive, measurements of the modulation of µ eff by E*T, namely the parameter θΤ = (1 ) ( ( * ) ⁄ ⁄ , which are in agreement with our model's analytical predictions that θΤ should vary as ∼ (E*T) -5/3 .
Our overall results are summarized in section VII, while the pertinent features of the Stern-Howard [5] and Schwarz-Russek [3] models are summarized in appendix A2, to whose models and data we compare and contrast ours throughout the present manuscript.
The analyses to follow aim to establish such relationships between the aforementioned phenomena as derivable from first principles. We thus bring conservation of energy and momentum laws to bear upon these from the outset, which are seldom, if at all, invoked explicitly in this context. This approach, thus underpinned and framed by a scaffolding of fundamental principles 4 , provides not only a variant pedagogical perspective on the complex effects attending the characteristics of the MOSFET's inversion layer, but such foundational safeguards as well, as are vital to assessing the validity of prospective physical constructs and interpretations thereof, especially in light of a considerable body of (often disparate) experimental observations extant in the literatureour reference section listing a mere representative sample thereof. The table in appendix A1 summarizes the correspondence between parameters of the gravitational and electrical fields pertaining, respectively, to the river and to the MOSFET's inversion layer. The variable z represents depth of river and depth (thickness) of inversion layer, as indicated in Fig. 1. include gravity, electrodynamics, and strong nuclear interactions, the latter weak nuclear interactions (β decay) [13]. direction at an angle α, the slope of the bed to the horizontal. Bottom: Cross sectional view of an inversion layer's volume: electrons flow in the positive x direction, from source towards drain. The "height" of an electron above the "floor" level (the gate oxide plane) is the thickness of the inversion layer z(x) << W; the direction of W is not shown. The ratio (E///E*T) corresponds to the slope of the "riverbed" above, with E// the longitudinal electric field. The wide arrows represent various energies entering or leaving the volume, only some of which are conserved.
A. River [6] The gravitational potential energy difference (loss) between depths z(x) and z(x+ ∆x) above the bed is mg∆z, where (∆z/∆x) = tan α , and (½)m(vdrift) 2 is the kinetic energy gain between x and (x +∆x). Energy loss to friction, or viscosity, along the riverbed is proportional to kinetic energy, and inversely to depth z(x). The coefficient of hydraulic friction KF ≥ 0, is a proportionality constant between the latter quantities. While potential and kinetic energies are conserved, energy lost to friction is not, and we have: dividing through by mg and taking the limit of the differences readily yields: If the flow of the river is steady at a mean velocity vdrift, such that Q is the volume, assumed constant, flowing across a given cross-section per unit of time, the following continuity equation holds: Substituting vdrift from (4) into (3) yields the slope (dz/dx) as a function of z(x): The relative magnitudes of KF and α determine the magnitude and sign of (dz/dx), and the integral curves of (5) yield the profile z(x) of the free surface.
Isoclines are straight lines parallel to the bed, which correspond to constant values of (dz/dx), a stream of uniform depth, laminar flow, and a river flowing into a reservoir. When z(x) is large enough so that the ratio inside the square brackets is near to unity, the flow is parallel to the slope of the riverbed, since (dz/dx) ≈ tan α.
If the ratio inside the square brackets changes sign, at shallow depths where z(x) is small for example, the flow may incur discontinuities in height and velocity, or hydraulic "jumps", thereby becoming turbulent or chaotic, such that the river becomes a torrent. Similar phenomena readily occur when dispensing a liquid from a bottle whose neck is narrower than its body, such as a typical (750 ml) wine bottle: as the neck is tipped downward gradually, the fluid first trickles out in a continuous stream until such an angle is reached as causes the fluid to fill the neck's cross section approximately. Upon exceeding this angle, part of the fluid "backs up" inside the bottle, thereby emerging irregularly, in "fits and starts", as its flow becomes turbulent or chaotic. Such effects are especially conspicuous when observed in slow motion, with a dense fluid such as warm honey.
Since laminar flow is the condition analogous to the standard, presumed, flow of electrons in MOSFETs' inversion layers under conditions of low longitudinal fields (E// << E*T), (5) may serve as reference for the forthcoming comparison. The case (dz/dx) = 0, corresponding to a level surface, or to a fluid in equilibrium at the bottom of a bucket, was previously invoked by C. Séquin [7], to promote visualization of the behavior of certain charge-controlled devices, including charge-coupled devices (CCD's) and MOSFETs, whereas non-laminar flow conditions were indicated as pertinent to MOSFETs operating in saturation, where they are typically biased in analog circuits [8]. Although the latter conditions entail more complicated dynamics [12], beyond the purview of the present analysis, an extension thereof in this respect may prove instructive in view of the ubiquity and practical importance of such circuits.

B. Inversion Layer
The potential energy difference for an electron traveling an incremental distance ∆x along the channel in the positive x direction, i.e., from source towards drain and "falling" a distance ∆z in the positive z direction inside the inversion layer (i.e., from the bulk towards the gate's "attractive" potential) is: For an inversion layer charge density varying nearly linearly with z, i.e., ρ(z) ∼ z , which will be established from observations of the ratio of mean to surface potentials (ψ*/ψs) in section IV below: we have, equivalently to the continuity (4) for the river: wherein Qi is the total mobile charge in a cross-sectional slice dx of the inversion layer, and the constant drain current ID, flowing at a mean velocity vdrift, is the counterpart of the river's constant volume Q flowing per unit of time and width. In this case: is the "classical" counterpart of (4), whose value is approximately 250m/s (when µeff ≈ 500 cm 2 /V•s , E// ≈ 50V/cm), near to the speed of sound in dry air at standard temperature and pressure, for (ID/W) ≈ 0.1µA/µm, z(x) ≈ 20 Å, and (ρo/zmax) ≈ 10 24 m -3 , the latter volume density corresponding to a surface density of ≈ 10 12 cm -2 for a commensurate charge-sheet model (see Fig.4).
If we account for an energy loss term proportional to (½)m(vdrift) 2 as in the case of the river above then, proceeding similarly, taking differences to their differential limits and dividing through by q, we have the following statement of conservation of energy: which may readily be put in the form: and is the counterpart of (5) in the foregoing river case. Equation (11) thus lends itself to the following interpretations: a) Large z (thick inversion layer ∼ deep river): (dz/dx) ≈ -(E// / E*T), which is the familiar "gradual channel" approximation when the ratio (E// / E*T), << 1, as typically the case in standard MOSFET circuit operation, and is in the range of (1/2000) to (1/6000) for our data (Figs. 3-5, and 7, below).
b) Small z → 0 (thin inversion layer ∼ shallow river): (dz/dx) tends to the limit (-KF), which indicates that electron flow is predominantly determined by surface scattering conditions, with KF corresponding to "friction" along the insulating interface (the "riverbed"). Hence the magnitude of (dz/dx) may vary locally, along x if KF does so, such that a "gradual" channel no longer obtains along the entire length of the conducting channel. If, furthermore, |KF| should be sufficiently large, or (worse), a function increasing as a power of (1/z), then (dz/dx) ≈ -KF(x,z) would behave in like manner, thereby yet again invalidating a "gradual" channel assumption and, importantly, result in mobility decreasing at a rate other than universal curves' power of (E*T) -1/3 . Such phenomena are observable in [4,I, for electrons and holes under both low temperature and high E*T conditions. c) The "pinch-off" condition corresponds to the latter situation whereby, in view of current continuity, the velocity of carriers along the channel (between source and pinch-off point) may vary as dictated by KF(x,z), while that between the pinch-off point and the drain increases owing to the local, accelerating, longitudinal field E//. Indeed, velocity "saturation" is observed under these circumstances, whose value is near to 10 7 cm/s at 300K for electrons and holes on <100> silicon [9], [12]. While the transverse electric field E*T no longer dominates the characteristics of the channel's conductance, the latter' modeling is complicated by the resulting twodimensional effects. d) Laminar flow, with z(x) not restricted to large values, occurs along isoclines, or curves along which (dz/dx) is constant: (5) and (11) may be solved for constant values of (dz/dx). In the case of special interest whereby E// << E*T, and the effects of KF negligibly small, the result is proportionality between z and (E*T) -1/3 , which is therefore consequence of our starting premise ρ(z) ∼ z, per (7). While the expression for the quantummechanical, average, extension zQM derived by Stern and Howard [5] resulted from an ostensibly different function, ρ(z) ∼ z 2 e -βz (see appendix A2), yet they obtained very nearly the same functional relationship < zQM > ∼ (E*T) -1/3 . In section II.C.2 below, we show that conservation of energy and momentum readily lead to to say, laminar flow is the physical condition underlying universal mobility behavior.

C. Similarities and Distinctions between River and Inversion Layer Cases
While (5) and (11) are similar in general form, the foregoing analogy between the river and inversion layer cases is apposite though limited, because the density of the "incompressible fluid" in the former is considerably higher than that of electrons, whose volume density is more akin to that of a diffuse "gas". Since, moreover, the motion of electrons in the inversion layer is governed by Schrödinger's equation as constrained by the boundary conditions of the system (see appendix A2), this motion may be construed as the diffusion of a probability amplitude from one point to the next along the x axis. Whereas ordinary diffusion of a gas along a thin tube gives rise to real exponential functions, Schrödinger's equation yields complex waves for an "oscillating gas" [13]. While their respective KF parameters, constant or otherwise, can thus be expected to be different qualitatively and quantitatively, further study of the effects contributing to KF, as formulated above, is relevant to the ongoing technological downscaling of integrated devices, circuits, and systems.

1) Dependence of inversion layer thickness on E*T and T
To the extent that the inversion layer may be regarded, conceptually, as the combination of a classical (zcl) and a quantummechanical (zQM) component [3], the following relationships obtain: where (3/2)kT is the average translational kinetic energy of a molecule of an ideal gas at a given temperature T, with k Boltzmann's constant, hence is the (rms) value of the thermal velocity vth ≈ 1.15x10 5 m/s at 300K, whereas the quantum-mechanical component zQM resulting from the uncertainty relationship (∆p ∆z) ≥ ℏ, where (ℏ = h/2π , h being Planck's constant), associated with the wave nature of electrons [13], [18], yields: which may be construed as an energy term when put in a form similar to (12a), such that is the counterpart of (12b), and a velocity the counterpart of (12c).
Since vth >> vdrift ≤ 250 m/s for our experimental conditions (see section VI), and since zcl and zQM, per (12b) and (13b) are related uniquely for a given value of E*T , we may put S = (zcl/zQM) and solve the conservation of energy constraint at fixed x and T: yielding the value of z corresponding to the total, conserved, energy ( ): which varies as (E*T) -1/3 , in correspondence with laminar conditions per II.B.d) above and, unlike the river, is subject to the minimum value imposed by the uncertainty relationship implicit in (13a), as accounted for in the next subsection, and without which 14(a) would allow arbitrarily small z with increasing E*T.

2) Maximum value of E*T for which laminar flow may be expected
The same uncertainty relationship ∆ ∆ ≥ ℏ associated with momentum in the z direction in turn yielding: at 300K, the inversion layer's thickness may be expected to "clear" the Si/oxide interface terrain (see Fig.2a) when: which may serve as an upper limit estimate for laminar flow under ideal conditions of negligibly small KF , and is a function only of terrain roughness amplitude Ao and fundamental constants.
The function (15) shown in Fig 2(b) with terrain "roughness" amplitude Ao as a parameter, takes on the approximate value 0.34 MV/cm (where S ≈ 1.15) at 300K, with Ao = 5 Å commensurate with silicon's lattice dimensions and with the ±5 Å interface amplitude observed for thermally oxidized <100> surfaces. The maximum electric field's value expected from (15), for Ao = 0 at 300K is near to 1.16 MV/cm, as indicated by the dashed arrows.   As far as may be discerned from their published figures [4,I,Figs.4,5], (15) appears to be in quantitative agreement with Takagi et al's data with respect to the temperature dependence of this maximum value for E*T, namely ∼ T 3/2 in the limit Ao = 0, and qualitatively as to its relative insensitivity to the different masses of electrons and holes, and to surface orientations [4,II].
Equation (15)'s close agreement with our observations is readily apprehended in Figs. 3,4,5,7, and in section VI wherein our experimental results are discussed.

3) Establishing that the variations of µ eff as ∼ (E*T) -1/3 , per universal curves, result from conservation laws, laminar flow conditions, and quantum-mechanical effects
To the total energy [qE*T (zcl + zQM)], which is the sum of (12a) and (13a), we may ascribe the equivalent velocity veq ∼ vth ∼ vQM >> vdrift : Momentum (strictly only the x component thereof), and energy conservation within an incremental inversion layer volume then yield: wherein the first term inside the square brackets is constant, per (17). Since vdrift = (µ eff E//), it follows that the product: must be constant. Since veq corresponds nearly to the total energy in the inversion layer's volume, for which z ∼ (E*T) so that µ eff varies as ∼ (E*T) -1/3 , as is indeed the widely reported behavior of "universal mobility" curves for (E*T) ≤ 0.5 MV/cm, for both electrons and holes [2], [4]. Hence µ eff ∼ (E*T) -1/3 is consequence, and manifestation, of the simultaneous concurrence of (i) laminar flow conditions, (ii) energy and momentum conservation, and (iii) quantum-mechanical quantization effects along the z direction resulting in z ∼ (E*T)

III. CONDUCTIVITY, MOBILITY, AND THE CHARGE SHEET MODEL
Under laminar flow conditions, the conductivity of a thin sheet of charge at depth z inside the inversion layer in a (W-x) plane, parallel to that of the gate insulator, is σ(x,z) = [qρ(x,z)µ(z)], where q is the electronic charge, ρ(x,z) the charge density at x along the channel, and µ(z) the average mobility of charge carriers along the x direction, which is related to the drift velocity of (9): where E// is the longitudinal electric field, along the x direction.
The channel's conductance is proportional to the integral of σ(x,z) over the depth of the inversion layer, namely to the sum of the conductivities of incremental sheets therein. The resulting drain current, in turn, being proportional to W, to conductance, and inversely to L, we have the following general functional relationship between the parameters of interest: wherein the integral over dz is taken between the limits of the inversion layer's thickness [0, zmax], and that over dx between those of the channel length [0, L]. This double integral is customarily simplified by assuming (a) constant mobility along the z axis, µ(z) = µ eff , thereby neglecting the effects of transverse fields E*T on mobility and, (b) an inversion layer of zero thicknessthe "charge-sheet" approximationsuch that ρ(x,z) = ρ(x) is a function of x only, or of the potential V(x), since (dV/dx) = E//. Under these simplifying assumptions, the drain current may be expressed in closed form, by integration along the channel length: + … higher order terms (23) wherein ρ(V) is a known, explicit, function of the terminal voltages applied to the transistor, VS = 0 and VD are the potentials at its source (x = 0) and drain (x = L) terminals respectively, C ' ox the gate oxide capacitance per unit area, a1, a2, … an are constants determined by terminal bias and signal potentials, and the higher order terms are negligibly small in this context. For a given value of ID, µ eff is then the only unknown quantity in (23), which is thereby deduced readily from standard measurements. We note that, as simplification (a) above sets µ eff (E*T) = µ eff in (23), the value of E*T corresponding to a given ID, which is essential to the determination of the function µ eff (E*T), is variously assumed to be an "average" [2], or an "effective" [2], [4], [14], value E*T = -(1/ε Si )[ηQi + Qb], with η = 1/2 for electrons [2]- [4], [14], η = 1/3 for holes [4], with Qi (corrected [17]), and Qb, the total inversion and bulk charges, respectively.

A. Limitations of the charge-sheet and constant mobility models
In view of Poisson's potential field equation associated with a given charge distribution (see appendix A1), the assumption of an inversion layer of zero thicknessentailing the tacit neglect of its spatial density profileresults in the loss of information about both the electric field E(z) = (-q/εsi)∫ ( , ) , and the potential ψ(z) = -∫ ( ) within the inversion layer, with εSi the permittivity of silicon, since these integrals are taken between identical limits [0,zmax(x)] when zmax = 0, as indeed the potential difference across any spatial region of zero extent is zero. The surface potential ψs at the Si/insulator interface, may nonetheless be calculated accurately from knowledge of the total inversion layer charge Qi, per (25) below, which is represented by qρ(V) in (23) above [8], [9].
If µ(z) is not assumed constant in (22), and an inversion layer of finite thickness zmax ≠ 0 is assumed, then E(z) and ψ(z) are, in principle, known functions given ρ(x,z), but the drain current in (22) may no longer be expressed in analytical closed form as in (23), because µ(z) may not properly be removed from the integrand of (22). Consequently, and crucially for purposes of quantifying the dependence of mobility on transverse fields (E*T) from observations of ID, the function µ eff (E*T) may no longer be "disentangled" from the result of the integration of the product σ(x,z) = [qρ(x,z)µ(z)] in (22), nor, thereby, extracted unambiguously from standard measurements of ID, unless the function ρ(x,z) is known to a sufficient degree of accuracy.
The following section is devoted to the description of a method for achieving the latter objective under laminar flow conditions of foregoing section II.B.d).

IV. AN ACCURATE METHOD FOR MEASURING THE MEAN POTENTIAL (ψ*) AND TRANSVERSE ELECTRIC FIELD (E*T) ACROSS THE INVERSION LAYER
The function ρ(z) in (22) may be deduced from such bias conditions as result in low/null harmonic distortion components (D2 ≈ D3 ≈ 0) for ID, when sinusoidal signals of suitable amplitudes are applied to the transistor's terminals simultaneously, thereby modulating E*T, as set forth in [10], [11]: under these experimental conditions, nulls of D3 occur uniquely because substrate effects counteract those of mobility, such that a3 = D3 = 0 in (23). When a2 ≈ D2 ≈ 0 simultaneously, ID is then (nearly) proportional to VD, hence the channel conductance constant, and the conducting channel obeys Ohm's Law, which corresponds to laminar flow since constant (dz/dx) = (dz/dV)(dV/dx) = (dz/dV)E//, per II.B.(d), yields proportionality between z(x) and V(x), hence between I D and V D per (8).
By contrast, traditional methods [2], [4] entail an extrapolation of the drain current characteristics at low drain-source potential, to a "threshold" voltage whose definition is neither well defined nor unique [8], thereby resulting in various inconsistencies. These, and other, derivative methods [14]- [17], have thus left unresolved the physical origins of the "universality" of mobility curves, among various related observations. The method implemented below sidesteps such inaccuracies and inconsistencies as incurred by the latter methods, by setting forth the objective criterion (D2 ≈ D3 ≈ 0) for determining the mean electric field E*T across the inversion layer from measurements of ψ*.
This method furthermore yields ρ(z), µ eff (E*T), and the latter' derivative function (θΤ) accurately, thereby rendering these quantities in a manner that is mutually consistent, and verifiable by direct observation, as will be shown in the sequel.

A. Measuring the mean potential ψ* across the inversion layer
The object of the following analysis is to determine the mean potential, ψ* = b*ɸt, across an inversion layer of extent zmax, (from which E*T , η , and θΤ follow), in terms of the known surface potential ψs = bsɸt of the charge sheet model, as given by (25) below. Further details about experimental conditions, methods, and θΤ are elaborated in section VI below. Confirmation that our observations correspond to laminar flow will be at hand after the necessary analytical and experimental evidence has been adduced (see section VI.B).
In the ensuing, we use standard MOSFET terminology for an n channel device, and denote VB = (VSB + 2ɸF + b*ɸt), yields the right-hand side of (24), which relates r to b*, thereby determining b* and ψ*. The observed variations of the mean potential ψ* across the inversion layer, and our analytical predictions for these (see section V below), are shown in Fig.3 for ready comparison with the empirical Schwarz-Russek model [3] (see appendix A2): the latter may be seen to be a reasonable approximation over the range of our data, namely the span of E*T ranging from onset of strong inversion through saturation of ψs.

B. Determining the mean transverse electric field E*T across the inversion layer
The mean electric field E*T (V), which is modulated by an applied drain-source potential V = Vds, is obtained from Qi(ψs) and Qb(ψs) [8, p.79
(d) the mean inversion layer thickness <z> = (ψ*/E*T) resulting from our observations of ψ* and E*T (per Fig.3) is commensurate with the predictions of Stern and Howard [5], and Schwarz and Russek [3]. whereby η = 1/(m+2) for the mean electric field E*T in (28), which is thus obtained as an intermediate result.
For 0 ≤ m(E*T) ≤ 1, the parameter η is thus expected to vary continuously between (1/2) and (1/3), per (28), as observed for standard devices with <100> surfaces, and predictably as a function of the relative magnitudes of inversion (Qi) and bulk (Qb) charges, hence as a function of the level of inversion (see Fig.5). Because this transition is especially prominent in the range of bias conditions whereby neither Qi nor Qb is overwhelmingly dominant over the other, our measurements are concentrated on this particular range, namely 0.4 ≤ (Qi/Qb) ≤ 1. which yields b* readily, given bs = (ψs/ɸt) as obtained from (25). The ratio (33) thus reduces to the expected value (3!)/(4!) = (1/4) in the charge-sheet limit b* → 0, and matches observations of (b*/bs) = (ψ*/ψs) nearly perfectly for (Qi/Qb) ≥ 1, where m ≈ 1 as observed in Fig.4, and otherwise to within ±5%, on an average, over the full observation span. The left-hand side of (33) indicates that the ratio (b*/bs) = (ψ*/ψs) decreases as ψs increases, i.e., as inversion level increases: physically, this means that as E* T increases, the increasing electron charge contributes mainly to an inversion layer's mobile Qi concentrated nearer to the oxide, while the fixed depletion charge Qb remains relatively unaltered. These trends are indicated by the arrows attached to the various quantities depicted in Fig.6(c). ; this condition is observed at low values of |Qi/Qb| < 1 where the "flat" depletion charge profile for Qb is dominant. The standard depiction of a "flat" inversion charge profile between 0 and zmax becomes physically unrealistic as |Qi/Qb| increases, because there must be a finite transition region near to the oxide interface wherein ρ(z) drops towards zero, which is a critical boundary condition of the system. The assumption of a fixed η = (1/2) thus becomes increasingly erroneous as |Qi/Qb| increases past unity (see Fig.7), which is consequential towards the accurate evaluation of µ eff (per section III).
The ratio of mean electric field (E*ρ) across the inversion layer corresponding to ρ(z) per (30), to that (E*u) of a uniform charge distribution supporting the same total charge The foregoing results thus indicate how ψ*, ψs, E*T, and η are related, which are determined by the level of inversion (i.e., the relative magnitudes of Qi and Qb), and by the charge density profile within the inversion layer ρ(z). The traditional assumption of a value η = (1/2) fixed a priorias illustrated in Fig.6(a), and corresponding to constant ρ(z) which does not account for these effects, may thereby be expected to give rise to observable discrepancies left wanting of clarification and reconciliation, as indeed noted in [2]- [4]. 5 Although θΤ may likewise be obtained from nulls of D2, those of D3 are much more sensitive to bias and

VI. TRANSVERSE-FIELD MODULATION PARAMETER (θΤ)
Conceptually, θΤ represents a first order Taylor series expansion of the expression for the mobility µ eff (E*T) associated with the MOSFET's drain current: the constant µ eff in (23) is replaced by is the potential along the channel's direction (x), and [θΤ • V(x)] << 1. The drain current ID is then obtained by integrating the function from 0 to L, in the conventional manner [10], [11]. As such, the single parameter θΤ effectively subsumes the "averaging", per (22), of the effect of E*T on the conductivities of the infinitesimal charge sheets in the z direction (between 0 and zmax).

A. Experimental conditions
All measurements were made on long n-channel devices (L ≈ 100µm) fabricated in a standard, industrial, CMOS process. The range of E*T for our data is centered near to 0.2 MV/cm, where universal mobility µ eff (E*T) ∼ (E*T) -1/3 behavior prevails, as reported in the literature [2], [4].
While our over-arching objective is to assess the validity and limits of the foregoing analytical model, we summarize the substantial quantitative agreement of our observations with this model in the next subsection.
which is indeed observed in Fig. 7. In view of our analogy with a river's flow, the same result may be deduced from (4) and/or (9) In view of (11), moreover, we may determine independently of section II.C.2, the absence of D3 = 0 observations for E*T ≥ 0.25 MV/cm for our devices, that is, why laminar flow conditions no longer prevail beyond this limit: at the upper end of our data range, where E*T ≈ 0.25 MV/cm, we have z ≈ 15Å, while z is decreasing at the nearly constant rate dz/dE*T ≈ −10Å/(0.1MV/cm), (see Fig.5). For this value of z, the non-unity term in the denominator of (11) is approximately 1/30, hence negligible. The same term multiplied by a factor of KF(E*T/E // ) appears in the numerator of (11), and increases as (1/z) with decreasing z. Since, moreover, (E*T /E // ) is of the order of 4x10 3 , and the viscosity of (uncompressed) air is 1.81x10 -4 poise at 20°C [18], the non-unity term is of the order of 0.7, hence the magnitude of the rate dz/dE*T increasing as (1/z) is near to 1.7 times its previous value. Hence z may be expected to drop by a further 10Å (i.e. from 15Å to 5Å) when E*T ≤ 0.35 MV/cm, in good agreement with the estimate provided by (15), and as illustrated in Figs.2(a),(b). While the foregoing argument is tentative, the requisite conditions for non-laminar flow may plausibly ensue, even neglecting (KF)'s scattering components -to the extent that the electron "gas" is compressed into a layer <z> of the order of silicon's lattice constant (≈ 5.4Å), and of the ≈ ±5Å amplitude of typical Si/oxide "terrain" irregularities, as observed by electron microscopy in good-quality devices.

VII. SUMMARY
We have elucidated physical origins of the dominant phenomena governing current conduction in the MOSFET's inversion layer, below saturation, by comparing the latter to a broad river's flow in a gravitational potential field, whereby energy conservation considerations are fundamental.
We have drawn upon such familiar concepts of fluid mechanics as readily apprehended by common sense and familiar experience, to derive analytical expressions promoting physical insight into subtle, intertwined, effects subsumed by the charge sheet, simplifying, assumptions of standard textbooks and circuit simulation models of the MOSFET's inversion layer. Such interrelationships have been made manifest as obtain between the inversion layer's charge density profile ρ(z), its mean <z> and maximum (zmax) extents, the low-field effective mobility (µ eff ), and the influence of scattering (KF), and quantum mechanical, effects.
A sensitive method for measuring the mean potential (ψ*) across the inversion layer was applied under conditions of harmonic distortion component nulls (D2 ≈ D3 ≈ 0) of the drain current under sinusoidal excitation, below saturation. We showed, thereby, how ρ(z) may be deduced accurately from the ratio (ψ*/ψs), as the surface potential (ψs) increases from onset of strong inversion towards saturation, and how such information underpins the physical origins of disparate observations of long standing. We found and established: (1) Non-uniform, or non-constant, ρ(z) causes variations of 1/3 ≤ η ≤ 1/2, which determine the mean transverse electric field E*T = -(1/ε Si )[ηQi + Qb] in terms of the inversion Qi, and bulk Qb, charges.
(2) Laminar current flow and maximum drift velocity, which prevail when D2 ≈ D3 ≈ 0, result from energy conservation, concurrently with the gradual channel approximation entailing <z> varying as ∼ (E*T) -1/3 . The latter relationship is nearly identical to that obtained by Stern and Howard [5] from solutions of Schrödinger's equationwhich embodies conservation principlesas constrained by the boundary conditions pertinent to the system. Moreover, whereas the latter solutions resulted in a fixed value η = (11/32), our measurements indicate that η varies with the level of inversion as per (1) above. The value (11/32) ≈ (1/3) nearly corresponds to the result of our model ρ(z) ∼ z, which is a close approximation of the Stern-Howard model, and otherwise substantially in accord therewith.
(3) The behavior of "universal mobility" curves, whereby µ eff also varies as ∼ (E*T) -1/3 , is consequence and manifestation of laminar flow conditions. We established this result, analytically, by invoking energy and momentum conservation, as well as the prevalence of quantum-mechanical effects.
(4) An analytical expression for an upper limit to the transverse electric field for which laminar flow may be expected, (E*T)MAX(300K) ≈ 0.35 MV/cm, which is a function of Si/oxide interface "terrain roughness" amplitude and fundamental constants only, in quasi-quantitative agreement with extant observations with respect to temperature and surface orientation for both electrons and holes, as available in the literature.
(5) The higher order, derivative, parameter θΤ quantifying the modulation of µ eff by E*T varies as ∼ (E*T) -5/3 under laminar flow conditions. The close agreement between our analytical predictions and experimental observations provide corroboration for the validity of our interpretations of the physical effects underlying the foregoing models, and for the consistency of the measurements thereto related.

A1.
Functional correspondence between parameters of the gravitational and electrical fields pertaining, respectively, to the river and to the MOSFET's inversion layer. Note that z represents depth of river and depth (thickness) of inversion layer, as indicated in Fig. 1.

Gravitational Electrical
Particle mass m Stern and Howard solved the timeindependent Schrödinger equation for the energy levels E, and envelope functions ξor measures of the diffusion of a probability amplitude from one point to the nextof inversion-layer electrons moving in a potential well in one dimension (our z dimension), which is similar to the harmonic oscillator problem wherein the allowed energy levels E are quantized [18]: where T is a kinetic energy operator and ɸ(z) the electrostatic potential obeying Poisson's equation (see appendix A1 above), with ρ(z) the sum of the densities of inversion and fixed charges in the depletion layer. The boundary condition ξ(0) = 0 was used for the oxide-silicon interface, where a small envelope amplitude |ξ(0)| is expected, while those on ɸ(z) were: ɸ(z) → 0 as z → ∞ , and  (  The density of states for electrons ρ(E) results from these considerations which, in turn, leads to a desired, self-consistent, approximate solution ρ(z) ∼ (½)β 3 z 2 e ˗βz , as shown in Fig.6(c

(A2.2)
and the factor (11/32) is a fixed value for the factor η in our expression for E*T per (28). The average value of the inversion layer thickness z weighted by the charge distribution ρ(z) is then <z> ≈ 3/(2β) ≈ 22.5Å, which is a maximum when Qi → 0, and considerably thinner than the depletion layer, which they evaluated at 1.2µm for their process, even in this limiting condition. The general agreement between the results of the Stern-Howard model and ours -except for the latter' variable η, per our experimental observationsis simply owed to the fact that (A.2.1) and (10) are equivalent starting premises embodying conservation laws for the system at hand.

Schwarz and
Russek formulated the following empirical expression for a uniform inversion layer thickness zmax, which they postulated to correspond to (zcl + zQM ≈ 2zcl) as per (12b) and (13b) above, and by matching z max ≈ 45Å per Stern and Howard ( where ɸt = kT/q ≈ 26mV at room T, and E*T, evaluated with fixed η = ½ is in units of V/cm. The mean potential across the inversion layer predicted by (A2.3), or (½)(zmax⸱E*T), is compared to our observations thereof, and to that of our analytical model in Fig.3, while our model's predictions for <z> are compared to (A2.3) in Fig.5. The second term on the right hand side of (A2.3), which they attributed to "quantum-mechanical broadening", and based on [5], served two purposes: (a) it introduced the (E*T) -1/3 dependence of z on E*T which they found necessary to match the observed variations of µ eff (E* T ) as ∼ (E* T ) -1/3 when this term dominates the first (i.e., at higher E*T values) and, (b) it was "calibrated" by the factor (1.24x10 -5 ) such that the first and second terms on the right hand side of (A2.3) are approximately equal. The overall result was that (A2.3) achieved, approximately, the desired doubling of the value of zcl, and a potential across the inversion layer near to 1.5ɸt ≈ 39mV throughout the data range, as illustrated by Fig.3. The arguments presented in section II.C.3 above, which invoke considerations of energy and momentum conservation (as "builtinto" Schrödinger's equation), namely z ∼ (E*T) -1/3 , are thus in quantitative and qualitative agreement with Stern and Howard's (A2.2), while only in quantitative agreement with Schwarz and Russek's empirical (A2.3), the latter' lack of theoretical justification notwithstanding (which they acknowledged).