Probing of quantum energy levels in nanoscale body SOI-MOSFET: Experimental and simulation results

For Silicon-On-Insulator Metal–Oxide-Semiconductor Field-Effect-Transistors (SOI-MOSFETs), the critical scaling of the channel thickness from deca-nanometer, for so-called Ultra-Thin Body (UTB) devices, to the nanometer node in so-called Nano-Scale Body (NSB) and non-planar Fin-FETs is still a bottleneck of the current VLSI technology.¹ A major disadvantage of such SOI NSB and Fin-FETs is a severe increase of parasitic source and drain resistances as silicon film thicknesses drop below the 10 nm limit.² In this perspective, several comparative studies of the electrical characteristics between UTB and NSB at room and low temperatures, as well as anomalous transport behavior, have recently been presented.^3–5 On the other hand, there is a clear advantage to using NSB transistors since their quantum well structure enables the discretization of the energy levels, leading to a step-like phenomenon in the transfer characteristics and possibly to radiative recombination in Inter Sub-Band Transition (ISBT). In this paper, measurements of the transfer characteristics of UTB device with those of NSB SOI-MOSFET devices, having respective channel thicknesses of 46 nm, 2.4 nm, and 1.6 nm, are presented and discussed. An analytic model, based on quantum mechanics and statistical physics, is developed to explain these important experimental results.

UTB and NSB devices have been simultaneously fabricated on the same SOI wafer, using a selective “Gate Recessed” (GRC) process.⁶ The starting material is a p-type doped 53 nm thick silicon layer on a 65 nm thick Buried Oxide (BOX) layer; other device specifications are detailed in a previous work.⁷ After a Local Oxidation of Silicon (LOCOS) process, two steps of oxidation were then performed: first, a 75 nm thick sacrificial oxide layer, called Channel Oxide (CHAN OX), was grown in order to significantly decrease the channel's thickness to about 10 nm, before removal. The second phase of the oxidation (26 nm thick), called Gate Oxidation (GOX), was a more accurate step, making possible to get the final channel thickness, in the nanometer range (down to 1.6 nm). Both UTB and NSB devices share the same Width to Length ratio (80/8 μm). In order to check the final silicon channel thickness, several measurements have been performed. One of them was a High Resolution Transmission Electron Microscopy (HRTEM) study of the quantum well structure, as shown in Fig. 1.

As part of the investigations, it was necessary to verify the electrical functionality of the UTB (46 nm channel thickness) and NSB (2.4 nm thickness) transistors using standard output characteristics curves, i.e., drain current I_DS vs. drain voltage V_DS, as shown below in Figs. 2 and 3, respectively, and both measured at 300 K. Although these characteristics are representative of a conventional MOSFET's, the NSB device presents huge attenuation of current (three orders of magnitude). In a previous work,⁵ we proposed to interpret such a behavior by a series resistance.

In order to seek for the quantum effects, it was necessary to study the behavior of the UTB and NSB devices at both room (300 K) and low temperatures (77 K and 4.7 K). This time, the transfer curves (I_DS-V_GS) were measured at a fixed low drain voltage (V_DS = 0.1 V) so the devices remain into the linear domain. Figures 4 and 5, respectively, present UTB and NSB electrical behaviors for the three temperatures. Again, three orders of magnitude separate the I_DS currents between UTB and NSB devices. When compared to the NSB, the UTB device did not present any step at any of the checked temperatures (Fig. 4). Also, the three UTB curves present the same current order of magnitude. Also, a decrease of the current with the temperature is observed (although for low V_GS, the current at 4.7 K is slightly above the current at high temperature). For NSB, a clear difference is observed between room temperatures curves and the lower ones (Fig. 5). 77 K and 4.7 K curves present steps, pointing to discrete values of filled energy levels, as will be discussed later. Such steps have been observed for NSB devices only, presenting possible quantum effects. In order to deny any artifact phenomenon, an additional NSB device thickness of 1.6 nm was measured at 77 K (Fig. 6). For this device, again we could observe clear “steps” areas.

Focusing on the lowest available states, we will consider the lowest energy conduction band of the silicon indirect electronic band structure, i.e., the sixfold degenerated valley in the Γ-X direction of the first Brillouin zone. The growth direction [001] is defined as the z direction which is also the confinement direction of the electrons in the quantum well. Since the silicon channel thickness t_si is low enough for NSB devices (t_si< 5 nm), it can be described as a finite square quantum well where the finite barriers are made of silicon oxide (Fig. 7). The bandgap for SiO₂ is 8.9 eV and for Si is 1.1 eV, and the energy difference between the conduction bands of SiO₂ and Si, i.e., the well energy barrier is qV₀ = 3.3 eV.⁸ Even so, it was reported that the bandgap value is increasing by decreasing the size ultrathin silicon layers or structures,^9,10 the bandgap value is not an explicit parameter of our model. We consider the n+ poly layer as a degenerated semiconductor (metal like) so the electron affinity merges with the work function.

Since the oxide barriers are sufficiently thick (>20 nm), the effective electron mass in the SiO₂ barrier will be assumed to be the bulk SiO₂ effective mass. The value of the effective mass in SiO₂ is not clear, not least because it is an amorphous material; due to disorder, it has a band structure only in an approximate sense. Some authors state that it is equal to the free electron rest mass m₀⁸ and others to 0.5m₀.¹¹ However, the exact value of the SiO₂ effective mass has a little influence on the energy levels (mainly less than 5%). The z confined electron effective mass m*_z will be taken as 0.92m₀ matching the twice degenerate valley noted Δ₂.² The fourfold degenerate valley (Δ₄) for which m*_z= 0.2m₀ will be also considered later to calculate the density of states. By solving the Schrödinger equation for a finite quantum well, a series of even and odd eigen values of energy levels E_n can be derived. Results for t_si 1.6 nm and 2.4 nm are presented in Table I.

An original approach to analyze the subthreshold regime to seek for quantum confinement effects was demonstrated,^12,13 however, we present here a different method to measure quantum effects in MOSFET Quantum Well. In order to correlate between the discrete energy levels of the NSB quantum well structure and the steps measured in the drain current, it is necessary to review the current equations and to develop some adapted relation to the quantum structure.

The UTB's output characteristics can be modeled by the well-established SOI MOSFET's equations developed for fully depleted devices.^5,14–16 For NSB, if the gate voltage V_GS is larger than the threshold voltage V_T, the channel can be considered as fully uniform volume inverted for channel thicknesses as low as 1.6 nm and 2.4 nm. For non-vanishing V_DS, the channel is not in thermodynamic equilibrium and a current develops. A priori, the system can be described using a simple Drude (drift-diffusion) model. This, it turns out, is insufficient to describe the observed step-like behavior; the details can be found in  Appendix A.

To compute the current density more precisely, we apply a well-known method for its success in calculating the conductivity of metals. The current density is computed as the thermal mean¹⁷ 

where n is the carrier density and $vx$ is the averaged component of the electron drift velocity along the channel (x axis). At equilibrium, the probability density is given by the Fermi-Dirac occupancy function$fFD[(v¯)]$⁠. The latter depends only on the energy level, and so the current density along the channel is expressed as the integral

which vanishes by symmetry. In the presence of an electric field, the system is not in equilibrium and the probability density must be computed using the Boltzmann Transport Equation (BTE). Under reasonable assumptions, such as isotropic scattering, and reasonably low fields, one may use the Relaxation Time Approximation (RTA). In essence, this says the following: after a collision, a carrier particle's distribution is “reset” to the equilibrium distribution. Between collisions, however, it experiences a uniform acceleration

due to the electric field E. $x^$ is a unit vector in the channel direction. The relaxation time τ can be identified as the mean time since the last collision; on average, the field imparts an additional

to the particle's velocity. The equilibrium distribution is thus shifted by $Δv¯$⁠: in evaluating the probability of a particular velocity $v¯$⁠, we must “rewind”—by subtracting $Δv¯$—to the instant immediately following the last collision, where the equilibrium distribution may be used. Since the FD distribution is only explicitly dependent on energy, the distribution in the presence of the electric field is given by

Since the method is well known, a more methodical derivation for the uninitiated reader is relegated to the  Appendix.

The shift is uniform only in $v¯$-space. In energy space, positive velocities are accelerated and thus shifted to higher energy, while negative energies are decelerated and thus shifted to lower energy. This is illustrated in Fig. 8. The distinction is critical, and it underlies the relation between the current and the density of states which we now derive.

For $aτ$ small, $fFDE(v¯−aτx^)$ can be approximated by expanding and truncating at first order

Inserting into the expression for current density $Jx$ gives

where the first term has vanished by symmetry as before. To evaluate the expression, we note first that

By symmetry $vx2$ in the integral may be replaced by $13v2$⁠. Noting next that

and transforming to an integral over energy, the expression for the current density becomes

It should be noted that for simplicity, the preceding discussion assumed an isotropic effective mass tensor, as pertains to the Γ-point valley. However, (10) holds also for the true conduction band edge at the Δ-valley, provided that the details of the anisotropy are contained in the acceleration $a(v)$ and the density of states (per unit energy per unit volume) $g(E)=g(v)(E)$ which must now depend on the v valley. The details of the derivation are left to the  Appendix B. We will return to the valleys in Sec. III G.

For temperatures which are not too high, the Fermi-Dirac function is close to a step function. It is thus reasonable to approximate

This is a direct reflection of the asymmetry of the shift cited above: for positive velocities, the distribution is shifted rightwards, for negative velocities, leftwards. Since f_FD is flat except in the vicinity of $EF$⁠, cancellation between the contributions from positive and negative velocities persists—except in the vicinity of $EF$⁠, where the distributions no longer overlap.

Inserting (11) into the expression for the current density, we see that if $EF<EC$⁠, the current is zero, while if $EC<EF$⁠, we obtain the important relation

In other words, the current samples the value of the density of states at the Fermi Energy. More specifically, we consider the effect of both terms in the pre-factor $EF−EC$⁠.

The first term, $EF$⁠, is constant. Since, as we shall see, varying the gate voltage serves to shift the density of states relative to $EF$⁠, the profile of this contribution to the current as a function of V_G should be, up to a constant, precisely that of the density of states. Therefore, it can be considered as a density of states probe. The latter in this case is step-like so that this term represents a step-like component of the transfer curve. The second term, involving $−EC$⁠, also appears constant; in fact, the conduction band edge undergoes band-bending so that the leading order $EC$ is linear in the gate voltage $VGS$⁠. The product of a straight line with a step-like profile gives a “broken-line,” or kinks, of increasing slopes. The I_DS-V_GS is thus a combination of two contributions, one step-like the other increasing linear slopes. Initially for non-degenerate doping, $EF<EC$ and there is no current. At the flat band voltage $EF=EC$⁠, the inversion layer begins to be populated and current begins to flow.

In order to correlate between the discrete energy levels of the NSB and the continuous energy integral in (10), the density of states must be specified. For a quasi-two-dimensional system, the global three-dimensional density of states (3D-DOS) noted $g(E)$ in the previous part, which is the number of electronic states per energy per unit volume, is of the form² 

where $N2D(E−Ej(v))$ is the two-dimensional DOS (in cm⁻² eV⁻¹ units) at the energy level $Ej(v)$ associated with the corresponding wave function $ψj(v)(z)$for a given Δ_ν valley. $tSi$ is the channel thickness.

The local DOS of the Δ_ν valley has the form of a series of step functions of equal heights

where $H(E−Ej(v))$ is the Heaviside step function and $g0(v)=1tSi2πℏ2md,υ∗$ is the height of the 3D-DOS step function relative to the Δ_ν valley of multiplicity ν (equals 2 or 4). The effective mass $md,υ∗$ for the DOS is 0.2 m₀ and 0.42 m₀ for Δ₂ and Δ₄ valley, respectively (with m₀ is the free electron rest mass). More details are provided in Sec. III G below.

The sum runs over the number of discrete energy states which is generally less than 10. Higher energy levels are not bound states which decay outside the well and are scattering states which are propagating outside the well. Any finite initial probability distribution of such states inside the well can be shown to decay exponentially with time, leaking out of the system. This has been confirmed numerically as well as can be seen in Figs. f9,10–13.

Inserting (14) into (13) and then into (10) and integrating by parts gives

where J₁ and J₂ are mathematical notations for the energy related components of the current density defined as follows:

and

The absolute energy levels $Ei(v)$ have been decomposed according to

where $ϵi(v)$ are the energy offsets relative to the bottom of the well, found above by solving Schrödinger's equation in the Effective Mass Approximation, that is, the energy level calculated relative to $EC$ as given in Table I. Also, the acceleration depends on the mass which in turn depends on the valley; hence, it has been brought within the sum over valleys. As alluded briefly in Sec. III C, and more fully in  Appendix A, the term $J1,i(v)$ in (16) is predicted by the drift-diffusion model; the second term, $J2,i(v)$⁠, in (17), then represents the specific contribution of the BTE-RTA model.

In principle, (15) gives a differential equation for the variation along the channel of the electrical potential V(x), related to the Fermi level via $EF(x)=qV(x)$⁠. The total current must be constant, whence, using

one arrives at the relation of the total current in the channel

The area of the cross section orthogonal to the current is given above by the product of the channel width W and the channel thickness t_Si. This expression can be integrated and expressed in terms of known functions to give a family of characteristic curves, I(V_DS)

where L is the channel length. It is evident by inspection of the integrand that saturation occurs at high $VDS$⁠, as expected for a transistor. However, for small values, one can probe the linear region of the transistor by assuming that the potential drop is linear over the channel's length and lower than the gate voltage V_GS, giving the following approximation:

Thus,

To determine the I–V transfer curve, we must find a relation between $EF−EC$ and the front gate voltage $VGS$⁠. The Fermi level is set by the value of the source potential

Meanwhile, the effect of the applied bias is that the conduction band edge $Ec$ deviates from its unperturbed value by a band bending potential $ϕbb,Ch$ as illustrated in Fig. 7 

To find the band-bending of the channel, we first consider the gate electrode, where a band-bending potential $ϕbb,G$ accounts for the difference between the metal work function and the applied bias (measured relative to the vacuum)

Since the electron charge density decreases by decreasing the channel thickness below 5 nm, one may assume that the voltage drop on the gate oxide is negligible as in Fig. 7. It is possible to investigate the significance of refining this assumption to take into account the linear potential drop across the oxide stemming from the inversion charge in the channel. At this stage, however, we approximate by taking the channel band-bending equal to the band-bending at the gate

The unperturbed conduction band edge is essentially identical to the silicon electron affinity

Inserting in (25) and combining with (26) and (28) gives

Subtracting this from $EF$in (24)—equivalent to taking the reference of the energies as the silicon channel Fermi Level E_F,s—gives

or

Finally, one defines the flat-band voltage $VFB$ such that $qVFB$ is the difference between the gate electrode work function $ΦG$∼4 eV for n+ polysilicon and the silicon electron affinity, equal to $χSi$ = 4.05 eV (bulk value)

This gives (Ref. 2, p. 107)

Substituting into expressions (16) and (17) for the current density gives the following expressions for the components of the current as a function of the front gate voltage:

and

To appraise the qualitative behavior of the current expression (23), we will take the limit of zero temperature, T → 0, where the expressions simplify greatly. Following this analysis, we will justify the relevance of this limit to the description at higher temperatures.

In the low temperature limit, the I–V curve is a combination of steps and a piecewise linear contribution. To see this, we consider the current density, first expressed in terms of $EF−EC$⁠, (31) and substitute later the dependence on V_GS. This avoids limiting the conclusions by specific assumptions relating the band-bending at the channel and at the gate, such as (27). To translate the form of E_C-dependence to V_GS-dependence, one need only recall that –E_C increases (approximately linearly) with V_GS.

In order to appreciate the qualitative behavior, it is sufficient to focus on the terms pertaining to one valley. Hence, we consider the expression (constants have been suppressed for simplicity)

in the low-temperature limit, discussing each of the two terms separately.

We start with the first term

The higher levels for which $Δi>0$ give a negative argument to the exponent; the latter thus tends to 0 leaving $ln⁡1=0.$ The lower levels for which $Δi<0$ lead to a positive argument for the exponent; the latter thus completely dominates the term equal to 1 leaving

In the limit of zero temperature, then this term tends to

As each $Δi$ turns negative, the subband “turns on” and begins to contribute $−Δi>0$ to the sum. As V_GS increases, −E_C increases linearly as does $−Δi.$This follows by substituting

giving (39) in terms of V_GS

The additional linear contribution implies that there is a discontinuous increase in the slope of the line segment. However, one notes that precisely at the gate voltage at which the i^th subband turns on $Δi=0$ and so the contribution to the sum starts at zero; there is thus no jump in the value of the function itself—the term $J~1$ is continuous. The graph is thus a “broken line,” with kinks—points where the slope of the line jumps. The increase in the slope is the same at all the kinks. (This is demonstrated ahead for $T≠0$ in Fig. 14.)

The second term is

In the limit T → 0, the exponent in the denominator either diverges or tends to zero depending on whether

is positive or negative. We see then that

The energies $ϵi$ increase with i; hence, for V_GS = 0, $Δi$ may be negative for the first few terms, but is positive thereafter. As V_GS increases E_C becomes more negative; for any given i, $Δi$ eventually turns positive. At this point, the energy level on an absolute scale, relative to the fixed vacuum, slips below the Fermi level, $EF$⁠, so that the subband begins to be populated. The corresponding term in the sum “turns on” and contributes $ϵi$ to the sum. This is illustrated in Fig. 9.

In the limit then, the graph of $J~2$ vs. E_C can be expected to consist of flat steps whose height increases with energy level $ϵi$⁠. The length of the steps is precisely the interval between successive energy levels. Considering for reference the levels of an infinite square well of width a,

the length of the steps increases linearly since

For the finite well under discussion, this quality can be expected to hold approximately. (This is exemplified ahead for $T≠0$ as shown in Fig. 15.)

It is also useful to consider the combination of terms

which has the zero-temperature limit

$J~2′$ is easily seen to represent steps of equal height $EF$ and thus reproduces the profile of the density of states. For the case of the infinite square well (45), the constant height of the steps in conjunction with the quadratic location of the energies (46) lends the profile the form of a square root function. This in fact discloses the true three-dimensional character of the underlying system. It will be seen to hold approximately for the system under investigation (see ahead for $T≠0$ in Fig. 15); generally, this is the case for the spectrum of a quasi-low dimensional system.

Finally, the zero-temperature limit of the total current is obtained by combining (39) [or (41)] and (44). The steps and slopes of the foregoing discussion will combine to produce piecewise linear segments of increasing slope, separated by discontinuities of increasing height. This is precisely what one finds explicitly by considering the limit

For later reference, $J~1$⁠, $J~2$⁠, and $J~2′$ are expressed in terms of V_GS

and

Finally, it is necessary to justify the validity of using the zero-temperature limit to describe the qualitative behavior at higher temperatures. One must estimate the range of temperatures for which this limit is a good approximation. The expressions presented in this section are not analytic in T, near T = 0. Instead, one may expand each term in the sums appearing say in (36) in terms of the small quantities

This leads one to identify

as the small parameter. The range of validity of the T = 0 limit is then determined by the requirement

One concludes first that deviations from the low temperature limit are to be expected primarily for low gate voltage. Furthermore, for gate voltages of order 1 V or more, substituting $kTrm=0.026eV$ leads to the estimate

We may thus anticipate that I–V curve will deviate very little from for the zero-temperature form, for temperatures beneath 100 K and that even at room temperature, the difference should be reasonably mild. The graphs of the following sections at T = 77 K, indeed realize these expectations: steps are clearly discernible, though with corners somewhat rounded. (At T = 4.7 K the energy related current component $J~1′$ clearly displays abrupt steps.)

After exploring the predictions of our model for a single valley by considering the zero-temperature limit, we proceed finally to incorporate the contributions from the different valleys in order to derive the precise expression for the total current. Referring to (23), the factor $μx(v)g(v)$must be evaluated. One assumes a pure parabolic approximation for the conduction band. As described in (14), the height of the 3D-DOS step function $g0(v)$ then contains a factor of the square root of the determinant of the valley-mass-matrix, restricted to the lateral, non-confined directions, . It thus depends on the geometric mean of the two effective masses in the lateral directions² 

This is multiplied by the valley-dependent electron mobility appearing in (19)

where the mass along the field, i.e., along the channel. This gives

For the two Δ₂ valleys, $mx(2)=my(2)=ml=0.92me$ and the factor in the square root is identically equal to one. For the two Δ₄ valleys oriented along the y-direction, $mx(4,y)=mt=0.19me,my(4,y)=ml=0.92me$ and we obtain

while for the two Δ₄ valleys oriented along the x-direction, $mx(4,x)=ml=0.92me,my(4,x)=mt=0.19me,$ and we obtain the reciprocal

The total current includes carriers from all valleys, which come in pairs according to their degeneracy

with

It is interesting that the channel thickness has dropped out. We conclude this section by noting that the terms $Ix(v)$ differ by more than just the coefficient (59). As touched upon above in Sec. III A, the energy levels—and the spacing between them—scale inversely with the effective mass in the confinement direction, $ϵi(v)∼1/mz(v)$⁠. The quantization mass of the two Δ₂ valleys is larger by a factor of $ml/mt≅4.8$ than that of the Δ₄ valleys. The lowest Δ₂ valley subband is thus lower by a factor of $≅4.8$⁠, and successive subbands are more closely spaced by the same factor. Using a parabolic approximation for $ϵi(v)$ allows one to estimate that the number levels lower than a given energy level increases as

Thus, at least four Δ₂ valley subbands (two energy levels) will turn on before the first (set of four) Δ₄ valley subbands.

We are in a position to apply the expressions developed above, (62) and (63), to the energy levels computed in Sec. III A, for the purpose of making precise predictions of the current as a function of gate voltage.

All the curves were computed at T = 77 K, with V_FB taken as 0.45 V and V_DS = 0.1 V. We first present results for the Δ₂ valleys. Figure 14 displays the two contributions $J~1$ and $J~2$ separately with arbitrary normalization, together with the total current contribution_.. The form of the curves exemplifies the characteristics above deduced from the T = 0 limit (Sec. III G), with corners suitably rounded: $J~1$ is continuous with nearly linear segments of increasing slope, while $J~2$ shows steps of increasing height, and the total current displays (nearly) linear segments of increasing slope separated by approximately vertical jumps. Next, the density of states can be glimpsed in Fig. 15, which displays the $J~2′$ term. The interval between steps increases roughly linearly leading to a square-root-like profile, also discussed above.

These Δ₂ valley curves are followed by a comparison of the (total) current for the Δ₂ and Δ₄ valleys, together with the total current obtained from all valleys in Fig. 16. This is to be compared to the measured behavior at 77 K presented earlier in Fig. 6. The foregoing graphs all display results for the thickness t_si = 1.6 nm. Figure 17 again displays a comparison of the current for the Δ₂ and Δ₄ valleys, together with the total current, this time for thickness t_si = 2.4 nm. Finally, in Fig. 18, we compare the total current obtained from all valleys for two values of the channel thickness, 1.6 nm and 2.4 nm.

The idea of a quantum-well based transistor has been raised in different configurations. Several architectures and designs of quantum-well based transistors have been developed in the last decade: part of them have been processed using III-V materials,¹⁸ some using Silicon-On-Insulator (SOI) starting wafers, and recently, nanowire FETs have been modeled.¹⁹ However, none of them presented the specifications of the device presented here nor conducted series of analyses in three representative temperatures (300 K, 77 K, 4.7 K). The steep decreasing of the channel conductance was already reported experimentally in the literature and interpreted by a decreasing of the electron mobility for small channel thicknesses. This trend was explained by several complex mechanisms like phonon scattering,²⁰ roughness interface effects,²¹ or quantum well fluctuations effects. The effect of conductance quantum steps was already reported.^22–24 Low temperature analyses (4.7 K) have been conducted and shown step-like phenomena in the current curves when studying electron mobility.²⁵ 

In Fig. 4, for UTB device, no steps were observed in the transfer curve since the channel thickness is too high to reveal quantum effects. We note a decreasing of both the threshold voltage and the drain current at lower temperatures. This can be explained, respectively, by the dominance of bipolar effects over thermal generation of carriers²⁶ and by the increasing of the series resistance due to a “freeze-out” effect of the dopants at cryogenic temperature.²⁷ 

By using the relaxation time approximation in the two dimensional density of state description of the electrons motion, two phenomena are described: current steps and kinks in the I_DS and V_GS curves at low temperature. These phenomena depict the scanning of the different discrete subband energy levels which are induced in narrow quantum wells. By varying V_GS, one actually probes the different energy levels in the well, that is, it is possible to directly measure the quantum well energy levels, with no consideration to selection rules or electron transition probability. However, as can be seen in Fig. 5, the steps in the measured graphs are less obvious than in the theoretical one. We believe that this is a consequence of some roughness in the interface between the well and the barriers.

The key parameters in quantum-based devices are not only the thickness of the relevant layer but also its interface with the upper and lower layers, creating the confinement. The roughness quality of such an interface is definitely a technology challenge, when growing up layers of few nanometers. As for the thickness itself, these are already two decades that studies are performed to evaluate and forecast the quantum effects on electrical and optical behaviors.²⁸ The mathematical model presented above enabled us to build a simulation of the steps using Matlab software together with COMSOL finite elements solver. The I_DS equation (20) was simulated, including the S factor and presented in Fig. 18. Two separated curves of steps, respectively, for 1.6 nm and 2.4 nm channel thicknesses, have been obtained. It is significant that the experimentally obtained levels are the Δ₂ valley (m_z* = 0.92m₀) levels only. This may be interpreted as experimental confirmation that Δ₂ valley states are lowest and most populated. Moreover, from Figs. 16 and 17, we see that the current contribution from the Δ₂ valley is much smaller than the contribution from the other valleys and they may be unresolved in our measurement. The fact that the electric field responsible of the quantum energy level acts only in the longitudinal direction may also play a role. Since the 77 K temperature shown most of the conductance steps, however, some were not sharp enough, and additional measurements have been performed at 4.7 K temperature, as shown in Fig. 5. In this case, the steps appear more clearly all along the curve and do reasonably fit the energy level values obtained in Table I.

In order to demonstrate the accuracy of the analytical model, assuming smooth processing phase, a comparative table of the numerical and analytical results is presented in Table II.

At the end, the last necessary comparison would be between the measured values of the steps in the transfer curves and the calculated values, as presented in Table III. The relative difference between the two series of values is not larger than 30%, which is reasonable for such low values, in particular when remembering that the analytical model does not take in consideration the processing issues, such as interface states, roughness, and series resistance.

Quantum steps have been clearly observed and demonstrated in transfer curves of NSB devices, when compared to UTBs. The combined experiment of quantum well structure, serving as the transistor channel, and of the cold temperature conditions, enabled the discretization of the drain current related to the filling process of the subband levels of energy. The locations of the steps are consistent with the predictions of an indirect-to-direct bandgap transition in ultra-thin silicon layers. Efforts should now be concentrated in order to optimize the device to achieve a higher rate of operation and better sensitivity, prior to future fabrication of the next generation.

In this section, a simple Drude model is presented to describe the channel current, together with a discussion of its insufficiencies. Assuming the classic gradual channel approximation, in which the field is uniform along the channel, the drift current is proportional to the drain voltage V_DS. The total current will consist of contributions from the different valleys, all of the form

where W and L are, respectively, the width and length of the channel, $μx(v)$ is the electron mobility along the channel due to electrons in valley ν, and n is the density of carriers. The calculation of n involves a thermal sum of the occupation numbers of each state. This will be carried out here in detail. The resulting expression is essentially identical to $J1,i(v)$ in expression (16). This term is pictured in Fig. 14 (in the green curve). As discussed in detail in Sec. III F, it is continuous, though its slope is not, leading to “kinks.” This alone, then, is not sufficient to explain the observed step-like behavior.

The calculation of n is given by a sum over the density of states, $g(ϵ)$⁠, multiplied by thermal occupancy of each state, given by the Fermi-Dirac distribution

The zero of the energy is taken at the band edge, i.e., $ϵ≡E−EC$ (and offsets of $EC$ are implicit in the arguments of the integrand). To be explicit

and the density of states was found in Sec. III C

Recall that $(v)$ is the valley index, $ϵj(v)$ are the quantized energy offsets within the well, and $Nυ$ is the highest confinement state lower than the height of the well. Inserting (68) and (67) into (66) gives

Performing the integral gives

Comparison confirms that, up to constant factors, this is indeed a sum over the terms $J1,i(v)$ of expression (16).

We show here that at first order, the solution to the Boltzmann Transport Equation (BTE)²⁶ in the Relaxation Time Approximation (RTA) is given by (5). It is desired to express this as an energy integral, taking into consideration the possibility of an anisotropic effective mass tensor

One notes first that

The total energy is given in the parabolic approximation by

To deal with the anisotropy, we absorb the effective masses into the momenta, changing variables

Thus,

and

As before, by symmetry, the latter may be replaced in the integral by $13p~2$⁠. Finally, using

and replacing (76) into the current density given by the relaxation time approximation (7)

Now, by transforming the integral over the momentum into an integral over then energy, via

The contribution of the valley v to the current density (79) becomes then

The valley index v is included in the acceleration term $a(v)$ and the density of states $g(v)(E)$ which depends on the effective mass components and hence on the valley. This justifies the transition from (7) to (10) even in the presence of anisotropy.