1.
Introduction
As the CMOS technology continues scaling down to sub-10 nm regime, the planar devices are gradually being replaced by non-planar devices[1, 2]. The introduction of a third dimension makes the gate interface orientation and transport orientation have more choices and the influence of crystal orientations more complex[3–5]. The flexible vertical gate interface makes the impact of crystal orientations on surface scattering more obvious as well. In nanoscale devices, short channel effect is more serious and quantum mechanical effects and non-equilibrium transport effects are more obvious[6]. Quasi-ballistic transport, in which carriers suffer only one or a few times scattering events in the channel, has been recognized in nanoscale devices[7, 8]. Traditional drift–diffusion model and velocity saturation model are no longer satisfied, and more accurate models and methods are required. Many works of experiments, simulation, and modeling have been done to study in depth the ballistic and quasi-ballistic transport[9–13].
The 3-D full band Monte Carlo method solves the Boltzmann equation statistically and would apply to simulation of quasi-ballistic transport in nanoscale devices. The transport characteristics of double gate MOSFETs (DG-MOSFET) with (100) and (110) surfaces in quasi-ballistic region are investigated using the 3-D Monte Carlo method. The anisotropic surface scattering mechanism is also investigated.
2.
Simulation method and device structure
An anisotropic 3-dimensional Monte Carlo simulator is used to study the transport characteristics of DG-MOSFETs with different surface orientations, which is based upon our previous full-band Monte Carlo simulator[14] taking into account the major scattering mechanisms including phonon, surface roughness, impact ionization, and ionized impurity scattering. The full band structure of silicon is calculated by the non-local empirical pseudo-potential approach. Four conduction bands and three valence bands are considered. Effective potential method is used for quantum correction. Different from the drift–diffusion method, the Monte Carlo method solves the Boltzmann equation statistically and could describe the non-equilibrium transport in the quasi-ballistic region. A rotation matrix is generated at the beginning based on the surface orientation and the channel orientation, which set up the relations between the device coordinate system (DCS) and the crystal coordinate system (CCS)[15]. The particle distribution in momentum space (first Brillouin zone) with different channel orientations are plotted in Fig. 1. Driven by <100>- and <110>-oriented electric field, the electrons are distributed anisotropically in the Brillouin zone. A silicon resistor with 1 μm length and 5 × 10 16 cm?3 n-type doping is simulated using the anisotropic 3D Monte Carlo simulator for verification. As shown in Fig. 2, the calculated velocity-electric field relations fit well to the experimental data[16] with temperature 160 and 300 K respectively.
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Figure2.
(Color online) Electron drift velocity as function of electric field at 160 and 300 K. Circles and squares refer to experimental data with the field parallel to (111) and (100) directions, respectively[16]. Solid and dashed lines indicate the calculated results using the anisotropic 3D Monte Carlo simulator.
The surface scattering model[17] consists of the surface roughness scattering (Eq. (1)) and surface phonons scattering (Eqs. (2)–(6)), which can be represented by an additional scattering rate in Monte Carlo simulation. Here Δ and L are the r.m.s. height and the correlation length of the silicon/oxide interface roughness. The Ξph is deformation potential of surface phonons. The γ and δ are the classical and quantum weight factors. The scattering rates depend on the effective field (Eeff) in both scattering. The r.m.s. height (Δ) and the correlation length (L) are considered in the surface roughness scattering, and the temperature and the local carrier screening density (ns) are included in surface phonons scattering. The values of parameters in the surface scattering model are listed in Table 1, which are referred to Ref. [17]. The values of parameters with different crystal orientations are the same.
| Parameter | Value |
| L (nm) | 2.0 |
| Δ (nm) | 0.1126755 |
| Ξph (eV) | 12.0 |
| γ | 1.5 |
| δ | 2.857703 |
Table1.
Parameters in surface scattering model[17].
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| Parameter | Value |
| L (nm) | 2.0 |
| Δ (nm) | 0.1126755 |
| Ξph (eV) | 12.0 |
| γ | 1.5 |
| δ | 2.857703 |
$${S_{{ m{sr}}}} = 2pi frac{{m_{ m{e}}^*{e^2}}}{{{hbar ^3}}}{left( {frac{{{varepsilon _{{ m{si}}}}}}{{{varepsilon _{{ m{ox}}}} + {varepsilon _{{ m{si}}}}}}} ight)^2}{Delta ^2}{L^2}E_{{ m{eff}}}^2,$$  | (1) |
$${S_{{ m{sac}}}}left( {{E_{{ m{eff}}}},{n_{ m{s}}},T} ight) = frac{{m_{ m{e}}^*}}{{ ho c_{ m{s}}^2}}frac{{{k_{ m{B}}}T}}{{{hbar ^3}}}{{varXi }}_{{ m{ph}}}^2frac{1}{{Wleft( {{E_{{ m{eff}}}},{n_{ m{s}}},T} ight)}} + frac{e}{{m_{ m{e}}^*}}frac{1}{{{K_{{ m{pha}}}}T_{ m{n}}^{{k_{ m{T}}}}}},$$  | (2) |
$${{W}}left( {{E_{{ m{eff}}}},{n_{ m{s}}},T} ight) = frac{{gamma {W_{{ m{cl}}}}left( {{E_{{ m{eff}}}},T} ight) + delta {W_{ m{q}}}left( {{E_{{ m{eff}}}}} ight)}}{{{p_{ m{f}}}left( {{n_{ m{s}}},T} ight)}},$$  | (3) |
$${W_{{ m{cl}}}}left( {{E_{{ m{eff}}}},T} ight) = T_{ m{n}}^{{ m{nbmod}}} frac{{{k_{ m{B}}}T}}{e} E_{{ m{eff}}}^{ - 1},$$  | (4) |
$${W_{ m{q}}}left( {{E_{{ m{eff}}}}} ight) = {left( {frac{{12em_{ m{e}}^*}}{hbar } {E_{{ m{eff}}}}} ight)^{ - 1/3}},$$  | (5) |
$${p_{ m{f}}}left( {{n_{ m{s}}},T} ight) = {p_{{ m{fT}}}} {T_{ m{n}}} + {p_{{ m{fN}}}} {left( {frac{{{n_{ m{s}}}}}{{{n_0}}}} ight)^{{n_{{ m{pf}}}}}} frac{1}{{{T_{ m{n}}}}}.$$  | (6) |
Fig. 3 illustrates the schematic view of the DG-MOSFET structure simulated in this work. Table 2 lists the parameters used to refer to the typical sub-20 nm node devices. The work function of gate material is adjusted to meet the conventional off-state current, which is 100 nA/μm. Typical (100)/<001> and (110)/<1–10> crystal orientation are chosen to study the anisotropic characteristics.
| Parameter | Value |
| Channel length (nm) | 16 |
| Silicon film thickness (nm) | 8 |
| Gate width (nm) | 40 |
| Equivalent oxide thickness (nm) | 1 |
| Channel doping (cm?3) | 1016 |
| Drain & source doping (cm ?3) | 1020 |
Table2.
Device parameters.
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| Parameter | Value |
| Channel length (nm) | 16 |
| Silicon film thickness (nm) | 8 |
| Gate width (nm) | 40 |
| Equivalent oxide thickness (nm) | 1 |
| Channel doping (cm?3) | 1016 |
| Drain & source doping (cm ?3) | 1020 |
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Figure3.
(Color online) The schematic view of DG-MOSFET structure.
3.
Results and discussion
3.1
The impact of crystal orientation on electrostatic and transport characteristics
Fig. 4 shows the I–V curves of both (100) and (110) cases. From the figure it can be seen that the (100) on-state current is significantly larger than that of (110), which is consistent with the situation of long channel devices. This difference is mainly due to the anisotropy in the band structure. Subjected to electric fields in different directions, the filling states of carriers in band structure are different (Fig. 1). Therefore, the energy and scattering probability of the carriers are different and lead to the anisotropy in device properties.
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Figure1.
(Color online) Electron distribution in momentum space (2-D cut in x–z & x–y plane) at ~100 kV/cm electric field along the (a) & (b) <100> and (c) & (d) <110> direction at 300 K. Red is high electron density and blue is low electron density.
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Figure4.
(Color online) Id–Vd characteristics for (100) and (110) surface devices with Vg varying from 0.6 to 1.2 V.
In the quasi-ballistic theory, the current is mainly affected by two aspects, the injection velocity and charge density at the top of potential barrier in the virtual source. The electric potential distributions with (100) and (110) directions are shown in Fig. 5, and the virtual source is marked near the source region in the channel. The electron density distributions in the channel with different gate voltages and different drain voltages are plotted in Fig. 6. The electron density significantly increases with the gate voltage increasing, while the drain voltage has little influence on the electron density above the threshold state, and only the carriers near the drain region deplete with the drain voltage increasing. The difference in electron density between the cases with (100) and (110) directions is small and mainly occurs in the center of the channel. Fig. 7 shows the electron longitudinal velocity distributions with different gate voltages and drain voltages. The difference in the cases with (100) and (110) directions is marked. The velocity with (100) direction is much larger than that with (110) direction. The influence of gate voltage on (100) direction is larger, while the drain voltage has same notable impact on the (100) case and the (110) case. With the drain voltage increasing, the longitudinal electric field in the channel increases and the carrier velocity increases as well.
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Figure5.
(Color online) Electrostatic potential along the channel direction for (100) and (110) surface with Vgs = 1 V and Vds = 1 V.
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Figure6.
(Color online) Electron density along the channel direction for (100) and (110) surface with (a) different gate voltages and (b) different drain voltages.
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Figure7.
(Color online) Electron velocity along the channel direction for (100) and (110) surface with (a) different gate voltages and (b) different drain voltages.
Therefore, the difference in crystal orientation is represented in velocity distribution. The velocity with (100) direction is much larger than that with (110) direction. The (100) case is more sensitive to the gate voltage than the (110) case and the drain voltage has large impact on the longitudinal velocity distribution with both directions.
3.2
The impact of surface scattering on different orientations
The influence of surface scattering on transport features with different crystal orientations has been simulated. The on-state current with and without surface scattering is shown in Fig. 8. When considering surface scattering, the on-state current for devices along both (100) and (110) direction reduced, and the degradation ratio (= (Ion(w/o ss)–Ion(with ss)/ Ion(w/o ss))) is plotted at the bottom of the figure. With the gate voltage increasing, the ratio increases, and the increment of the ratio with (100) direction is larger than that with (110) direction.
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Figure8.
(Color online) The on-state current and degradation with Vg varies from 0.4 to 1.2 V with different surface orientation and with and without surface scattering.
The electron longitudinal velocity at the top of the potential barrier is called the inject velocity which contributes to the quasi-ballistic transport characteristics. The inject velocity under different conditions is plotted in Fig. 9. Without considering the surface scattering, the inject velocity with (100) and (110) direction increases with the gate voltage increasing, and the one with (110) direction is saturated at high gate voltage. After considering the surface scattering, the inject velocity with both (100) and (110) direction is saturated at high gate voltage. With the gate voltage increasing, the electron density in the channel increases, and the carriers occupy the higher energy level and thereby achieve higher velocity. Therefore, the inject velocity increases. With further increase of gate voltage, the carrier–carrier scattering and surface scattering events increase, the inject velocity is saturated or even decreasing. At low drain voltage, the inject velocity increases with drain voltage increasing, and then saturates. The surface scattering leads to the inject velocity degradation, and the degradation ratio decreases with the drain voltage increasing as plotted at the bottom of Fig. 9. According to the quasi-ballistic theory, the carrier transports in the channel include the positive flux and negative flux. While the drain voltage increased, the positive flux is much larger than the negative flux, and then the saturation occurs. Due to the different channel surface orientations, the influence of surface scattering on different crystal orientations is different. The degradation of on-state current and inject velocity with (100) direction is larger than that with (110) direction with considering the surface scattering.
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Figure9.
(Color online) The inject velocity and degradation with different surface orientation and with and without surface scattering.
The longitudinal velocity distribution is plotted in Fig. 10. Under different crystal orientations and with and without considering the surface scattering, the electron density is almost constant, while there is a significant difference in longitudinal velocity. The velocity with (100) direction is larger than that with (110) direction, but the influence of surface scattering on velocity with (100) direction is larger as well.
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Figure10.
(Color online) Cross section of electron velocity at the potential barrier with different surface orientation and surface scattering.
In conclusion, the impact of crystal orientation and surface scattering on transport features mainly reflects in the carrier velocity distribution. The electron transport features with (100) direction are greater than that with (110) direction, but are more likely to be affected by the surface scattering.
4.
Conclusion
The influences of crystal orientation and surface scattering on transport characteristics in quasi-ballistic region are investigated using 3D full-band Monte Carlo method. The (100) case is affected by gate voltage more than the (110) case, while the (100) case is affected by drain voltage the same as the (110) case. The impact of crystal orientation and surface scattering on transport features mainly reflects in the carrier velocity distribution. The electron transport features with (100) direction are greater than those with (110) direction, but are more likely to be affected by the surface scattering.
