In accordance with Moore's Law, metal oxide semiconductor field effect transistors (MOSFETs) have continuously shrunk, leading to the development of gate-all-around (GAA) architecture, which have superior gate controllability than FinFETs1,2,3,4,5,6,7,8,9,10. Hence, quantum mechanical effects such as tunneling and confinement are becoming increasingly crucial, and there has been extensive research on quantum conduction in the GAAFETs11,12,13,14. However, most previous studies have focused only on simplified, uniform-shaped device structures, as illustrated in Fig.1a. The possibility of significant quantum mechanical effects arising from the non-uniform complex geometry has generally been overlooked. In the GAAFETs, the inherent constriction geometry with the bulk source/drain (S/D) and stacked nanosheet (NS) channels necessitates an in-depth investigation into quantum transport through the narrowing junction. Ironically, although “realistic fabrication” of GAAFETs has already become feasible, “numerical experiment” still remains a challenging field due to the excessive turnaround time of large-scale quantum transport simulation.
It has been reported that quantum conduction can be largely influenced by the constriction both in the experimental15,16 and theoretical17,18 studies. As illustrated in Fig.1d, transverse energies occupied by electrons change abruptly at the constriction, resulting in injection barrier and carrier reflection17. This phenomenon can also occur in GAAFET, as it has inherent constriction geometry. For instance, IBM group firstly showed that a dumbbell-shaped device (Fig.1b) exhibits much lower on-current compared to a uniform-shaped device (Fig.1a) based on the quantum transmitting boundary (QTBM) method in ballistic limit19,20,21,22. Semiclassically, the dumbbell-shaped device seems to have a higher on-current than the uniform-shaped device, since it has the bulk S/D with low resistance. However, they demonstrated that carrier reflection at the constriction can cause the unexpected performance degradation of the dumbbell-shaped devices. After that, Purdue group also investigated same phenomena based on the non-equilibrium Green’s function (NEGF) formalism23,24. They found that, contrary to the ballistic cases, once scattering effects are included, the inter/intra-valley interactions enhance the carrier injection through the constriction, and the on-current degradation becomes much less significant.
Although GAAFETs have a geometry that is vulnerable to quantum access resistance (QAR), previous papers already concluded that the on-current degradation due to carrier reflection at the constriction will be not significant in presence of the scattering mechanisms19,20,21,22,23,24,25,26,27. However, they only focused on “ideal” devices with uniform high doping density at the bulk S/D regions, whereas “realistic” doping profiles are non-uniform and can be less than 1019/cm3 at the constriction. In practical designs, the entrance of each stacked NS channel is exposed to different doping values ranging from high to low doping. At the extremely scaled dimension, these low doping cases are inevitable to simultaneously optimize various performance characteristics such as leakage current, variability, and DC/AC performance. For example, in the case of high S/D doping, if only a few dopants happen to diffuse into the NS regions, it can lead to very high leakage current, resulting in device failure28,29. In addition, a bottom parasitic channel with a fin or planar shape, which has poor gate controllability, makes GAAFETs more vulnerable to short channel effects4. Therefore, it is desirable to design the device so that the doping density decays far enough away from the NS channels. Furthermore, even in cases where the low doping is not intended, low active doping cases frequently occur at the constriction, because dopants tend to segregate and inactivate at the oxide interface30,31,32.
In this Article, we reported strong carrier reflection at the constriction in realistic scenarios, which can reduce the on-current by more than two-fold and exacerbate the device variability. In low constriction doping cases, low energy electrons are highly filtered by the constriction-induced injection barrier. It causes a weird phenomenon where the on-current increases with the scattering rate, as it enhances the carrier injection through the constriction. Utilizing the supercomputing resources to ultra-large-scale simulation (36 x 62 nm2), we found that the QAR largely affects the realistic GAAFET optimization. By introducing 2/2.5 dimensional simulation highly parallelized with 19,200 CPU cores, we have significantly reduced the turnaround time to a few days for each bias point. Unique carrier injection characteristics, accompanied with complex current path and the Schottky contact-induced depletion regions, result in peculiar device operations and optimization methods. We benchmarked our results with recently reported experimental data7,8,9,10, and demonstrated that simulations for realistic geometry yields reliable on-current level without any nonphysical parameter tuning.
Low S/D doping devices
We considered the uniform-shaped, dumbbell-shaped, and dog-bone-shaped devices as illustrated in Fig.1a-c. All other parameters, such as device length, width, and doping concentration, remain the same. The only difference for each case is the channel access geometry in the S/D regions. The bulk S/D have a cross-section of 12nm x 12nm, and the NS regions have a cross-section of 4nm x 8nm. The doping profile follows a Gaussian distribution, and has low doping density less than 1019/cm3 around the constriction. It mimics both “intended low doping case” for suppressing short channel effects, and “unintended low doping case” due to dopant segregation and inactivation at the oxide interface. We considered two cases: (100)-devices, which have (100) surface orientation and <110> transport direction, and (110)-devices, which have (110) surface orientation and <110> transport direction.
The drain-current versus gate-voltage characteristics reveal that the dumbbell-shaped device exhibits a noticeably smaller on-current (-55% for the (100) device and -47% for the (110) device) compared to the uniform device, as depicted in Fig.2a-b. It demonstrates that if the NS device has a low doping density in the constriction regions, QAR considerably degrade the DC performance. The low S/D doping devices possess a low Fermi level around the constriction, which causes strong carrier reflection17,33. The lower the Fermi level, the less the longitudinal kinetic energy relative to the barrier (mode energy mismatch), hence the carrier rarely propagates through the constriction. The dog-bone-shaped devices can mitigate the abrupt change of cross-section, and both the (100)-device and the (110)-device exhibit a higher drain current compared to the dumbbell-shaped device. For all scenarios, the on-current degradation is more pronounced in the (100) devices, indicating that the QAR is a more significant problem in the GAAFETs, which typically have (100) surface orientation, than in the FinFETs with (110) surface orientation. In addition to channel orientation, the GAAFETs are also more prone to QAR structurally as they have a narrower channel area and stronger confinement effects than the FinFETs. Furthermore, these low junction doping scenarios predominantly occur in GAAFETs with aggressively scaled gate length and channel thickness.
NS channels have a lower density of state (DOS) than bulk S/D (due to sparse mode energy as shown in Fig.1d) and thus a higher fermi level for same electron density. Therefore, at the narrowing junction, built-in potential occurs due to fermi level mismatch in order to align the fermi level24. The (100)-device has a lower fermi level than the (110)-device, as it has lower subband energy and higher DOS due to heavier confinement mass. Consequently, as shown in Fig.2e, the self-consistent calculation results show that the (100)-device has a lower electron density at the constriction due to the smaller built-in potential at the junction, which implies lower fermi level relative to the lowest mode energy. Since the (100)-device has the lower Fermi level at the constriction, it clearly shows the low energy dominant electron density as shown in Fig.2f. The higher the Fermi level relative to the transverse mode energy, the smaller the carrier reflection, as the longitudinal kinetic energy is greater relative to the subband-mismatch-induced barrier17. Consequently, the (110)-device exhibits a higher carrier injection compared to the (100)-device.
We confirmed the on-current characteristics according to the effective oxide thickness of the spacer (EOTspacer). The effect of the gate field on the bulk S/D regions may vary depending on the EOTspacer. We only tuned the dielectric constant of the spacer layer to control the EOTspacer in the exactly same geometry. As shown in Fig.2c, the smaller the EOTspacer, the larger the gate fringing field and the higher the drain current. It can be seen that the thinner EOTspacer has better gate controllability for the same structure. However, even if the EOTspacer becomes extremely thin as 1.2 nm, it still has a much smaller on-current than the uniform-shaped device (high QAR), and such a design greatly increases the parasitic capacitance and degrade the AC performance. Therefore, reducing the EOTspacer is not the best way to suppress the QAR.
Fig.2d shows the simulation results for various scattering rates. A weird situation occurs where the on-current is smaller in the ballistic case than in the scattering cases in dumbbell-shaped devices. It can be interpreted that carrier reflection is maximized due to the absence of scattering that enhances carrier injection through intra-mode interaction. Recently, GIST group also reported a similar phenomenon25. They developed a geometrical scattering model that describes mode coupling effects at the narrowing junction. It was confirmed that the geometrical scattering also enhances carrier injection through intra-mode interaction, increasing the on-current. The absence of scattering improves mobility (ballisticity), but it also largely increases carrier reflection at the constriction24. Hence, when QAR is dominant, a higher scattering rate results in a higher on-current. However, if the scattering rate becomes excessively high (>1), the effect of mobility reduction becomes predominant and the on-current decreases again. In contrast, in the uniform-shaped devices, as generally expected, the on-current decreases monotonically and more rapidly with increasing scattering rate as shown in Fig.2d. This is a unique characteristic of the GAAFETs that can only be uncovered through the quantum transport simulations, which deviates from the semiclassical prediction, where the subband-splitting/mixing effects cannot be considered.
Simulation for various doping cases
We performed simulations on quite ideal doping scenarios, similar to those in the previous studies19,20,21,22,23,24,25,26,27. Unlike the previous systems shown in Fig.1a-c, a high doping concentration (>1020/cm3) is employed around the constriction as depicted in Fig.3a-b. In this scenario, we noted that the dumbbell-shaped device exhibits nearly identical drain current to the uniform-shaped device, as shown in Fig.3c. Therefore, QAR will be negligible in high S/D doping cases. If the doping density is sufficiently high, the Fermi level is also high, leading to dominance by the subbands with a high energy level. Consequently, carrier reflection is easily suppressed by the inter/intra-valley interaction, and the dumbbell-shaped device exhibits operation close to the classical Ohm’s law.
We also conducted simulations for various S/D doping cases. To examine the effects of the doping density at the constriction, we carried out simulations for various doping depth, assuming a constant Gaussian decay rate. As shown in Fig.3d, uniform-shaped devices are not significantly impacted by the S/D doping profile. However, in the dumbbell-shaped device, the QAR is highly sensitive to the doping density, leading to considerable on-current degradation as the doping density decreases. In both ballistic and scattering cases, the current of the uniform-shaped and dumbbell-shaped devices converge to the nearly same value as the doping increases. Since carrier reflection is maximized in the ballistic case, the on-current is lower than in the scattering case for the low doping density. In addition, the ballistic cases exhibit a more step-functional on-current characteristic depending on the doping density, as there is no energy relaxation of the source-injected current. In both cases, the doping density required to suppress QAR is at least 1020/cm3, and such high doping densities at the constriction are generally not desirable in practical device design. Thus, we can infer that QAR is indeed occurring in many practical situations. Previous papers have demonstrated larger on-current reductions than our results in high junction doping cases: ~60% for ballistic and ~10% for scattering24. We note that this is because they used a much narrower channel thickness of 1.6nm, and the detailed doping profile and scattering model are also quite different. Moreover, the ballistic current in their work used the electrostatic potential obtained from the scattering calculation, not the self-consistent potential. Similarly, we also calculated the ballistic current based on the electrostatic potential obtained from the scattering calculation, as shown in Ext.Fig.1. In this case, consistent with the previous result24, large on-current differences were observed even at the high constriction doping cases. This demonstrates that ballisticity causes significantly higher carrier reflection at the exactly same electrostatic potential due to the absence of the inter/intra valley interaction.
Fig.3e-f display a 1-D cut of the conduction band at the center of devices. In the case of low junction doping, as shown in Fig.3e, the dumbbell-shaped device exhibits a higher relative barrier height and wider effective channel length than the uniform-shaped device, indicating a higher reflection of carriers at the entrance of the NS channel. Strong carrier reflection greatly influences the carrier density in the channel and results in a significantly different conduction band profile. The dumbbell-shaped device has a lower conduction band energy around the channel, implying a lower electron density for the same external bias condition. However, in the case of high junction doping, as depicted in Fig.3f, the conduction bands are nearly identical around the channel, leading to nearly identical on-currents. The difference in the conduction band at the S/D regions arises because the bulk S/D of the dumbbell-shaped device has a lower Fermi level than the NS S/D of the uniform-shaped device. It is confirmed that the built-in potential occurs due to the Fermi level mismatch at the narrowing junction. In the uniform-shaped device, the low doping density of the NS extension regions forms the wide barrier, which can effectively suppress short channel effects with minimal on-current loss34. However, when the constriction is exposed to the low doping density, as in the dumbbell-shaped device, the large on-current reduction occurs due to strong carrier reflection.
Impurity and surface roughness effects
For a more realistic and statistical analysis, we conducted simulations that explicitly include atomistic dopant and surface roughness. We randomly generated discrete dopant (RDF) and surface roughness (SR) based on the continuum device structure depicted in Fig.1a-c. As most dopants are located in the S/D regions far from the channel, the drain current exhibits a small standard deviation of 5.6% in the uniform-shaped devices, as shown in Fig.3g. However, in dumbbell-shaped devices, QAR is substantial and carrier injection is significantly affected by the atomistic dopant location26. Hence, the standard deviation of the drain current is considerably larger at 19.9%. When averaging each sample (represented by the black and red solid lines in Fig.3g), the average-on-current degradation is 60.5%, which is not significantly different from the continuum cases in Fig.2a. Therefore, simulations without RDF and SR can provide a sufficiently accurate description of average device behavior. In the dumbbell-shaped device, if many dopants happen to be located around the constriction, it exhibits a very high on-current (best case, Fig.3h), and if there are almost no dopants around the constriction, it exhibits a very low on-current (worst case, Fig.3i). Fig.3j displays the current spectrum at the source contact for both cases, clearly indicating high injection (low reflection) in the best case. This results in a broad distribution of DC performance, and we found that even if the channel is close to intrinsic, the discrete dopants in the bulk S/D can significantly impact the reliability of GAAFETs.
Stacked NS GAAFET simulation
Finally, we simulated an ultra-large-scale device (36 x 62 nm2) with six-stacked NS channels to account for a more realistic device geometry, as shown in Fig.4a. We assumed that all six NS channels have the same doping density and dimensions. We used the real-space NEGF method due to the complex device shape and non-uniform current path. This requires significantly longer turnaround time than coupled mode space (CMS) method, making three-dimensional simulation practically impossible. Thus, assuming that the width is sufficiently large compared to the channel thickness, we performed a two-dimensional simulation with the assumption of periodicity in the depth direction. As the channel thickness continues to thin and approaches the 2D material in the future technology nodes, this methodology, which can effectively reduce the simulation dimension, is expected to become increasingly attractive.
The large-scale simulation results are compared with the result of simulating each channel individually, as shown in Fig.4d. Here, the simulation result for a uniformly shaped single-channel device were multiplied by a factor of six and compared to the simulation result for full-structure. The simulation, assuming a uniform-shaped device - which is commonly used to focus on the channel area to reduce the turnaround time - greatly overestimates the device performance as shown in Fig.4d. This clearly demonstrates that the parasitic effects in bulk S/D further degrade device performance, which primarily determines the overall device operation. By introducing parasitic resistance terms or tuning the scattering rates in the simplified uniform-shaped device simulation, we may be able to achieve a current level similar to hardware data. However, this is merely a fitting and cannot be utilized for practical device optimization considering the complex geometrical effects.
For more rigorous simulations, SR and RDF can also be included. However, generating SR and RDF in a 2D simulation results in an artificial cylindrical dopant and geometric shapes due to the periodicity in depth direction. Since 3D simulation is practically impossible due to excessive turnaround time, we proposed the 2.5D simulation. We created a thin (2.4 nm in this case) slab in the depth direction and applied a periodic boundary condition to it as shown in Ext.Fig.2a. Unlike 2D simulations, it makes available to describe point charges and a reliable SR configuration. Ext.Fig.2b shows that there is no significant on-current difference between including SR and RDF compared to not including them. Although SR and RDF obviously cause additional scattering, the average difference in on-current would not be significant because scattering (Fig.2d) and discrete dopants (Fig.3g) can also enhance the carrier injection in some cases. Our large-scale calculations with SR/RDF take about a month for obtaining the I-V curve with 19,200 CPU cores of Intel® Xeon® Gold 6342 processor. We note that, with a small CPU (<100), it will take at least few years of calculation time even for a single bias point. If it is not for statistical analysis, new efficient modeling efforts will be desirable to simply include various scattering effects in continuum device configuration35.
Fig.4b shows that the current path sharply bends in an "L" shape, as electrons are injected in a direction perpendicular to the NS channels. These 90° turns in the current path necessitate a sharp momentum change and create additional parasitic resistance. This was also demonstrated by Intel group using the Monte Carlo method36, where they reported that the parasitic resistance due to the L-shaped current path can significantly degrade the on-current. They showed that the decrease in on-current becomes more severe as the scattering rate and doping density decrease, interestingly showing the same trend as the QAR we discovered. Various phenomena in the bulk S/D commonly indicate that sufficient scattering rate and high doping density enhance carrier injection. Ext.Fig.3 shows the impact of the current path on device characteristics. We conducted a simple test for two cases - the contact (open boundary condition) position is on the side or top. The on-current is lower in the top contact case compared to the side contact case in the dog-bone-shaped device, which clearly indicates that the L-shaped current path results in additional parasitic resistance. We also incorporated a Schottky barrier at the S/D contact area. We can create complex contact shapes that are curved as desired, as shown in Fig.4c, and account for contact resistance and depletion regions depending on the Schottky barrier height and doping density. As described above, our simulation includes realistic parasitic effects in the bulk S/D, all of which are complexly coupled to determine device characteristics in the full device structure.
Another unusual phenomenon is that the bottom three channels exhibit a higher current density (illustrated as red arrows) than the top three channels, as shown in Fig.4b. Among them, especially the middle channel, sandwiched between Schottky contacts and other channels, has the lowest current density. Based on the semiclassical concept, it is expected that the further the channel is from the injection boundary condition, the greater the S/D parasitic resistance, and hence the lower the current. However, as the bulk S/D region has aggressively shrunk, electron depletion regions, induced by the Schottky barrier, largely degrade the carrier injection into the NS channels when the channel is too close to the S/D Schottky contact. Ext.Fig.2a shows the spatial electron density in the on-state. It can be observed that a strong depletion region is formed near the Schottky contact, resulting in low electron density and low injection at the entrance of the top three channels. This also impacts the optimization of the contact depth. Fig.4e shows that the on-current increases until the contact depth reaches 30nm, as the contact area increases and the contact resistance decreases. However, if the contact depth becomes too deep, the reduction in carrier injection due to the depletion region becomes more dominant than the increase in contact area, leading to a decrease in the on-current.
We evaluated the effect of the NS extension length (Lext) in Fig.4a. While longer Lext generally reduces parasitic capacitance, it is also known to degrade the on-current by reduced gate fringing field (as shown in Fig.2c) and increased extension resistance37-38. However, we discovered some interesting device behavior that contradicts conventional predictions. Fig.4f shows that the on-current effectively increases up to Lext=4nm and decreases again with further length. As illustrated in Fig.4g, this phenomenon can be interpreted as follows:
(1) In the case of very short Lext, low junction doping causes strong carrier reflections.
(2) Longer Lext suppresses the carrier reflection due to an increase in junction doping, resulting in a significant increase in on-current.
(3) However, when the junction doping exceeds 1020/cm3, the narrowing junction approaches “reflectionless contact”. Therefore, further increasing the Lext does not improve carrier injection, and the on-current decreases again due to increased extension resistance and decreased gate fringing field.
(4) As the Lext becomes much longer, the entrance of the NS region encounters Schottky barrier-induced depletion regions as shown in Ext.Fig.4. This largely degrades the carrier injection, causing the on-current to drop sharply.
As demonstrated above, quantum mechanical analysis can be utilized to determine the optimal Lext. If doping cannot be pushed deep into the channel to suppress short channel effects, junction doping can be increased by forming longer NS extension regions to achieve “reflectionless contact” at the constriction. If we additionally introduce the dog-bone-shaped extension shape as illustrated in Fig.4a, we can further enhance the performance of the device as shown in Fig.4f, as it suppress the carrier reflection and much decreases the parasitic resistance. Fig.4c shows the full I-V curves both for the non-optimized device with dumbbell-shaped NS extension (Lext=2nm, Lcontact=30nm) and optimized device with the dog-bone-shaped NS extension (Lext=4nm, Lcontact=30nm). Considering its immunity to carrier reflection at the accidental low constriction doping cases, as shown in Fig.4a, the dog-bone-shaped NS extension becomes more attractive.
Fig.5a-b shows the drain current according to the number of stacked NS channels. The drain current increases with the channel number, but it does not increase proportionally to the number of channels due to various parasitic effects. As the number of NS channels increases, not only the on-current but also the off-current increases, so the on-current gain for the same off-current becomes much smaller. If we simply increase the number of stacked NS channels without any other device optimization, we cannot achieve a significant on-current gain after 3-stacks. We benchmarked our simulation results against recent experimental results7,8,9,10 and the International Roadmap for Devices and Systems (IRDS)39. As shown in Fig.5c, our simulation results show reliable current values compared with hardware data, without any nonphysical parameter tuning. A non-optimized device with a dumbbell-shaped short NS extension (Lext=2nm) has an on-current much lower than the IRDS HP (the optimistic drive current for high-performance application in IRDS). However, an optimized device with a dog-bone-shaped long NS extension (Lext=4nm), which is designed to maximize the carrier injection, has an on-current close to the IRDS HP, even with a single NS channel. Our results clearly show that increasing NS extension length can effectively boost the device performance. We note that the TSMC device, which has the highest on-current among the references, is the only one that uses an additional inner spacer to lengthen the extension length, which has a dog-bone-like shape8, whereas the others have very short NS extension7,9,10. While the detailed doping density and the specs of the fabricated devices are not known, they are consistent with our results in that the devices with sufficiently long dog-bone-shaped NS extension are necessary for obtaining superior device performance.