modeling and simulation of nanowire mosfets …i would like to appreciate to internal committee in...
TRANSCRIPT
DOCTORAL DISSERTATION
Modeling and Simulation of Nanowire MOSFETs
by
Yeonghun Lee
Submitted to the Department of Electronics and Applied Physics
in partial fulfillment of the requirement for the degree of
Doctor of Philosophy
at
Tokyo Institute of Technology
March 2012
Advisors: Professor Iwai Hiroshi and Professor Kenji Natori
DEPARTMENT OF ELECTRONICS AND APPLIED PHYSICS INTERDISCIPLINARY GRADUATE SCHOOL OF SCIENCE AND ENGINEERING
TOKYO INSTITUTE OF TECHNOLOGY
i
Acknowledgement
First of all, I would like to greatly appreciate to my two advisors—Prof. Hiroshi Iwai
and Prof. Kenji Natori—for giving me valuable advice and guidance to pursue the
Ph.D. degree. I am deeply grateful to Prof. Kuniyuki Kakushima for many stimulating
discussions and for all of his help. I would also like to express my gratitude to Prof.
Kenji Shiraishi at the University of Tsukuba for his continuous support since I was an
undergraduate student. I would like to express my gratitude to Dr. Toyohiro Chikyow
for support during my internship at the National Institute for Materials Science
(NIMS). I would like to appreciate to the Prof. Takeo Hattori, Prof. Nobuyuki Sugii,
Prof. Akira Nishiyama, Prof. Yoshinori Kataoka, Prof. Kazuo Tsutsui, and Prof. Parhat
Ahmet for the helpful advice at the laboratory meetings. I would like to express my
gratitude to Dr. Daniel Berrar for guiding technical writing and proofreading my
journal papers.
I would like to appreciate to internal committee in charge of my dissertation
defense: Prof. Hiroshi Iwai, Prof. Kenji Natori, Prof. Kazuo Tsutsui, Prof. Masahiro
Wanatabe, Prof. Shun-ichiro Ohmi, and Prof. Kuniyuki Kakushima. I would also like
to thank Prof. Ming Liu (Institute of Microelectronics Chinese Academy of Sciences),
Prof. Hei Wong (City University of Hong Kong), Prof. Zhenan Tang (Dalian
University of Technology), Prof. Zhenchao Dong (University of Science and
Technology of China), Prof. Junyong Kang (Xiamen University), Prof. Weijie Song
(Ningbo Institute of Material Technology and Engineering), Prof. Chandan Sarkar
(Jadavpur University), Prof. Baishan Shadeke (Xinjiang University), Prof. Wang Yang
(Lanzhou Jiaotong university), and Prof. Kenji Shiraishi (University of Tsukuba) for
giving valuable comments on the manuscripts at the final examination of this
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dissertation.
I would like to express my gratitude to Prof. Giorgio Baccarani for giving me a
chance to study at the University of Bologna, Italy. I would also like to appreciate to
Prof. Elena Gnani for useful advice during collaborative work, which is associated
with chapter 7. I also appreciate to Dr. Roberto Grassi and other colleagues for all of
their help in Bologna.
I would like to thank my laboratory colleagues for discussion about general
knowledge of semiconductor devices and for all of their help in Japan; especially, Dr.
Jaeyeol Song, Dr. Kiichi Tachi, Dr. Soshi Sato, and Dr. Takamasa Kawanago. I
appreciate to Mr. Darius Zade for proofreading the dissertation, and I thank to Mr.
Michihiro Hosoda for his help to perform a part of mobility calculations. I also thank
my old laboratory colleagues at the University of Tsukuba for their continuous
support. I would like to express my gratitude to laboratory secretaries—Ms. Akiko
Matsumoto and Ms. Masako Nishizawa—for all of their help. Finally, I would like to
give special thanks to my family.
My doctoral study was supported by New Energy and Industrial Technology
Development Organization (NEDO) and Grant-in-Aid for Fellows of Japan Society
for the Promotion of Science (JSPS). A part of this work was also supported by Global
COE Program “Photonics Integration-Core Electronics” and Innovative Platform for
Education and Research. The first-principles calculation was performed by Tokyo
Ab-initio Program Package (TAPP), which has been developed by a consortium
initiated at the University of Tokyo; I would like to thank the developers.
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Abstract
Downscaling of the conventional planar MOSFETs has required continuing efforts
to suppress short-channel effects (SCEs). Nanowire (NW) MOSFETs have been
the focus of intensive research because of their superior SCE immunity. In this
study, we interpret device physics of ultra-scaled NW MOSFETs through
comprehensive modeling of gate capacitance and quasi-ballistic transport, and we
investigate what parameters control performance through numerical simulations.
To represent electronic structures, we use quantum mechanical approaches based
on the effective mass approximation and the first-principles calculation. To take
into account carrier transport, we use a semiclassical approach based on a direct
solution of the Boltzmann transport equation.
In chapter 3, we develop a comprehensive gate capacitance model to distinguish
the contributions of the quantum effects with respect to the finite inversion layer
centroid and the finite density of states. The finite inversion layer centroid caused
the positive effect on the increase in the total gate capacitance for small NW
MOSFETs. In chapter 4, with the developed gate capacitance model, we
investigate size-dependent performance of Si and InAs NW MOSFETs based on
the semiclassical ballistic transport model. In Si NW MOSFETs, performance in
terms of the intrinsic delay time depended on the injection velocity, which
generally increased with shrinking diameter. On the other hand, in InAs NW
MOSFET, the performance was insensitive to the injection velocity. We also found
out that the desirable diameter simultaneously giving high performance and low
power dissipation were around 5 nm for Si NWs and 10 nm for InAs NWs. In
chapter 5, we investigate the band structure effect on electrical characteristics in
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substantially small Si NWs based on the first-principles calculation. Since
effective mass of cylindrical Si NWs widely fluctuated with curvature variation,
we should carefully take into account the band structure effect. By adopting
rectangular Si NWs, we could suppress the fluctuation of the effective mass,
where the nonparabolic EMA was a useful metric to investigate size-dependent
electrical characteristics. In chapter 6, we investigate size and corner effects on the
phonon-scattering-limited mobility of rectangular Si NW MOSFETs based on
spatially resolved mobility analysis. As a result, the size effect without considering
the strain effect does not lead to a drastic mobility increase in experimental results
because the electronic structure hardly changes. We also found out that the
mobility drastically modulated in width smaller than 6 nm. In chapter 7, finally,
we develop a semi-analytical model of the quasi-ballistic transport in an
ultra-short channel. For the modeling, we used a combination of one-flux
scattering matrices and a semi-analytical solution of the Boltzmann transport
equation. The developed quasi-ballistic transport model was in quantitatively good
agreement with a numerical simulation. Here, one-dimensional source and drain
lengths were important parameters to adjust the drain current. In chapter 8, we
summarize conclusions of each chapter and describe future works. From the
developed models and simulation results, we could interpret device physics and
find out what parameters control performance in ultra-scaled NW MOSFETs.
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Contents
Acknowledgement .........................................................................................................i Abstract....................................................................................................................... iii Contents ........................................................................................................................v List of Tables............................................................................................................. viii List of Figures..............................................................................................................ix Chapter 1 Introduction.............................................................................................1 1.1 Nanowire MOSFET..............................................................................................1 1.2 Issues on Nanowire MOSFETs.............................................................................2
1.2.1 Gate Capacitance Modeling........................................................................2 1.2.2 Size-Dependent Performance......................................................................3 1.2.3 Band Structure Effect..................................................................................4 1.2.4 Size and corner effects on electron mobility in rectangular cross sections .4 1.2.5 Quasi-Ballistic Transport ............................................................................5
1.3 Modeling and Simulation of Nanowire MOSFETs ..............................................6 1.4 Dissertation Outline ..............................................................................................7 _Toc319987732 Chapter 2 Methodologies .........................................................................................8 2.1 First-Principles Calculation ..................................................................................8
2.1.1 Density Functional Theory .........................................................................8 2.1.2 Local Density Approximation...................................................................11
2.2 Gate Capacitance ................................................................................................12 2.2.1 Self-Consistent Solution of the Schrödinger and Poisson Equations .......12 2.2.2 Inversion Layer Capacitance.....................................................................15
2.3 Boltzmann Transport Equation ...........................................................................17 2.3.1 One-Dimensional Multisubband Boltzmann Transport Equation.............18 2.3.2 Low-Field Mobility Calculation by the Kubo-Greenwood Formula........20 2.3.3 Ballistic Boltzmann Transport Equation...................................................21 2.3.4 Device Simulation Based on the Deterministic Numerical Solution of the One-Dimensional Multisubband Boltzmann Transport Equation.........................24
Chapter 3 Gate Capacitance Modeling of Nanowire MOSFETs ........................26 3.1 Introduction.........................................................................................................26 3.2 Gate Capacitance Modeling................................................................................28 3.3 Discussion ...........................................................................................................32
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3.4 Conclusions.........................................................................................................37 Chapter 4 Size-Dependent Performance of Nanowire MOSFETs .....................38 4.1 Introduction.........................................................................................................38 4.2 Simulation Methods ............................................................................................39
4.2.1 Effective Mass Approximation with a Nonparabolic Correction..............40 4.2.2 Top-of-the-Barrier Ballistic Transport Model...........................................42
4.3 Results and Discussion .......................................................................................43 4.4 Conclusions.........................................................................................................53 Chapter 5 Band Structure Effect on Electrical Characteristics of Silicon Nanowire MOSFETs with the First-Principles Calculation ..................................54 5.1 Introduction.........................................................................................................54 5.2 Simulation Methods ............................................................................................56
5.2.1 First-principles band structure calculation................................................56 5.2.2 Compact model of ballistic nanowire MOSFETs .....................................57
5.3 Results and Discussion .......................................................................................60 5.4 Conclusions.........................................................................................................70 _Toc319987767 Chapter 6 Size and Corner Effects on Electron Mobility of Rectangular Silicon Nanowire MOSFETs...................................................................................................71 6.1 Introduction.........................................................................................................71 6.2 Simulation Methods ............................................................................................72 6.3 Results and Discussion .......................................................................................75
6.3.1 Size and Orientation Effects .....................................................................76 6.3.2 Corner Effect.............................................................................................81
6.4 Conclusions.........................................................................................................91 Chapter 7 Modeling of Quasi-Ballistic Transport in Nanowire MOSFETs.......92 7.1 Introduction.........................................................................................................92
7.1.1 Quasi-Ballistic Transport ..........................................................................92 7.1.2 Natori’s Model for Quasi-Balistic Transport ............................................94
7.2 Modeling of Quasi-Ballisic Transport ................................................................97 7.2.1 Expression by one-flux scattering matrices ...................................................1 7.2.2 Solution of the Boltzmann Transport Equation ......................................100
7.2.2.1 Barrier and Elastic Zones..............................................................101 7.2.2.2 Relaxation Zone ............................................................................103 7.2.2.3 Source and Drain Zones................................................................105
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7.3 Validation by Numerical Simulation.................................................................109 7.4 Discussion .........................................................................................................114 7.5 Conclusions.......................................................................................................120 _Toc319987789 Chapter 8 Conclusions..........................................................................................121 8.1 Summary of Conclusions..................................................................................121 8.2 Future Work ......................................................................................................122 References .................................................................................................................124 Appendix A: Self-Consistent Calculation of the Top-of-the-Barrier Semiclassical Ballistic Transport Model ..........................................................................................133 Appendix B: Solution of the Poisson Equation in a Cylindrical Coordinate System 135 Appendix C: Gate Capacitance Modeling of Planar and Double-Gate MOSFETs ...137 Appendix D: Carrier Degeneracy and Injection Velocity ..........................................139
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List of Tables
Table 1.1: Calssification of transport models ................................................................1
Table 3.1: List of main symbols related to the gate capacitance....................................1
Table 5.1: Convergence of the cutoff energy of 12.25 Ry .............................................1
Table 6.1: Parameters for intravalley acoustic phonon scattering .................................1 Table 6.2: Parameters for intervalley phonon scattering ...............................................1 Table 6.3: Effective mass tensor ....................................................................................1 Table 6.4: Mobility for each subband group..................................................................1
ix
List of Figures
Figure 1.1: Required physical gate length for high performance logic technology in the ITRS 2010 update. ......................................................................................1
Figure 2.1: Flowchart of the self-consistent gate capacitance calculation with the given charge distribution function. If the contribution of the longitudinal electric field is neglected, it could be reduced to a 2D problem...............................1
Figure 2.2: Flowchart of a device simulation based on the deterministic numerical solution of the 1D MSBTE. .........................................................................1
Figure 3.1: Concept of the two quantum effects, which contribute to the Cinv. Because of the quantum effects, we need additional voltage drop to charge electrons.......................................................................................................................1
Figure 3.2: Capacitance-voltage characteristics of the (a) 3-nm-diameter [100] Si NW, (b) 12-nm-diameter [100] Si NW, (c) 5-nm-diameter InAs NW, and (d) 30-nm-diameter InAs NW. In the 5-nm-diameter InAs NW MOSFET, Ce and Ccentroid were almost the same and hardly varied, and the values were 9.4 to 9.5 µF/cm2. Here, Voff is the off-state gate voltage shown in Fig. 4(b). (Cox = 3.45 µF/cm2, Vd = 0.5 V)...................................................................1
Figure 3.3: Effective gate capacitance as a function of diameter. (Cox = 3.45 µF/cm2, Vd = 0.5 V). ..................................................................................................1
Figure 3.4: Diameter-dependent effective capacitances in (a) [100] Si NWs. and (b) InAs NWs. Here, Ccentroid is t:he value at on-state. (Cox = 3.45 µF/cm2, Vdd = 0.5 V). .................................................................................................1
Figure 3.5: (a) Inversion layer centroid as a function of diameter. (b) Effective oxide thickness devided by the oxide thickness versus diameter. (Cox = 3.45 µF/cm2, Vdd = 0.5 V). ................................................................1
Figure 3.6: Radial electron density with various diameter at on-state in (a) [100] Si and (b) InAs NWs. (Cox = 3.45 µF/cm2, Vdd = 0.5 V).........................................1
Figure 4.1: Schematic of a cylindrical NW MOSFET. ..................................................1 Figure 4.2: Nonparabolic transport effective mass at the minimum of the lowest
subband in (a) [100] Si and (b) InAs NWs. Solid lines indicate the results from the EMA with the nonparabolic correction, and open symbols indicate the results from TB model for [100] Si NWs in [29] and for [111] InAs NWs
x
in [102]. The nonparabolic correction could give a reasonable E-k dispersion of the lowest subband. (Cox = 2.45 µF/cm2, Vd = 0 V, Vg = 0 V).......................................................................................................................1
Figure 4.3: Concept of the top-of-the-barrier semicalssical ballistic transport model. Carrier distribution with positive velocity (dE / dk > 0) is described by source Fermi-Dirac distribution, and carrier distribution with negative velocity (dE / dk < 0) is described by drain Fermi-Dirac distribution. ........1
Figure 4.4: (a) Adjusted tox is required for fixing Cox because Cox with fixed tox depends on diameter in the cylindrical capacitor. (b) Voff as a function of diameter. The Voff denotes the Vg when drain-current per unit wire periphery is 100 nA/µm. (c) Id-Vg characteristics. Threshold voltages are set to (Voff + 0.15 V). (Cox = 3.45 µF/cm2, Vd = 0.5 V). ........................................1
Figure 4.5: (a) On-current as a function of diameter. (b) Effective total gate capacitance at on-state as a function of diameter, where the overdrive gate voltage is 0.35 V. (c) Injection velocity at on-state as a function of diameter. (Cox = 3.45 µF/cm2, Vdd = 0.5 V). ................................................................1
Figure 4.6: (a) Diameter-dependent CG / CDOS, which is a metric of carrier degeneracy for the lowest subband. (b) Diameter-dependent Eµ – Efs (Eµ’ – Efs), which gives the degree of the carrier degeneracy for each subband. The E1’ is the minimum of the lowest primed subband. Carrier degeneracy is maximized when the second lowest subband minimum is around Efs. (Cox = 3.45 µF/cm2, Vdd = 0.5 V). ................................................................1
Figure 4.7: (a) Intrinsic gate delay as a function of diameter. (b) Power delay product as a function of diameter. The power delay product is normalized by wire periphery. (Cox = 3.45 µF/cm2, Vdd = 0.5 V, Lg = 14 nm).............................1
Figure 5.1: Convergence check at the cutoff energy of 12.25 Ry for (a) total energy, (b) bulk modulus, (c) lattice constant, (d) band gap, and (e) longitudinal effective mass along Γ–X in bulk Si............................................................1
Figure 5.2: (a) Id-Vd and (b) Id-Vg characteristics of a Si NW MOSFET from the compact model. The compact gives reasonable I-V characteristics. (d = 2.69 nm, Cox = 3.45 µF/cm2). ...............................................................1
Figure 5.3: Cross-sectional atomic array of the 2-nm-diameter [100] Si NW with circular cross section. The curvature variation is no longer negligible in substantially small diameter. Large atoms are silicon and circumferential small atoms are hydrogen. ...........................................................................1
Figure 5.4: Effective mass of the lowest unprimed subband as a function diameter. In diameters smaller than 2 nm, we adopted every possible circular Si NWs
xi
with the center of a Si atom. The result from the nonparabolic EMA has been calculated in the previous chapter. ......................................................1
Figure 5.5: Cross-sectional atomic arrays of square Si NWs with [100]-directed channel and (110)-oriented surface. Large atoms are silicon and circumferential small atoms are hydrogen...................................................1
Figure 5.6: Calculated band structures of Si NWs with various widths. In small width, valley splitting of four-fold degenerate unprimed subabnds is observed. ...1
Figure 5.7: Band gap as a function of width. Strong quantum confinement broadens band gap. Dotted line is the bulk band gap from the first-principles calculation. Although the result from the DFT with the LDA does not give valid value of the band gap, tendency could be a good guide. ....................1
Figure 5.8: Calculated effective masses of the lowest unprimed subband and the lowest primed subband. The bulk effective masses from the first-principles calculation are overestimated from the known values, which are 0.19 and 0.916 m0 with transverse and longitudinal effective masses in bulk Si, respectively. .................................................................................................1
Figure 5.9: Width dependences of (a) drain current, (b) effective gate capacitance, and (c) injection velocity. Results of the nonparabolic EMA correspond to the cylindrical Si NW MOSFETs with the same periphery, where the electrostatic capacitance per unit wire surface holds. (Cox = 3.45 µF/cm2, Vg – Vt = 0.35 V, Vd = 0.5 V)........................................................................1
Figure 5.10: Width dependences of subband minima of the lowest four unprimed subband and the lowest two primed subband based on the source Fermi level. (Cox = 3.45 µF/cm2, Vg – Vt = 0.35 V, Vd = 0.5 V). ............................1
Figure 6.1: Schematic model of a rectangular NW MOSFET. ......................................1 Figure 6.2: Schematic models of adopted Si NWs with various directions and
orientations. [100] and [110] are NW directions, and (100) and (110) are orientations of wafer. ...................................................................................1
Figure 6.3: Width dependence of cross-sectional local lectron density in (a) [100]/(100) Si NWs, (b) [110]/(100) Si NWs, and (c) [110]/(110) Si NWs. (h = 10 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2)...............................................................1
Figure 6.4: Width dependence of phonon-scattering-limited mobility in [100]/(100), [110]/(100), and [110]/(110) Si NWs. (h = 10 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).................................................................................1
Figure 6.5: Width dependence of cross-sectional specially resolved mobility in (a) [100]/(100) Si NWs, (b) [110]/(100) Si NWs, and (c) [110]/(110) Si NWs. We can distinguish orientation and corner effects based on the specially
xii
resolved mobility analysis. Corner mobility is always lower than the (100)-surface mobility (h = 10 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2). .....1
Figure 6.6: Cross-sectional local electron density in (a) 4- and (b) 12-nm-width SiNW MOSFETs. The corner electron density is approximately twice as high as the side electron density in the 12-nm-width Si NW MOSFET. (h = w, tox = 1 nm, Ninv = 0.8 × 1013 /cm2). ..............................................................1
Figure 6.7: Cross-sectional spatially resolved phonon-scattering-limited mobility in (a) 4- and (b) 12-nm-width SiNW MOSFETs. The corner is lower than the side mobility in the 12-nm-width Si NW MOSFET. (h = w, tox = 1 nm, Ninv = 0.8 × 1013 /cm2)..................................................................................1
Figure 6.8: The number of electrons occupying each subband group at the corner and side. At the corner, the large rate of electrons belongs to the 2nd subband group. (h = w = 12 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2). ........................1
Figure 6.9: Sum of probability densities for (a) the 1st subband group and (b) the 2nd subband group. The most electrons of the 2nd subband group distributes near the corners. (h = w = 12 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2). ................1
Figure 6.10: (a) Density of states, (b) group velocity, and (c) total phonon scattering rate (d) intravalley acoustic phonon scattering rate for subband µ. The horizontal axes are based on the quasi Fermi level, Efn. (h = w = 12 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2)..................................................................................1
Figure 7.1: We describe the kT layer within a schematic potential profile. The critical distance LkT is the distance between the top of the barrier and the position of the potential drop by kBT. The effect of the backscattering beyond the critical distance is neglected. .......................................................................1
Figure 7.2: Concept of Natori’s quasi-ballistic transport model. Contrary to the kT-layer theory, the critical distance is set to the length between the top of the barrier and the position where carriers can emit the optical phonon energy, ħω. He took into account the effect of the backscattering beyond the critical distance. .......................................................................................................1
Figure 7.3: f+(z1,E), f–(z1,E), f+(z2,E), and f–(z2,E) can be described by a one-flux scattering matrix for a slab between z1 and z2. R1 and T1 correspond to the carriers injected from the left side, and R2 and T2 correspond to the carriers injected from the right side. .........................................................................1
Figure 7.4: A device is divided into five zones in the model developed here. The ideal source and drain are located at each end of the device. ...............................1
Figure 7.5: Scattering rate under nondegenerate equilibrium in a cylindrical Si NW MOSFET. Solid line is the result considering elastic acoustic phonon
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scattering only, and dotted line is that considering both elastic acoustic phonon scattering and inelastic optical phonon scattering. The inelastic optical phonon scattering could be neglected below ħω = 63 meV. (d = 3 nm, tox = 1 nm, Vg = 0.6 V, Vd = 0 V). ...............................................1
Figure 7.6: Backscattering coefficient, R(ε), for (a) saturation region with Ld = 10 nm, (b) subthreshold region with Ld = 10 nm, and (c) saturation region with long drain, Ld = 100 nm. Open symbols are results from the numerical simulation, solid lines are those from this model, and dotted lines are those considering elastic acoustic phonon scattering only. The modeling of the drain zone is available even in substantially long drain. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Lg = 30 nm, Nd
s = Ndd = 2 × 1020 /cm3). ......................................................1
Figure 7.7: Distribution functions at the top of the barrier in (a) saturation region with short source (Ls = 10 nm) and (b) subthreshold region with long source (Ls = 100 nm). Open symbols are the results from the numerical simulation, solid lines are those form this model, and dotted lines are Fermi-Dirac distribution function within the ideal source. The modeling of the source zone is available even in substantially long source. (d = 3 nm, tox = 1 nm, Lg = 30 nm, Ld = 10 nm, Vd = 0.3 V, Nd
s = Ndd = 2 × 1020 /cm3). ................1
Figure 7.8: Id-Vd characteristics from the numerical simulation and the drain current from this model. This model is in quantitatively good agreement with the numerical simulation. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Lg = 30 nm, Ld = 10 nm, Nd
s = Ndd = 2 × 1020 /cm3). ......................................................1
Figure 7.9: Adopted potential profile, U(z), for discussion of quasi-ballistic transport. There is no barrier zone for simplicity. Us and Ud are determined by source and drain donor impurity densities, Nd
s and Ndd, under equilibrium, where
Ud = Us – qVd when Nds = Nd
d. .....................................................................1 Figure 7.10: Ballisticity as functions of (a) gate length with Ld = 1 nm, (b) gate length
with Ld = 20 nm, and (c) drain length with Lg = 1 nm. Although the Ld and Lg of 1 nm are not realistic, it makes sense as eliminating their influences. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Vd = 0.3 V, U(z0) = Efs, Nd
s = Ndd = 2 × 1020 /cm3). ..........................................................................1
Figure 7.11: Drain current as functions of (a) gate length with Ld = 10 nm and (b) drain length with Lg = 10 nm. Close circles are results from this model, open squares are those from Natori’s model, and open triangles are those considering elastic acoustic phonon scattering only. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Vd = 0.3 V, U(z0) = Efs, Nd
s = Ndd = 2 × 1020 /cm3). ..................1
Figure 7.12: Drain current as a function of source length. Close circles are results from this model, , and open squares are those considering elastic acoustic phonon
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scattering only. (d = 3 nm, tox = 1 nm, Lg = 10 nm, Ld = 10 nm, Vd = 0.3 V, U(z0) = Efs, Nd
s = Ndd = 2 × 1020 /cm3). .......................................................1
1
Chapter 1
Introduction
1.1 Nanowire MOSFET
Performance of conventional planar MOSFETs has been improved by downscaling,
which has required continuing efforts to suppress short-channel effects (SCEs), such
as the drain-induced barrier lowering (DIBL) [1], [2], [3], [4]. According to the
International Technology Roadmap for Semiconductors (ITRS) 2010 update, the gate
length will be downscaled as small as 10 nm by 2021 as shown in Figure 1.1 [1].
Nanowire (NW) MOSFETs showed SCE immunity due to the superior gate
controllability with the surrounding gate; thereby, it is a promising candidate with
substantially short channel and without serious SCEs [5], [6], [7]. In the NW
MOSFETs, therefore, large on-current can be achieved without degrading the
electrostatic control of the channel carriers even in ultra-short channel. Recently
fabricated Si NW MOSFETs have shown to effectively suppress SCEs and improve
on-current/off-current ratio [8], [9], [10], [11], [12], [13], [14], [15], [16].
2
1.2 Issues on Nanowire MOSFETs
This subsection briefly describes theoretical issues on ultra-scaled nanowire MOSFET
operation: gate capacitance modeling, size-dependent performance, band structure
effect, size and corner effects on electron mobility, and quasi-ballistic transport.
1.2.1 Gate Capacitance Modeling
One of the most distinguishing features of NW MOSFETs is the subband quantization
in a strongly confining potential, which causes the strong quantum effect on the gate
capacitance. The gate capacitance is usually described by a series connection of the
gate oxide capacitance and the inversion layer capacitance. The inversion layer
capacitance is the capacitance due to two quantum effects: finite inversion layer
centroid and finite density of states (DOS).
In planar MOSFETs, the inversion layer capacitance can be modeled with the
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2005 2010 2015 2020 2025Year of production
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Figure 1.1: Required physical gate length for high performance logic technology in the ITRS 2010update.
3
charge centroid with respect to inversion and depletion charges [17]. In double-gate
(DG) and NW MOSFETs, the inversion layer capacitance cannot be described only
with the charge centroid. Nevertheless, a model adopted in [18], [19] described the
inversion layer capacitance with only the inversion layer centroid. Another model
adopted in [20], [21], [22] described the inversion layer capacitance with a series
connection of two capacitances due to the finite inversion layer centroid and due to
the finite DOS, where the former capacitance was defined as the capacitance except
for the series connection of the latter capacitance; thus, the inversion layer
capacitance due to the finite inversion layer centroid has remained ambiguous.
1.2.2 Size-Dependent Performance
NW MOSFETs shows size-dependent properties accompanied with the volume
inversion [23]:
Total gate capacitance can be increased because the inversion charges approach to
the surface. [18], [19];
DOS can be reduced with subband quantization and can degrade the gate
capacitance through quantum capacitance [24], [25];
The increased total gate capacitance and the reduced DOS can increase carrier
degeneracy and velocity [26], [27].
These properties are general features regardless of channel materials. By considering
those effects on performance, we can reveal the general size-dependent performance.
The ballistic transport regime is a useful metric to estimate performance even in the
quasi-ballistic and diffusive regimes [28], [29]. In the ballistic MOSFETs, the
injection velocity of the carriers from source has been focused on, to estimate the
drain current. The small effective mass increases the injection velocity but decreases
4
the total gate capacitance through quantum effects. How about the relation between
the injection velocity and the total gate capacitance in the size dependence? With the
effect of the effective mass, the size-dependent performance needs to be analyzed
with the features in the previous paragraph.
1.2.3 Band Structure Effect
Derivation of the confined electronic structures by the effective mass approximation
(EMA) is very attractive from computational point of view. However, the EMA just
gives us a rough outline when external potential rapidly varies compared to the period
of atoms or the periodicity is lost by confinement. Since the deviation from the EMA
is inevitable within strongly confining potential [30], atomistic calculations are
necessary for Si NWs. Size-dependent subband structures have been investigated
based on the tight-binding method [31], [32], [33], [34], [35] and the first-principles
calculation [36], [37]. They show that the EMA had to be corrected in extremely
downscaled NWs:
In [100] Si NWs, the transport effective mass increased with shrinking size;
The degenerate valleys split.
Therefore, band structure effects on carrier transport in Si NW MOSFETs have been
investigated with atomistic approaches, e.g., the tight-binding (TB) model [32], [34],
[35], [38] and the first-principles calculation [37].
1.2.4 Size and corner effects on electron mobility in rectangular cross
sections
The low-field mobility has been actively investigated with fabricated Si NW
MOSFETs [8], [10], [12], [15], [16], [39], [40], [41], [42], [43],and computational
5
studies [44], [45], [46], [47], [48], [49], [50]. Sakaki [44] reported that the GaAs NW
shows high mobility at low temperature owing to the suppression of Coulomb
scattering by reduced density of states. In small Si NW MOSFETs, unfortunately, the
benefit of the reduced density of states is eliminated by the increase in the
electron-phonon wave function overlap for the phonon scattering [45]. Nevertheless,
the mobility was enhanced in several fabricated Si NW MOSFETs [8], [10], [39], [40],
[42]. Koo et al. [39] and Sekaric et al. [42] suggested that the mobility enhancement
in small Si NW MOSFETs would be due to stack-induced stress. On the other hand,
Moselund et al. [40] suggested that the local volume inversion at the corner would
cause the mobility enhancement under on-state. To analyze the local volume inversion
effect, the size and corner effects need to be interpreted in the rectangular Si NW
MOSFETs.
1.2.5 Quasi-Ballistic Transport
An ultra-short gate length of the state-of-the-art logic devices raises transport issues,
such as high electric field and quasi-ballistic transport. M. Lundstrom [51], [52] has
developed a scattering theory with the kT layer for the high-field transport. Although
the kT-layer theory has been empirically validated by a Monte-Carlo simulation [51],
it has not been clearly postulated. The backscattering coefficient in the kT-layer theory
is derived with assuming near-equilibrium analogous to diffusive regime, which is not
valid under high field even around the top of the barrier where the longitudinal
electric field is small [53]. Furthermore, the critical distance in the framework of the
kT-layer theory was discrepant from the LkT [54], [55], [56], [57]. With renouncing the
crude assumptions of the kT-layer theory, E. Gnani et al. [58] and K. Natori [59]
developed a quasi-ballistic transport model for Si NW MOSFETs by directly solving
6
the BTE with constraint of dominant elastic scattering due to acoustic phonon. Further
work by K. Natori [60], [61] took into account the inelastic scattering due to the
optical phonon emission, where he assumed endless drain for simplicity. The
assumption of the endless drain interrupts interpretation of the quasi-ballistic transport
because the carriers end up relaxing their energy.
1.3 Modeling and Simulation of Nanowire MOSFETs
We can classify the transport regimes to: diffusive transport, quasi-ballistic transport,
and ballistic transport as shown in Table 1.1. Roughly speaking, when the gate length,
Lg, is longer than 0.1 µm, the carriers traverse within the diffusive transport regime,
where we can describe the carrier transport with the drift-diffusion model [62], [63],
[64], [65]. When the Lg is shorter than 10 nm, the carriers traverse within the ballistic
transport regime. In the intermediate Lg, the carriers traverse within the quasi-ballistic
transport regime. The ballistic and quasi-ballistic transports can be semiclassically
simulated by the Monte-Carlo (MC) simulation [66], [67] or the deterministic solution
of the Boltzmann transport equation (BTE) [68], [69], [70], [71]. They can also be
TABLE 1.1
CLASSIFICATION OF TRANSPORT MODELS
Lg > 0.1 µm Intermediate Lg < 10 nm
Diffusive transport Quasi-ballistic transport Ballistic transport
MC simulation
Direct solution of the Boltzmann transport equation (BTE)
Drift-diffusion model
Non-equilibrium Green function (NEGF) formalism
7
quantum mechanically simulated by the non-equilibrium Green function (NEGF)
formalism [72], [73], [74], [75], [76], [77], [78], [79].
In this study, to interpret device physics of the NW MOSFETs with modeling and
simulation, we adopted quantum mechanical approach for the cross-sectional electron
distribution and semiclassical approach for the carrier transport by solving the
one-dimensional (1D) multisubband Boltzmann transport equation (MSBTE).
1.4 Dissertation Outline
In this dissertation, we study on the interpretation of device physics in NW
MOSFETs and on what parameters control performance of the NW MOSFETs.
Chapter 2 briefly introduces the adopted methodologies to simulate electrical
characteristics of the NW MOSFETs. In chapter 3, we develop a gate capacitance
model to distinguish the contributions of the quantum effects to the total gate
capacitance. In chapter 4, we investigate size-dependent performance of Si and
InAs NW MOSFETs. In chapter 5, we investigate the band structure effect in
substantially small thickness based on the first-principles calculation. In chapter 6,
we clarify the size and corner effects on the mobility in rectangular Si NW
MOSFETs with the Kubo-Greenwood formula. In chapter 7, we develop a
comprehensive quasi-ballistic model based on the direct solution of the BTE.
Finally, chapter 8 concludes this dissertation and introduces future work.
8
Chapter 2
Methodologies
2.1 First-Principles Calculation
In chapter 5, the band structures of Si NWs were calculated by first-principles
calculation based on density functional theory (DFT) with local density
approximation (LDA) [80], [81], [82], [83], [84], [85]. The band calculations are
performed with Tokyo Ab-initio Program Package (TAPP) [86]. The DFT and LDA
are briefly introduced in this section.
2.1.1 Density Functional Theory
DFT is the First-principles calculation to solve the non-relativistic time-independent
Schrödinger equation. If the total energy is a function of the electron density n, then a
many-body problem can be the simple problem with respect to the electron density.
Because the wave function corresponding to an arbitrary electron density is not
9
unique, the total energy cannot be directly derived from an arbitrary electron density.
According to the N-representability [81], single electron density is described by an
antisymmetric wave function ψ as described in
∫= NN dxdxdxxxNn LL 212
211 |),,,(|)( ξψr , ),( iiix ξr≡ , (2.1)
where ri is the spatial coordinates, and ξi is the spin coordinates. Because the external
potential energy is uniquely determined by the ground state electron density, nGS, (v-
representability) [80], the wave function can be determined by nGS with the external
potential energy, i.e., the relation of (2.1) can be reversed for nGS:
][ GSGS nψψ = . (2.2)
P. Hohenberg and W. Kohn [80] showed that the ground state energy is determined
by nGS. If we define an energy function of n, E[n], as described in
GSext ][)()(][ EnFdnnE ≥+= ∫ rrrυ , (2.3)
consequently the minimum of E[n] becomes the ground state energy EGS according to
[80]. Here, υext(r) is an external potential at position r = (x,y,z) as described in
∑ −≡I
II )()(ext Rrr υυ , (2.4)
where υI and RI are the potential energy caused by nucleus I and the nucleus position,
respectively. F[n] is described in
][ˆˆ][][ mineemin nVTnnF ψψ += , (2.5)
where T̂ is the electron kinetic energy operator, eeV̂ is the electron-electron
interaction energy operator, and ψmin[n] is the wave function that yields the minimum
of ( eeˆˆ VT + ) from the given n. Because EGS is the same as E[nGS] according to [80], we
can obtain the ground state energy and electron density by using the variational
principle. Eventually, a many-body problem of N electrons with 3N spatial
10
coordinates could be reduced to the problem as a function of the electron density with
3 spatial coordinates.
To calculate the ground state energy based on the DFT, W. Kohm and L. J. Sham
provided a solution taking into account a non-interacting particle system [82]. The
many-body problem is changed to an effective single particle problem is described in
)()()(2 eff
22
rrr iiimψεψυ =⎥
⎦
⎤⎢⎣
⎡+∇−
h , (2.6)
∑=N
iin 2|)(|)( rr ψ , (2.7)
where the Kohn-Sham orbital ψi(r) denotes wave function of the non-interacting
particle, and υeff(r) is an external effective potential energy. The sum over i of (2.7) is
carried out by an order of εi where i involves the spin degree of freedom. In this
system, F[n] of (2.3) can be divided into three components:
][|'|)'()('
2][][ XC
2
s nEnnddqnTnF +−
+= ∫∫ rrrrrr , (2.8)
where the second term of the right hand side (RHS) is potential energy of Coulomb
interaction between electrons, the third term of RHS is the exchange-correlation
energy regarding entire many-body effects, and the first term of RHS indicates kinetic
energy in effective non-interacted system described in
∑∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛∇−=
N
iii d
mnT rrr )(
2)(][ 2
2*
s ψψ h . (2.9)
From (2.6), (2.9) can be rewritten as
∫∑ −= rrr dnnTN
ii )()(][ effs υε . (2.10)
υeff(r) of (2.6) can be derived from the ground state electron density nGS as following.
By substituting (2.10) into (2.8), E[n] is rewritten as
11
][|'|)'()('
2)()()()(][
2
exteff nEnnddqdndnnE XC
N
ii +
−++−= ∫∫∫∫∑ rr
rrrrrrrrrr υυε . (2.11)
Variation of (2.11) is described in
rr
rrrrrr
rrrrrrrr
dn
nEdnqn
dndndnnE GS
∫ ∫
∫∫∫
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−++
−−=
)(]['
|'|)'()()(
)()()()()()(][
GS
GSXCGS2extGS
GSeffeffGSGSeff
δδυδ
δυυδδυδ r, (2.12)
where we have used the below equation,
∑ ∫=N
ii dn rrr )()( GSeffδυδε . (2.13)
In order that δE[nGS] of (2.12) becomes zero, υeff(r) should be described in
)(]['
|'|)'()()(
GS
GSXCGS2exteff r
rrrrrr
nnEdnq
δδ
υυ +−
+= ∫ , (2.14)
where the third term of RHS,
)(][)(
GS
XCXC r
rn
nE GS
δδµ = , (2.15)
is called exchange-correlation potential energy. Equations (2.6), (2.7) and (2.14) are
called Kohn-Sham equations, which need to be self-consistently solved. Firstly, by
substituting an arbitrary wave function into (2.7), n(r) is obtained. Secondly, by
substituting the calculated n into nGS of (2.14), υeff(r) is obtained. Thirdly, by
substituting the obtained υeff(r) into (2.6), a new wave function is obtained. These
flows iterate until the new υeff(r) is the same as the old υeff(r). If we know EXC, then
we can exactly calculate the electron density and the total energy of the ground state.
2.1.2 Local Density Approximation
The exchange-correlation potential term µXC(r) of (2.15) is necessary to solve the
Kohn-Sham equation. However, it is difficult to directly calculate the
12
exchange-correlation energy EXC[n]. The LDA is a method to approximately deal with
the EXC[n]. When the spatial variation of the electron density is gradual, it can be
regarded as homogeneous electron gas. EXC[n] is described as
∫= rrr dnnnE )())((][ XCXC ε , (2.16)
where εXC(n) denotes exchange-correlation energy density of the homogeneous
electron gas. By substituting (2.16) into (2.15), the µXC(r) of (2.15) is rewritten as
dnndnn ))(()())(()( XC
XCXCrrrr εεµ += . (2.17)
εXC[n] could not be derived analytically, but it has been numerically calculated by the
quantum Monte Carlo method [83]. The results are imported in the LDA [84].
2.2 Gate Capacitance
We implemented the code to calculate electrostatics and subband structure of NW
MOSFETs. The subsection 2.2.1 describes the self-consistent solution for gate
capacitance of MOS devices. In ultra-scaled MOS devices, the quantum effects on the
total gate capacitance though inversion layer capacitance become important. The
subsection 2.2.2 describes the inversion layer capacitance.
2.2.1 Self-Consistent Solution of the Schrödinger and Poisson
Equations
We self-consistently solve Schrödinger and Poisson equations to calculate the
subband structure and electrostatics of NW MOSFETs. The differential equations
were solved by the finite difference method (FDM). Fundamental concept of the
self-consistent calculation referred to in the text of S. Datta [78]. Figure 2.1 shows the
13
flowchart of the self-consistent gate capacitance calculation with the given charge
distribution function. If the contribution of the longitudinal electric field is neglected,
it could be reduced to a two-dimensional (2D) problem. Actually, we can neglect the
longitudinal electric field at the top of the barrier or under the low-field limit.
We handle the MOS capacitor with a p-type substrate in this subsection. We took
Schrödinger equation: E>, >>
Charge calculation: n
Poisson equation: U
Convergence check
U = U0
Start
End
Figure 2.1: Flowchart of the self-consistent gate capacitance calculation with the given chargedistribution function. If the contribution of the longitudinal electric field is neglected, it could bereduced to a 2D problem.
14
into account the Schrödinger equation with spatially varying effective mass [47], [78],
[87], [88] and the Poisson equation with spatially varying dielectric constant [88] at
the boundary between the NW and oxide. In the EMA, the time-independent
Schrödinger equation with a spatially varying effective mass is described in
)()()()(
12 c
2
rrrr µµµ χχ EU
m=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛∇⋅∇−
h , (2.18)
where µ is the subband index, mc(r) is the spatially varying effective mass, U(r) is the
total potential energy, Eµ is the eigenvalue, and χµ(r) is the envelope wave function
[78]. Equation (2.18) assumes parabolic spherical band with effective mass mc.
To simply discuss the solution with FDM, we take into account a 1D system with
spatially uniform parabolic effective mass, where the Schrödinger equation is written
as
)()()(2 2
2
c
2
xExxUdxd
m µµµ χχ =⎥⎦
⎤⎢⎣
⎡+−
h . (2.19)
In terms of the FDM, the second derivative term can be represented as
)]()(2)([11122
2
−+
=
+−→⎟⎟⎠
⎞⎜⎜⎝
⎛nn
xx
xxxadx
d
n
µµµµ χχχχ , (2.20)
where a is the spacing between discrete lattices. This allows us to write (2.19) as a
matrix equation,
}{}]{[ µµµ χχ EH = , (2.21)
where
1,01,0,0 ]2)([ −+ −−+= mnmnmnnnm tttxUH δδδ , (2.22)
2c
2
0 2 amt h
≡ .
(2.23)
If the effective mass changes from m1 to m2 at a boundary in a 1D system, the
15
resulting 1D Hamiltonian matrix [H] in terms of the FDM is represented as
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−++−
−+=
O
O
22
2211
11
20
02][
tUttttUt
ttUH , (2.24)
where
21
2
1 2 amt h
≡ , (2.25)
22
2
2 2 amt h
≡ . (2.26)
Since the Hamiltonian matrix of (2.24) is hermitian, we can obtain real number
energy eigenvalue. The non-uniform mesh was also used in our calculation because
the potential and wave function rapidly change near the surface [89]. The Poisson
equation with a spatially varying dielectric constant is described in
])([)]()([0
2
r aNnqU +−=∇⋅∇ rrrε
ε , (2.27)
where εr(r) is the spatially varying dielectric constant, ε0 is the vacuum permittivity,
n(r) is the electron density, and Na is the uniform accepter concentration. We can
solve (2.27) based on FDM in the same way as the Schrödinger equation. By solving
coupled equations of (2.18) and (2.27), we can obtain the electronic structure and the
gate capacitance. Appendix A describes the self-consistent calculation of the
electronic structure in cylindrical nanowire MOSFETs based on the top-of-the-barrier
semiclassical ballistic transport model [90].
2.2.2 Inversion Layer Capacitance
The total gate capacitance, Cg, is described by a series connection of the gate oxide
16
capacitance, Cox, and the inversion layer capacitance, Cinv [3]. The Cinv consists of two
quantum effects: the finite inversion layer centroid and the finite DOS. In ultra-scaled
MOS devices, the quantum effects regarding Cinv largely affect Cg. In partial depletion
of the single-gate planar MOSFET, the Cg is described in
dinvox
di
s
di
ox
di
g
g
11)()(
)(1
CCC
QQdd
QQdd
QQddV
C
++=
−−+
−−=
−−=
ψφ , (2.28)
where
)(1
di
ox
ox QQdd
C −−≡
φ, (2.29)
)(1
d
s
d Qdd
C −≡
ψ, (2.30)
)()()(
)(1
i
d
i
ds
i
s
inv Qdd
Qdd
Qdd
C −+
−−
=−
≡ψψψψ
. (2.31)
Qi is the inversion charge, Qd is the depletion charge, ψs is the surface potential, and
ψd is the potential drop due to the depletion charge. In strong inversion, we can
approximate the Cg as
invox
i
s
i
ox
i
g
g
11)()(
)(1
CC
Qdd
Qdd
QddV
C
+=
−+
−=
−≈
ψφ . (2.32)
Let us discuss DG and NW MOSFETs. Partial depletion of DG and NW MOSFETs is
the same as that of the single-gate planar MOSFET. In full depletion of DG and NW
MOSFETs, Cg is exactly described in
17
invox
i
g
g
11
)(1
CC
QddV
C
+=
−=
, (2.33)
where
)(1
i
ox
ox Qdd
C −≡
φ, (2.34)
)()()(
)(1
i
c
i
cs
i
s
inv Qdd
Qdd
Qdd
C −+
−−
=−
≡ψψψψ
. (2.35)
ψc is the central potential of the body. According to (2.31) and (2.35), the effect of
depletion-charge-induced potential drop is analogous to that of central potential drop,
which is adjusted by DOS. The physical interpretation of the effect of the central
potential drop is discussed in chapter 3.
2.3 Boltzmann Transport Equation
In this study, we semiclassically handled the carrier transport by using the BTE.
According to [91], the BTE in a six-dimensional position-momentum space is given
by
stffqf
tf
kr +∂∂
=∇⋅+∇⋅+∂∂
coll
1 Fυh
, (2.36)
where
zzfy
yf
xxff zyx
r ˆˆˆ∂∂
+∂
∂+
∂∂
≡∇ , (2.37)
zkfy
kf
xkff
z
z
y
y
x
xk ˆˆˆ
∂∂
+∂
∂+
∂∂
≡∇ . (2.38)
f is the distribution function, and s is the ‘generation-recombination’ term. The first
18
term of the right hand side of (2.36) is the net rate of increase of f due to collisions.
The group velocity, υ, and the external electric field, F, can be respectively given by
Ek∇=h
1υ , (2.39)
Eq r−∇=F , (2.40)
where E is the total energy. In the isotropic parabolic EMA, the E can be described in
c
2222
c 2)(
mkkk
EE zyx +++=h
, (2.41)
where Ec is the conduction band minimum, and mc is the isotropic effective mass. The
total potential energy corresponds to Ec.
2.3.1 One-Dimensional Multisubband Boltzmann Transport Equation
In NW MOSFETs, we can handle the carrier transport as the quasi-1D transport. In
this subsection, we take into account the one-dimensional multisubband Boltzmann
transport equation (1D MSBTE) in Si NW MOSFETs. We assume the steady state and
neglect the generation-recombination term. By substituting (2.39) and (2.40) into
(2.36), the 1D MSBTE is given by
coll
),(1),(1t
fk
kzfzE
zkzf
kE
∂
∂=
∂
∂
∂∂
−∂
∂
∂∂ µµµ
hh, (2.42)
where fµ is the distribution function for subband µ. The subband index, µ = (η,ξ,l),
consists of the valley index, η, the principle quantum number, ξ, and the angular
quantum number, l. The net rate of increase of fµ due to collisions is described by the
difference between in- and out-scattering as
),(),( outin
coll
kzCkzCtf
µµµ −=
∂∂ . (2.43)
According to [92] [93], the in- and out-scattering integrals are given by
19
∑ ∫′
′′ ′′−=µ
µµµµµ πkdkzfkkSLkzfkzC ),(),(
2)],(1[),( ,
zin , (2.44)
∑ ∫′
′′ ′−′=µ
µµµµµ πkdkzfkkSLkzfkzC )],(1)[,(
2),(),( ,
zout , (2.45)
where the transition rate from k to k’ is described in
∑ ′+′=′ ′′′j
j kkSkkSkkS ),(),(),( ,ac
,, µµµµµµ . (2.46)
The transition rate of elastic acoustic phonon scattering, ac,µµ ′S , and the transition rate
of six-type inelastic phonon scatterings, jS µµ ′, , are given by
)(),( ,,2Siz
2ac
, EEFugL
TkΞkkSl
B ′−=′ ′′′ δδρ
πµµηη
υµµ
h, (2.47)
)(21
21)(
2)(
),( op,,Siz
2
, jjj
j
jtj EENFggL
KDkkS ωδω
ωρπ
µµηηυ
µµ hmh ′−⎟⎠⎞
⎜⎝⎛ ±+=′ ′′′ , (2.48)
where E’ is the total energy associated with the primed state, ρSi is the density of Si, ul
is the sound velocity, Ξ and (DtK)j are the deformation potentials, Nop and ħωj are the
phonon number and energy, η is the valley index, δη,η’ is the Kronecker delta, and
jg ηη ′, is δη,η’ for g-type process and 2(1 − δη,η’) for f-type process. The number of
phonons Nop is given by
11)( /op
−= Tkj Bje
N ωωh
h . (2.49)
Form factor, Fµ,µ’, is given by
∫∫ ′′ = dxdyyxyxF 22, |),(||),(| µµµµ σσ , (2.50)
where σµ(x,y) is the transverse envelope wave function. Using the radial envelope
wave function, Rµ(r), the form factor is given by
20
∫∞
′′ =0
22, |)(||)(|2 drrRrRrF µµµµ π , (2.51)
where
φµµµ φσσ ilerRryx )(),(),( == . (2.52)
In the parabolic EMA, the total energy, Ε, and group velocity, υµ, are given by
µµ m
kEE2
22h+= , (2.53)
µµυ
mkh
= , (2.54)
where Eµ is the minimum of subband µ, and mµ is the effective mass for subband µ.
2.3.2 Low-Field Mobility Calculation by the Kubo-Greenwood Formula
In the low electric field limit, we can assume a spatially homogeneous distribution
function and the relaxation time approximation (RTA). Under the steady state and the
low electric filed limit, we can rewrite the 1D MSBTE of (2.42) as
)()()()(
)( 00
EEfkf
dEEdf
kqFµ
µµµ τ
υ−
−=− , (2.55)
where τµ is the momentum relaxation time, and f0 is the equilibrium distribution
function [51], [79]. f0 with the Fermi level Ef is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
≡
TkEE
Ef
B
f0
exp1
1)( . (2.56)
Equation (2.55) can be rearranged as:
dEEdfkEqFEfkf )()()()()( 0
0 µµµ υτ+= , (2.57)
By integrating (2.57) over k, we can derive the current associated with subband µ as
described in
21
∫
∫
∫
∞
∞
∞
∞−
−−=
∂∂
−=
−=
µ
µ
µµµµ
µµµ
µµυµ
υτρ
υτρ
υπ
EB
E
dEEfEfEEETkFq
dEEEfEEEFq
dkkkfqgI
)](1)[()()()(
)()()()(
)()(2
2
002
2
022 , (2.58)
where the group velocity and the DOS, υµ and ρµ, are described in
µ
µµυ
mEE
kk )(2
||−
= , (2.59)
)(22
µ
µυµ π
ρEE
mg−
=h
,
(2.60)
and gυ is the valley degeneracy. From (2.58), we can derive the mobility µµ as described
in
∫∞
−=µ
µµµµ
µ υτρµE
B
dEEfEfEEETk
qn
)](1)[()()()(100
2 , (2.61)
where nµ is the number of electrons associated with subband µ. Equation (2.61) is
called Kubo-Greenwood formula [94], [95].
2.3.3 Ballistic Boltzmann Transport Equation
The ballistic transport regime is a useful metric to the estimate the performance limit.
By using relations,
EEzf
zE
zEzf
zkzf
∂∂
∂∂
+∂
∂=
∂∂ ),(),(),( µµµ , (2.62)
EEzf
kE
kkzf
∂
∂
∂∂
=∂
∂ ),(),( µµ , (2.63)
(2.42) can be rewritten as
22
0),(
|),(| =∂
∂±
±
zEzf
Ez µµυ , (2.64)
where
µ
µµυ
mzEE
kkEz
)]([2||
),(−
= . (2.65)
The collision integral terms are eliminated because the ballistic transport is assumed.
+µf and −
µf are distribution functions with positive and negative velocity components.
Here, we consider a longitudinal potential profile along the source, channel, and drain
in which the device range is from 0 to L, and the position of the top of the barrier is z0.
Boundary conditions are that fµ for z < 0 is the source equilibrium distribution
function fS and that fµ for L < z is the drain equilibrium distribution function fD:
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
≡
TkEE
Ef
B
fsS
exp1
1)( , (2.66)
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
≡
TkEE
Ef
B
fdD
exp1
1)( , (2.67)
where Efs and Efd are source and drain Fermi levels, respectively. By solving (2.64) as
shown in [79] and [96], the resulting distribution functions are given by
⎪⎩
⎪⎨
⎧
<<<<<<<<>
=+
,and)(for),(,0and)(for),(
,0and)(for),(),(
00D
00S
0S
LzzzEEEfzzzEEEfLzzEEEf
Ezf
µ
µ
µ
µ
(2.68)
⎪⎩
⎪⎨
⎧
<<<<<<<<>
=−
.and)(for),(,0and)(for),(,0and)(for),(
),(
00D
00S
0D
LzzzEEEfzzzEEEfLzzEEEf
Ezf
µ
µ
µ
µ (2.69)
23
Therefore, the distribution functions at the top of the barrier are given by
)(),( S0 EfEzf =+µ ,
(2.70)
)(),( D0 EfEzf =−µ . (2.71)
Longitudinal BTE: fβ
Charge calculation: n
Poisson equation: U
Convergence check
Transverse Schrödinger equation: Eβ, ββ
Start
End
U = U0
Figure 2.2: Flowchart of a device simulation based on the deterministic numerical solution of the 1DMSBTE.
24
2.3.4 Device Simulation Based on the Deterministic Numerical Solution
of the One-Dimensional Multisubband Boltzmann Transport Equation
Figure 2.2 shows a flowchart of a device simulation based on the deterministic
numerical solution of the 1D MSBTE [92], [93]. After the transverse Schrödinger
equation is solved in each slab along z direction, the longitudinal 1D MSBTE is
solved to extract distribution functions. The device simulator used in this study was
developed by M. Lenzi et al. at the University of Bologna [92].
By using (2.62) and (2.63), the (2.42) can be rewritten as
),(),(),(
|),(| outin EzCEzCz
EzfEz ±±
±
−=∂
∂± µµ
µµυ , (2.72)
where the 1D MSBTE has been divided into two parts with respect to positive and
negative velocity components. The group velocity υµ is given by (2.65). With taking
into account the elastic scattering, the inelastic scattering with energy relaxation, and
the inelastic scattering with energy excitation, the in- and out-scattering integrals are
described as
∑
∑
∑
′
−′
+′
′′
±
′
−′
+′
′′
±
′
−′
+′
′′
±±
−+−−
−+
++++
+−+
+−=
µµµ
µµµµ
µµµ
µµµµ
µµµ
µµµµµ
ωωωρ
ω
ωωωρ
ω
ρ
,op,
,op,
ac,
in
]),(),([2
),()(]),(1[
]),(),([2
),(]1)([]),(1[
]),(),([2
),(]),(1[),(
jjj
jj
j
jjj
jj
j
EzfEzfEz
NMEzf
EzfEzfEz
NMEzf
EzfEzfEz
MEzfEzC
hhh
h
hhh
h,
(2.73)
∑
∑
∑
′
−′
+′
′′
±
′
−′
+′
′′
±
′
−′
+′
′′
±±
+−++−+
+
−−+−−−
++
−+−=
µµµ
µµµµ
µµµ
µµµµ
µµµ
µµµµµ
ωωωρ
ω
ωωωρ
ω
ρ
]}),(1[]),(1{[2
),()(),(
]}),(1[]),(1{[2
),(]1)([),(
]}),(1[]),(1{[2
),(),(),(
op,
op,
ac,
out
jjj
jj
jjj
jj
EzfEzfEz
NMEzf
EzfEzfEz
NMEzf
EzfEzfEz
MEzfEzC
hhh
h
hhh
h.
(2.74)
Here, a coefficient associated with the elastic acoustic phonon scattering ac,µµ ′M and a
coefficient associated with the inelastic phonon scatterings jM µµ ′, are respectively
given by
25
µµηηυ
µµ δρ
π′′′ = ,,2
Si
2ac
, FugTkΞM
l
B
h, (2.75)
µµηηυ
µµ ωρπ
′′′ = ,,Si
2
, 2)(
Fgg
KDM j
j
jtj , (2.76)
and the DOS ρµ is given by
)]([22),(
zEEmgEz
µ
µυµ π
ρ−
=h
. (2.77)
26
Chapter 3
Gate Capacitance Modeling of
Nanowire MOSFETs
3.1 Introduction
Since gate capacitance is closely related to the operation of MOSFETs, the gate
capacitance should be clearly modeled to analyze the performance of the MOSFETs.
The gate capacitance, Cg, is usually described by a series connection of the gate oxide
capacitance, Cox, and the inversion layer capacitance, Cinv. The Cinv is a capacitance
due to two quantum effects as schematically shows in Figure 3.1: finite inversion
layer centroid and finite DOS. Thus far these two contributions have not been clearly
distinguished yet in DG and NW MOSFETs.
In planar MOSFETs, the Cinv can be modeled with the charge centroid with respect
to inversion and depletion charges [17]. S.-i.Takagi and A. Toriumi described the Cinv
with a series connection of two inversion layer capacitances due to the finite inversion
27
layer centroid and the finite DOS [97], where the effect of finite DOS was regarded as
that of the depletion charges in weak inversion.
In DG and NW MOSFETs, the Cinv cannot be described only with the inversion
layer centroid. Nevertheless, a model adopted in [18], [19] regarded the Cinv as the
inversion layer capacitance due to the finite inversion layer centroid; however, it is
not valid in substantially small DOS because the variation of the central potential is
not negligible even in strong inversion. Another model adopted in [20], [21], [22]
described the Cinv with a series connection of two inversion layer capacitances due to
the finite DOS and due to the finite inversion layer centroid, where the latter
capacitance was defined as the capacitance except for the series connection of the
former capacitance; thus, the inversion layer capacitance due to the finite inversion
layer centroid was still ambiguous. Our gate capacitance model is more
comprehensive than the former model, where we regard the effect of the varying
central potential as the inversion layer capacitance due to the finite DOS. As a result,
our model could clarify the diameter-dependent confinement effects on the gate
capacitance.
Oxide
Silicon Subbandminimum
Finite inversion layer controid
Oxide
Silicon
Fermi level
Electron distribution
Finite density of states (DOS)
Parasiticvoltage drop
Parasiticvoltage drop
Oxide
Figure 3.1: Concept of the two quantum effects, which contribute to the Cinv. Because of the quantum effects, we need additional voltage drop to charge electrons.
28
3.2 Gate Capacitance Modeling
Since gate capacitance is closely related to the operation of MOSFETs, the gate
capacitance should be clearly modeled to analyze the performance of the MOSFETs.
In our model, we assume the ideal gate control, and neglect the charge within the
oxide layer. Table 3.1 describes symbols related to the gate capacitance. Figure 3.2(a)
shows a schematic radial band diagram at the flat band state. Here, U(r) is the radial
potential, and Ufb is the uniform radial potential at the flat band state. Figure 3.2(b)
shows a schematic radial band diagram when gate voltage, Vg, is applied.
TABLE 3.1
LIST OF MAIN SYMBOLS RELATED TO THE GATE CAPACITANCE
Symbol QUANTITY
Cg total gate capacitance
Cinv total inversion layer capacitance
Ce electrostatic capacitance due to the finite inversion layer centroid
Ccentroid inversion layer capacitance directly described by the inversion layer centroid
Cq quantum capacitance due to the finite DOS
Cdos inversion layer capacitance directly described by the DOS
CG effective total gate capacitance
CE effective electrostatic capacitance due to the finite inversion layer centroid
CQ effective quantum capacitance due to the finite DOS
CDOS effective inversion layer capacitance directly described by the DOS
xeff effective inversion layer centroid
xavg exact inversion layer centroid
teff effective oxide thickness
ψs surface potential
ψc central potential
E1 energy level of the lowest subband
ψst surface potential at Vg = Vt
ψct central potential at Vg = Vt
E1t energy level of the lowest subband at Vg = Vt
29
Figure 3.2(c) shows the equivalent circuit in an intrinsic channel. On the basis of [98],
the gate capacitance is divided into three parts according to the position of the
potential drops. The gate capacitance with an intrinsic channel is described in
)()()(
)()(1
i
c
i
cs
i
ox
i
g
g Qdd
Qdd
Qdd
QddV
C −+
−−
+−
=−
=ψψψφ ,
(3.1)
where Cg is the gate capacitance per unit wire surface, and Qi is the electron inversion
charge density per unit wire surface:
R
rndrqQ
R
∫−≡ 0
i , (3.2)
ψs is the surface potential described in qψs = Ufb – U(R), and ψc is the central
potential described in qψc = Ufb – U(0). The oxide capacitance per unit wire surface,
Cox, can be calculated as
]/)ln[()(
ox
0ox
ox
iox RtRRd
QdC+
=−
≡εε
φ,
(3.3)
where tox and εox are the thickness and the dielectric constant of the SiO2, respectively.
ε0 is the vacuum permittivity. The second and third terms of right hand side in (3.1) is
related to the Cinv.
Ce, the second series component of Cg in (3.1), is the electrostatic inversion layer
capacitance described with spatial distribution of inversion charge. We can derive the
Ce with solving the Poisson equation with the cylindrical coordinate system:
nqdrdr
drd
r 0r
1ε
ψε =⎟⎠⎞
⎜⎝⎛ ,
(3.4)
where ψ(r) is [Ufb – U(r)] / q, n(r) is the radial electron density per unit volume, and
εr(r) is the dielectric constant. By solving (3.4) as shown in the appendix B, we derive
the ψs – ψc, which is ψ(R) – ψ(0), as described in
30
centroid
ics
)(C
Q−=−ψψ .
(3.5)
Ccentroid is the inversion layer capacitance due to the finite inversion layer centroid:
)]/(ln[ eff
0nwcentroid xRRR
C−
≡εε
, (3.6)
where εnw is the dielectric constant within the NW region, and xeff is the effective
inversion layer centroid from the NW-SiO2 surface as described in
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−≡
∫∫
R
R
rndr
rndrrRx
0
0eff
)ln(exp .
(3.7)
The average distance of electrons from the surface, xavg, can be defined as
∫∫−≡ R
R
rndr
ndrrRx
0
0
2
avg , (3.8)
which gives the exact inversion layer centroid [19]. Equations (3.6), (3.7), and (3.8)
imply that, to estimate the Ccentroid, we cannot adopt the xavg instead of the xeff in
cylindrical symmetry since there is large discrepancy between the xeff and xavg as
shown in Figure 3.5(a). Differentiating (3.5) with respect to (ψs – ψc), we derive the
Ce as described in
1
i
eff
eff0nw
i
centroidcs
ie )(
)(1)(
)(−
⎥⎦
⎤⎢⎣
⎡−−
−+=
−−
≡Qd
dxxR
RQCd
QdCεεψψ
. (3.9)
The Ce is closely associated with the Ccentroid although the second term in the bracket
of the right hand side in (3.9) is not negligible as shown in Figure 3.2.
Cq, the third series component of Cg in (3.1), is the quantum inversion layer
capacitance (quantum capacitance), which is described with the varying central
potential, ψc:
31
1
i
c1
dosc
iq )(
)(1)(−
⎥⎦
⎤⎢⎣
⎡−−−
−=−
≡QqdqEd
CdQdC ψ
ψ,
(3.10)
where E1 is the energy level of the lowest subband, and Cdos is the inversion layer
capacitance due to the finite DOS. According to [24] and [97], the Cdos is described by
)()(
1
idos Ed
QqdC−−
≡ . (3.11)
In the strong volume inversion, the Cq can be approximated by Cdos as shown in
Figure 3.2(c) since the strong volume inversion causes hardly varying potential in
space, where dψc / dψs ≈ 1 and d(–E1) / qdψc ≈ 1. Therefore, we can approximate the
Capacitance (λF/cm
2)10
8
6
4
2
04
3
2
1
0
Capacitance (λF/cm
2)
Vg – Voff (V)0 0.1 0.40.30.2 0.5
Vg – Voff (V)0 0.1 0.40.30.2 0.5
(a)[100] Si NWd = 3 nm
(b)[100] Si NWd = 12 nm
(d) InAs NWd = 30 nm
(c) InAs NW d = 5 nm
Ce
Cox
Cq Cdos
Ccentroid
Cg=d(-Qi)/dVg
Cg=1/(1/Cox+1/Ce+1/Cq)
Figure 3.2: Capacitance-voltage characteristics of the (a) 3-nm-diameter [100] Si NW, (b) 12-nm-diameter [100] Si NW, (c) 5-nm-diameter InAs NW, and (d) 30-nm-diameter InAs NW. In the 5-nm-diameter InAs NW MOSFET, Ce and Ccentroid were almost the same and hardly varied, and the values were 9.4 to 9.5 µF/cm2. Here, Voff is the off-state gate voltage shown in Fig. 4(b). (Cox = 3.45 µF/cm2, Vd = 0.5 V).
32
Cg as 1 / Cg ≈ 1 / Cox + 1 / Cdos as in [24], [99], [100], [101]. Although the Ce
comparable to Cq makes the relation between Cq and Cdos ambiguous, Cq still reflects
the effect of the subband quantization on the Cg.
The numerical difference between Ce and Ccentroid and that between Cq and Cdos are
shown in Figure 3.2. Here, the off-state gate voltage, Voff, is defined at the first
paragraph of the section 4.3. Figure 3.2 also shows that our gate capacitance model is
well established and that Ce is larger in smaller diameter because of the decrease in
the xeff as shown in Figure 3.5(a). The small InAs NW almost operates in the quantum
capacitance limit [99]. Appendix C shows the gate capacitance modeling of planar
and DG MOSFETs in the same concept.
3.3 Discussion
Figure 3.3 shows diameter dependences of effective total gate capacitance, CG, in
Diameter (nm)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Effective gate capacitance,
CG(ζF/cm
2)
InAs NW
[100] Si NW
@ on-state
Diameter (nm)
Figure 3.3: Effective gate capacitance as a function of diameter. (Cox = 3.45 µF/cm2, Vd = 0.5 V).
33
[100] Si and InAs NW MOSFETs. The CG is defined as:
tg
iG VV
QC−
−≡ ,
(3.12)
at on-state under the ballistic transport regime, where the gate overdrive voltage,
(Vg – Vt), is 0.35 V. Here, the threshold voltage, Vt, is defined at the second
paragraph in the section 4.3. To analyze the diameter-dependent CG, we recall the gate
capacitance model described in the previous section. Since the capacitances depend
on Vg, we take into account the effective capacitances, CG, CE, CQ, and CDOS.
Analogous to CG, the effective capacitances at on-state are defined with difference
between potential drops at Vg = Von and Vg = Vt. Hence, the effective capacitances
described as follows:
QEox
i
tcc
i
tc
tscs
i
ox
G
111)()(
)()()(
1
CCC
QQQC
++=
−−
+−
−−−+
−≈
ψψψψψψφ
, (3.13)
where ψst and ψc
t are the ψs and ψc at Vg = Vt, respectively, and the φox at Vg = Vt is
approximated by 0 V. The CDOS can now be defined as
1t
1
iDOS EE
QC−
−≡ . (3.14)
where E1t is the E1 at Vg = Vt.
Figure 3.4 shows that the diameter dependences of CG, CE, CQ, CDOS, and Ccentroid at
on-state. When the (ψst – ψc
t) is negligible, the CE is the same as Ccentroid. Figure 3.4(a)
shows that the diameter dependence of the xeff reflects that of the xavg, which decreases
with shrinking diameter. Figure 3.5(b) shows that the xeff eventually decreases the
effective oxide thickness, teff, in small diameter despite the change in the quantum
capacitance. Here, the teff includes the effects of the Ce and Cq, and is calculated by
34
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+=
G
0oxoxeff exp1)(
RCtRt εε . (3.15)
The decrease in xavg is due to the volume inversion, and the strong volume inversion
drastically decreases the xavg in small diameter because the spread of inversion
carriers is limited by wire size as shown in Figure 3.6. López-Villanueva et al. [18]
showed that the volume inversion decreases the inversion layer depth also in DG
MOSFETs. Therefore, the CE monotonically increases with shrinking diameter in
0
2
4
68
10
12
14
16
0 5 10 15 20 25 30
0
0.5
1
1.52
2.5
3
3.5
4
0 10 20 30 40 50
(a) [100] Si NW
Diameter (nm)
Effective capacitance (λF/cm
2)
Diameter (nm)
Effective capacitance (λF/cm
2)
(b) InAs NW
CE
Cox
CQ
CDOS
Ccentroid
CG
CE
Cox
CQ
CDOS
Ccentroid
CG
@ on-state
@ on-state
Figure 3.4: Diameter-dependent effective capacitances in (a) [100] Si NWs. and (b) InAs NWs. Here,Ccentroid is t:he value at on-state. (Cox = 3.45 µF/cm2, Vdd = 0.5 V).
35
both Si and InAs NWs, which is a positive effect to achieve large CG in small NWs.
The CDOS can decrease with shrinking diameter owing to decreasing DOS due to
drastic subband quantization. The DOS per unit wire length is proportional to d2 in the
small diameter limit with strong volume inversion, whereas the DOS is proportional
to d in the large diameter limit with surface inversion, because the DOS per unit wire
length is approximately proportional to the cross-sectional area of the inversion layer.
Figure 3.4(b) shows that the CDOS in InAs NWs decreases with shrinking diameter. In
0123456789
10
0 10 20 30 40 50
@ on-state
xeff
xavgIn
vers
ion
laye
r cen
troid
(nm
)
[100] Si NW
InAs NW
xeff
xavg
(a)
00.5
11.5
22.5
33.5
44.5
5
0 10 20 30 40 50Diameter (nm)
[100] Si NW
InAs NW
t eff/ t
ox
@ on-state
(b)
Figure 3.5: (a) Inversion layer centroid as a function of diameter. (b) Effective oxide thickness devidedby the oxide thickness versus diameter. (Cox = 3.45 µF/cm2, Vdd = 0.5 V).
36
a diameter smaller than 6 nm, on the other hand, the CDOS drastically increases
because the effect of decreasing DOS no longer affects the CDOS, where the electrons
occupy only the lowest subband. In InAs NWs, the diameter dependence of the CQ
reflects well that of the CDOS as shown in Figure 3.4(b). Although there is a difference
between the diameter dependences of the CQ and CDOS in Si NWs, we do not need to
take this account because the CQ – which is substantially larger than Cox as shown in
Figure 3.4(a) – hardly affects the diameter-dependent CG. Even in Si NWs, the CQ
eventually closes to the CDOS when CE drastically increases in small diameter.
With the CE and CQ, we could interpret the diameter dependence of the CG as
follows. In Si NWs, the CG monotonically increases with shrinking diameter owing to
the increase in the CE as shown in Figure 3.4(a). In InAs NWs, the diameter
dependence of the CG is determined with a trade-off between the CE and CQ as shown
in Figure 3.4(b) because the degradation of the CG due to the decrease in the DOS,
expected in NWs, can be compensated by the decrease in the xavg. In substantially
small Si NWs, Cg was approximated by Cox; on the other hand, in substantially small
Figure 3.6: Radial electron density with various diameter at on-state in (a) [100] Si and (b) InAs NWs. (Cox = 3.45 µF/cm2, Vdd = 0.5 V).
37
InAs NWs, Cg was approximated by Cdos.
3.4 Conclusions
Since gate capacitance is closely related to the operation of MOSFETs, the gate
capacitance should be clearly modeled in order to analyze the performance of the
MOSFETs. We developed a gate capacitance model, where the effect of the varying
central potential is regarded as the capacitance due to the finite DOS. Our gate
capacitance model was well established and that Ce was larger in smaller diameter
because of the decrease in the xeff. In substantially small Si NWs, Cg was controlled
by Cox; on the other hand, in substantially small InAs NWs, Cg was controlled by Cdos.
The finite inversion layer centroid caused the positive effect on the total capacitance in
small NW MOSFETs. Our results could show that this model helped with clarifying
the confinement effects on the gate capacitance.
38
Chapter 4
Size-Dependent Performance of
Nanowire MOSFETs
4.1 Introduction
Since the deviation from the EMA is inevitable within strongly confining potential
[30], band structure effects on carrier transport in Si NW MOSFETs have been
investigated with atomistic approaches, e.g., the tight-binding (TB) model [32], [34],
[35], [38] or the first-principles calculation [37], [102]. Band structure effects on
carrier transport in InAs NW MOSFETs have also been investigated with the TB
model [103].
Small NW MOSFETs shows not only the deviation from the EMA but also other
important features accompanied with the volume inversion [23]:
Gate capacitance can be increased when the inversion charge is close to the
surface. [18], [19];
39
Density of states (DOS) can be reduced with subband quantization and can
degrade the gate capacitance through quantum capacitance [24], [25];
The increased gate capacitance and the reduced DOS can increase carrier
degeneracy and velocity [26], [27].
These phenomena are general features regardless of channel materials. Hence, the
detailed band structure effects, such as the deviation from the EMA, should be
considered after taking into account the three features mentioned above.
As a case study, we used [100]-directed Si NWs ([100] Si NWs) and InAs NWs as
the channel materials, which have large and small effective masses, respectively, such
that we identify the effects of the effective mass on the capacitances. We used an
EMA with a nonparabolic correction [30], [47], [93] and a semiclassical ballistic
transport model [35], [90] to calculate subband structures and electrical characteristics
of NW MOSFETs. The ballistic transport regime is a useful metric to estimate
performance even in the quasi-ballistic and diffusive regimes [28], [29]. To
investigate the diameter-dependent performance of NW MOSFETs, we evaluated gate
capacitance, injection velocity, on-current, intrinsic gate delay, and power delay
product.
4.2 Simulation Methods
We used gate-all-around cylindrical n-type NW MOSFETs with [100] Si and InAs
channels. To calculate the subband structures and electrical characteristics of the NW
MOSFETs, we used a combination of the nonparabolic EMA [30], [47], [93] and the
semiclassical ballistic transport model [35], [90].
40
4.2.1 Effective Mass Approximation with a Nonparabolic Correction
Subband structures of Si and InAs NWs were reproduced with the EMA. Table 4.1
describes band parameters for the EMA, where the mc and mtrans are confinement and
transport effective masses in bulk, respectively. For conduction band of [100] Si NWs,
we considered three two-fold degenerate ellipsoid valleys with anisotropic effective
masses, which are (mt, mt, ml), (mt, ml, mt), and (ml, mt, mt). The mt and the ml are 0.19
m0 and 0.916 m0, respectively, which correspond to the transverse and the longitudinal
effective masses of ∆ valleys of bulk Si, where the m0 is the free electron mass. Note
that we define unprimed subband as the subband quantized from the ∆4 valley, and
primed subband as the subband quantized from the ∆2 valley. The lowest subband is
always unprimed. As we adopted a cylindrical coordinate system, we assumed the
anisotropic confinement mass of the ∆4 valleys as the isotropic confinement effective
mass 2mtml /(mt + ml) [37], [47], [93]. For the conduction band of InAs NWs, we
considered the Γ valley with an isotropic effective mass, 0.023 m0, without
degeneracy.
Figure 4.1 shows a schematic model of a cylindrical NW MOSFET. We used
intrinsic NW channels and SiO2 gate insulator without any fixed interface charges.
The flat band voltage, Vfb, was set to 0 V. The electron effective mass in the insulator
was 0.5 m0. The band offset of Si and SiO2 conduction bands corresponded to 3.15 eV,
TABLE 4.1
CONDUCTION BAND PARAMETERS
NW Type Valley Type mc mtrans Degeneracy
[100] Si NW ∆4 0.315 m0a 0.19 m0 4
∆2 0.19 m0 0.916 m0
2
InAs NW Γ 0.023 m0 0.023 m0 1
aThe isotropic confinement effective mass of the ∆4 valley is for the cylindrical coordinate system.
41
and that of InAs and SiO2 conduction bands corresponded to 3.8 eV. We set Si, InAs,
and SiO2 permittivities to 11.8 ε0, 15.1 ε0, and 3.9 ε0, respectively, where ε0 is the
vacuum permittivity. The nonparabolicity factor, α, of Si NWs is 0.5 /eV. The α of
InAs NWs was calculated with 1 / Eg [1 –(m* / m0)]2, where m* and Eg are the
effective mass of 0.023 m0 and the band gap of 0.36 eV, respectively.
According to the nonparabolic EMA in [30], [47], [93], the nonparabolic E-k
dispersion Eµnp(k) is approximated by
α
α µµ
µµ 22
411)(
p
trans
22
np⎟⎟⎠
⎞⎜⎜⎝
⎛−+++−
+≈
UEm
k
UkE
h
. (4.1)
where ħ is the reduced Planck constant, Eµp is the minimum of subband µ derived
from the parabolic EMA, and Uµ is the expectation value of potential for subband µ.
Here, Uµ is calculated within the NW region since the nonparabolic correction in
confined potential assumes that the wave function vanishes at the
oxide-semiconductor surface [30]. The anisotropic confinement effective masses in Si
are considered with Eµp. In the nonparabolicity correction, the anisotropy is
considered through the ground subband energy that is well approximated by the
2mtml /(mt + ml) [104]. Figure 4.2(a) shows that the E-k dispersions are reasonably
represented with the adopted nonparabolic correction in cylindrical [100] Si NWs as
tox
d
SiO2
NW
Metal gate
Figure 4.1: Schematic of a cylindrical NW MOSFET.
42
well as in square-well [93]. In InAs NWs, the nonparabolic correction gives good
approximation as shown in Figure 4.2(b). Here, the nonparabolic transport effective
mass at the minimum of the lowest subband is derived from [1 + 2α(Eµ – Uµ)]mtrans,
where Eµ is the minimum of subband µ derived from the nonparabolic EMA.
4.2.2 Top-of-the-Barrier Ballistic Transport Model
With assuming ideal gate control to focus on intrinsic diameter dependences without
SCEs such as drain-induced barrier lowering, we could take into account the subband
structure and electrostatics only at the top of the barrier. Equations (2.70) and (2.71)
give the distribution functions at the top of the barrier in term of the ballistic transport
regime. Hence, states of dE / dk ≥ 0 and dE / dk ≤ 0 feed forward and backward
currents, where E and k are the energy eigenvalue and wave vector, respectively; the
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8Diameter (nm)
Tran
spor
teffe
ctiv
e m
ass
(m0)
Diameter (nm)Tr
ansp
ort e
ffect
ive
mas
s (m
0)
(a) [100] Si NW (b) InAs NW
TB modelNonparabolicParabolic
TB modelNonparabolicParabolic
Figure 4.2: Nonparabolic transport effective mass at the minimum of the lowest subband in (a) [100] Siand (b) InAs NWs. Solid lines indicate the results from the EMA with the nonparabolic correction, andopen symbols indicate the results from TB model for [100] Si NWs in [32] and for [111] InAs NWs in [103]. The nonparabolic correction could give a reasonable E-k dispersion of the lowest subband. (Cox = 2.45 µF/cm2, Vd = 0 V, Vg = 0 V).
43
former and latter states are occupied according to source Fermi level, Efs, and drain
Fermi level, Efd, respectively, as schematically shown in Figure 4.3. The Efd is
described by Efd = Efs – qVd, where Vd is the drain voltage. According to this
distribution, the subband structures and electrostatics at the top of the barrier were
calculated with a self-consistent solution of the Schrödinger and Poisson equations as
shown in appendix A.
Ren et al. [105] showed that device operation in the semiclassical ballistic transport
model is in good agreement with that in the quantum ballistic transport model until
channel length longer than 10 nm.
4.3 Results and Discussion
To compare the performance in various diameters, we fixed the Cox and Vdd, where
Vdd is the power supply voltage. Because the Cox with fixed tox increases with
shrinking diameter owing to the increase in curvature, we adjusted tox to keep the
Source
Drainz
ChannelE
Efs
k
E
Efd
Figure 4.3: Concept of the top-of-the-barrier semicalssical ballistic transport model. Carrier distribution with positive velocity (dE / dk > 0) is described by source Fermi-Dirac distribution, and carrier distribution with negative velocity (dE / dk < 0) is described by drain Fermi-Dirac distribution.
44
Cox = 3.45 µF/cm2 as shown in Figure 4.4(a). On the basis of (3.3), the adjusted tox is
45
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 300
1
2
3
4
5
6
00.10.20.30.40.50.60.70.80.9
1
0 10 20 30 40 50
-5-4-3-2-101
0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2-5-4-3-2-10
00.20.40.60.81
Diameter (nm)
Off-
stat
e ga
te v
olta
ge, V
off(V
)
InAs NW
[100] Si NW
(b)
d=1.5 nmd=12nm
Vg – Voff (V)
[100] Si NW
InAs NW
I d(m
A/µ
m)
Log(
I d) (m
A/µm
)
I d(m
A/µ
m)
Log(
I d) (m
A/µm
)
d=2nmd=30nm
20
1-5
(c)
Diameter (nm)
Cox
with
fixe
d t ox
(µF/
cm2 )
Adj
uste
d t ox
for f
ixin
g C
ox(n
m)
tox = 1 nm
Cox = 3.45 µF/cm2
(Parallel metal plateswith 1-nm-thick SiO2)(a)
Figure 4.4: (a) Adjusted tox is required for fixing Cox because Cox with fixed tox depends on diameter in the cylindrical capacitor. (b) Voff as a function of diameter. The Voff denotes the Vg when drain-current per unit wire periphery is 100 nA/µm. (c) Id-Vg characteristics. Threshold voltages are set to (Voff + 0.15 V). (Cox = 3.45 µF/cm2, Vd = 0.5 V).
46
calculated by
⎥⎦
⎤⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛= 1exp
ox
0oxox RC
Rt εε . (4.2)
The Cox corresponds to the capacitance of parallel metal plates with 1-nm-thick SiO2.
Figure 4.4(b) shows the off-state gate voltage, Voff, where the Voff is the Vg when the
drain-current per unit wire periphery, Id, is 100 nA/µm. Here, Vfb is 0 V, and the
intrinsic Fermi level is fixed by (Ec – 0.56 eV), where Ec is the conduction band
minimum of bulk Si. The diameter-dependent Voff implies the threshold voltage shift
due to quantum confinement [37]. Note that the off-state is the state with Vd = Vdd and
Vg = Voff, and the on-state is the state of Vd = Vdd and Vg = Von ≡ Voff + Vdd.
The threshold voltage, Vt, is set to (Voff + 0.15 V), which was derived from the
second derivative method [106] in both Si and InAs NW MOSFETs. Although the Vt
from the second derivative method was slightly deviated from (Voff + 0.15 V) in
diameter smaller than 10 nm, the Vt of (Voff + 0.15 V) is reasonable in the viewpoint
of the constant current method [106] as shown in Figure 4.4(c).
Figure 4.5 shows diameter dependences of drain current, Id, effective gate
capacitance, CG, and injection velocity, υinj, at on-state in [100] Si and InAs NW
MOSFETs. The υinj is calculated as the average velocity over all carriers injected
from the source to the channel. The Vd larger than (Vg – Vt) is enough to saturate Id not
only in diffusive transport regime but also in ballistic transport regime [90]. The
on-current normalized by periphery, Ion, is proportional to the product of υinj and CG
as described in
)]([ offtddGinjon VVVCI −−= υ , (4.3)
where υinj and CG are the values at on-state. The diameter-dependent CG has been
47
already shown in Figure 3.3. Even in extremely small diameter, Ion of InAs NWs is
48
Diameter (nm)
00.20.40.60.8
11.21.41.61.8
0 10 20 30 40 50
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Drain current, Id(mA/m)
InAs NW
[100] Si NW(a)
Effective gate capacitance,
CG(F/cm
2)
InAs NW
[100] Si NW
(b)
@ on-state
Injection velocity, inj(10
7cm/s)
@ on-state
3
4
5
6
0 10 20 30 40 50
1
1.1
1.2
1.3
0 10 20 30 40 50
@ on-state
InAs NW
Diameter (nm)
[100] Si NW(c)
31.3
Diameter (nm)
Figure 4.5: (a) On-current as a function of diameter. (b) Effective total gate capacitance at on-state as a function of diameter, where the overdrive gate voltage is 0.35 V. (c) Injection velocity at on-state as a function of diameter. (Cox = 3.45 µF/cm2, Vdd = 0.5 V).
49
still larger than that of [100] Si NWs owing to the much higher υinj of InAs NWs for
all diameters despite smaller CG.
In Si NWs, Ion increases with shrinking diameter until υinj reaches a peak, whereas
in InAs NWs, Ion gradually decreases with shrinking diameter, but as some point
below 5 nm it drastically increases. The different tendency of the diameter
dependences in Ion is due to the fact that the small effective mass of the InAs NW
seriously affects the diameter-dependent CG through the quantum capacitance. In Si
NWs, the CG monotonically increases with shrinking diameter; whereas, in InAs NWs,
the CG gradually decreases then increases with shrinking diameter. On the other hand,
υinj gradually increases with shrinking diameter and reaches a peak at diameter around
5 nm in both NWs. Note that the diameter with the highest υinj was decreased with
increasing Cox(Vg – Vt).
To interpret the diameter dependence of the υinj, we analyzed the three factors
affecting the υinj:
Modulation of the transport effective mass;
Degree of carrier degeneracy;
Carrier occupancy ratio for subbands with different effective masses.
Although the transport effective mass monotonically increases with shrinking
diameter as shown in Figure 4.2, the υinj increases until a certain diameter in both Si
and InAs NWs is reached, as shown in Figure 4.5(c).
The increasing υinj can be explained with the second factor, the degree of carrier
degeneracy. Appendix D shows the relation between the carrier degeneracy and the
injection velocity. If we assume that E1t hardly depends on diameter, the CG / CDOS
can be a metric of carrier degeneracy for the lowest subband since it reflects the shift
50
of the lowest subband minimum as described in
)( tg
1t
1
DOS
G
VVqEE
CC
−−
= . (4.4)
Figure 4.6(a) shows that the effective degree of the carrier degeneracy for the lowest
subband increases with shrinking diameter, and it reaches a peak. Figure 3.4 shows
00.10.20.30.40.50.60.70.80.9
0 10 20 30 40 50
(a)
InAs NW
[100] Si NW
CG
/ CD
OS
@ on-state
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0 10 20 30 40 50Diameter (nm)
Eµ
–E f
s(e
V)
[100] Si NW (b)
E2 – Efs
E1 – Efs
E2 – EfsE1 – Efs
E1’ – Efs
InAs NW
@ on-state
Diameter (nm)
Figure 4.6: (a) Diameter-dependent CG / CDOS, which is a metric of carrier degeneracy for the lowest subband. (b) Diameter-dependent Eµ – Efs (Eµ’ – Efs), which gives the degree of the carrier degeneracyfor each subband. The E1’ is the minimum of the lowest primed subband. Carrier degeneracy ismaximized when the second lowest subband minimum is around Efs. (Cox = 3.45 µF/cm2, Vdd = 0.5 V).
51
that the increase in the CG / CDOS with shrinking diameter is due to the increase in the
CE and the decrease in the CDOS although the CDOS in Si NWs with diameter larger
than 15 nm gradually increases.
The CG / CDOS does not directly represent the degree of the carrier degeneracy
because of the modulation of the DOS. Nevertheless, Figure 4.6 shows that the degree
of the carrier degeneracy for the lowest subband, (Efs – E1), is closely related to the
CG / CDOS in the both Si and InAs NWs. Figure 4.6(b) also shows that, eventually,
electrons do not occupy the second subband. When the electrons occupying the
second lowest subband are almost vanished, the diameter dependence of υinj peaks as
shown in Figure 4.3(c). The peak structure of the injection velocity, because of the
coincidence of the peak carrier degeneracy for the lowest subband and the empty
second lowest subband, would be the universal feature regardless of the channel
material. In Si NWs, the increase in the occupancy ratio of the unprimed subbands
also helps with increasing the υinj. Neophytou et al. [38] have shown the peak
structures of the diameter-dependent υinj in Si NW MOSFETs despite the modulation
of the effective masses.
Finally, Figure 4.7 shows the diameter-dependent intrinsic gate delay, τ, and power
delay product, P･τ, where P･τ is normalized with wire periphery. As calculated in
[99], [100], [101], the τ and P･τ are estimated with
ddon
ggiddoff
off
)(
VI
dVLQVV
V∫+
−=τ , (4.5)
∫+
−=⋅ ddoff
offggi )(
VV
VdVLQP τ ,
(4.6)
respectively, where the gate length, Lg, is set to 14 nm. Although Khayer and Lake
[101] have already shown the diameter dependences of the τ and P･τ in the condition
52
of the constant Efs – E1 with respect to InAs NW MOSFETs, our approach is more
practical to adjust diameter. The τ is inversely proportional to the υinj in Si NWs. On
the other hand, the relation between the υinj and τ is ambiguous in InAs NWs because
the diameter-dependent DOS affects not only the υinj but also the CG through the CQ.
With respect to the intrinsic delay time, the best performance is obtained at a diameter
with the highest υinj in Si NWs, whereas the best performance is obtained at the
diameter slightly larger than that with the highest υinj in InAs NWs. The diameter
353739
4143
0 10 20 30 40 50
7
9
11
13
0 10 20 30 40 50
InAs NW
[100] Si NW
Intri
nsic
gat
e de
lay,
τ(fs
)3513
(a)
02468
101214161820
0 10 20 30 40 50Diameter (nm)
Pow
er d
elay
pro
duct
, P･τ
(10-
20J/
nm)
InAs NW
[100] Si NW
(b)
Figure 4.7: (a) Intrinsic gate delay as a function of diameter. (b) Power delay product as a function ofdiameter. The power delay product is normalized by wire periphery. (Cox = 3.45 µF/cm2, Vdd = 0.5 V, Lg = 14 nm).
53
dependence of the P･τ is roughly similar to that of CG. If we consider both high
performance and low power dissipation, then diameter around 5 nm and 10 nm are
desirable in Si NWs and InAs NWs, respectively. If a higher Ion, however, was
necessary, then a smaller diameter would be desirable.
4.4 Conclusions
Based on the gate capacitance model developed in the previous chapter, we
investigated diameter-dependent performance of Si and InAs NW MOSFETs. To
calculate the electrical characteristics, we used the combination of the nonparabolic
EMA and semiclassical ballistic transport model. Performance of a
large-effective-mass NW like the Si NW depends on the injection velocity, which
generally increases as shrinking diameter; on the other hand, that of a
small-effective-mass NW like the InAs NW is insensitive to the injection velocity. As
a result, we could figure out that a desirable diameter that achieves both low intrinsic
gate delay and low power dissipation was around 5 nm for Si NWs and 10 nm for
InAs NWs. Finally, our results also imply that the optimal performance is
accompanied with strong volume inversion.
54
Chapter 5
Band Structure Effect on Electrical
Characteristics of Silicon Nanowire
MOSFETs with the First-Principles
Calculation
5.1 Introduction
It was reported that electrical characteristics of an ultrathin body transistor made on a
silicon-on-insulator (SOI) wafer strongly depend on subband modulation by quantum
confinement [107]. Si NWs also have subband structures owing to confinement within
the width, and so the confinement influences electrical characteristics in a Si NW
MOSFET. Practically, derivation of the confined electronic structures by the EMA is
very attractive from computational point of view. However, EMA just gives us a
rough outline when external potential rapidly varies compared to the period of atoms
55
or the periodicity is lost by confinement. Therefore, atomistic calculations are
necessary for Si NWs with strong confinement. Several works, which have been done
based on the tight-binding method [31], [32], [33], [34], [35] and the first-principles
calculation [36], [37], have shown the width-dependent subband structures. They also
showed that the EMA had to be corrected in small nanowires.
One of the prospects of Si NW MOSFETs is a large drive current by the ballistic
transport [108]. Si NW MOSFETs can suppress SCE even in an ultra-short channel
where ballistic transport can be obtained. Attempts to calculate the electrical
characteristics of the ballistic Si NW MOSFETs have been reported so far [32], [33],
[34], [35], [36], [37], [109].It has been revealed that the electrical characteristics
strongly depend on the wire direction and the width of the wire. The direction of wire
induces changes in the subband structure, whereas the width of the wire changes both
the subband structure and the gate oxide capacitance.
In this chapter, we mainly analyze the size-dependent potential performance of
ballistic Si NW MOSFETs aligned along [100] direction. The subband structures of Si
NWs were derived by the first-principles calculation [80], [82], which is the most
refined method now available. The electrical characteristics are calculated by a
compact model [109], [110] for the ballistic NW MOSFET. The compact model is
very useful to analyze the factors determining the on-current because the
size-dependent subband structures can be directly handled, and it is one of the
highlights in this work. We discuss the size-dependent subband structure of Si NWs.
The derived subband structures were compared with other works. On the basis of the
obtained band structures, we also assess the on-current of Si NW MOSFETs under
ballistic transport.
56
5.2 Simulation Methods
From the calculated band structures by the first-principles calculation, ballistic
transport characteristics of the Si NW MOSFET were derived on the basis of a
compact model in [109] and [110]. The compact model is based on the
top-of-the-barrier ballistic transport model.
5.2.1 First-principles band structure calculation
The band structures of SiNWs were calculated by the first-principles calculation on
the basis of the DFT with the LDA using pseudopotential [80], [82]. All the band
calculations were performed with Tokyo Ab-initio Program Package (TAPP) [86]. The
adopted models of the Si NW are those with channel aligned along the [100] direction
([100] Si NWs). The dangling bonds of Si atoms at the periphery were passivated by
hydrogen atoms. The pseudopotentials were made by setting the cutoff radii of wave
functions of silicon and hydrogen atoms at 2.2 and 0.7 a.u., respectively. To ensure the
periodic boundary condition, supercells with neighboring wires separated by 0.7 nm
were adopted. It was confirmed that wires with 0.7 nm separation were sufficient to
eliminate the interaction between the neighboring wires. Brillouin zone integration to
evaluate the total energy was calculated by two sample k-points. The cutoff energy of
the plane-wave basis set was 12.25 Ry. Figure 5.1 and Table 5.1 show that the
quantities under 12.25 Ry exhibited small errors although the adopted cutoff energy is
not large enough to obtain excellent convergence. Atomic reconstruction was not
carried out.
57
90
92
94
96
98
100
0 10 20 30 40
5.3755.38
5.3855.39
5.3955.4
5.4055.41
5.4155.42
0 10 20 30 400.5
0.52
0.540.56
0.58
0.6
0.62
0.64
0 10 20 30 40
0.9
0.910.920.93
0.94
0.950.96
0.97
0 10 20 30 40
-216.1-216
-215.9-215.8-215.7-215.6-215.5-215.4-215.3-215.2
0 10 20 30 40
Total energy (eV)
Cutoff energy (Ry)
Bulk modulus (GPa)
Cutoff energy (Ry)
Lattice constant ()
Cutoff energy (Ry)
Band gap (eV)
Cutoff energy (Ry)
Longitudinal effective mass (m0)
Cutoff energy (Ry)
(b)
(c) (d)
(e)
(a)
Figure 5.1: Convergence check at the cutoff energy of 12.25 Ry for (a) total energy, (b) bulk modulus,(c) lattice constant, (d) band gap, and (e) longitudinal effective mass along Γ–X in bulk Si.
58
5.2.2 Compact model of ballistic nanowire MOSFETs
Based on the top-of-the-barrier ballistic transport model introduced in the subsection
4.2.2, the drain current per single-wire, Id is evaluated as
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
−+⎟⎟⎠
⎞⎜⎜⎝
⎛=
i Bifd
Bifsi
Bd TkEE
TkEEg
qTk
GI]/)exp[(1]/)exp[(1
ln0,
(5.1)
where the total current is the sum of carrier flows in each subband i. A numerator and
a denominator in natural logarithm indicate elements of the forward and backward
currents, respectively. The G0 denotes the quantum conductance of 77.8 µS. The gi
and Ei denotes the degeneracy of the subband and the minimum eigenvalue of the i-th
subband. Equation (5.1) has been simplified by neglecting subband maxima which are
much higher than Efs. When the Vd, gate overdrive (Vg – Vt), and linear gate
capacitance per unit channel length, Cox, are given, the Efs in (5.1) can be calculated
by
)(11 1
qEE
VVQCC
fstg
centroidox
−−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛+ ,
(5.2)
where |Q| denotes the linear charge density per unit channel length, and quantum
capacitance Cq is considered by the second term on the right-hand side and the second
TABLE 5.1
CONVERGENCE OF THE CUTOFF ENERGY OF 12.25 RY
Quantity Discrepancy from
cutoff energy of 36 Ry
Discrepancy from
known value Known value
Total energy 0.1%
Bulk modulus 2.4% 7.6% 101.97 GPa
Lattice constant 0.4% 0.5% 5.43 Å
Band gapa 4.3% 1.12 eV
Longitudinal Effective mass 0.8% 2.7% 0.916 m0 aThe DFT with the LDA does not represent the real band gap.
59
term of the bracket on the left-hand side and. For an estimation of the gate capacitance
with square cross-section, we adopt an approximate model of [111]. In the rectangular
cross section, the Cox is approximated as
Figure 5.2: (a) Id-Vd and (b) Id-Vg characteristics of a Si NW MOSFET from the compact model. Thecompact gives reasonable I-V characteristics. (d = 2.69 nm, Cox = 3.45 µF/cm2).
60
⎟⎠⎞
⎜⎝⎛ +
=
wt
C ox
ox
0ox
451ln
5 εε,
(5.3)
where tox denotes an insulator thickness, and εox denotes dielectric constant of the
insulator. ε0 is the vacuum permittivity. The |Q| can also be described as
∑ ∫Γ
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
+
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
=i
Z
B
fdi
B
fsii dk
TkEkE
TkEkE
gqQ )(
exp1
1)(
exp1
1π
. (5.4)
Integrating the Fermi distribution function associated with each subband within half
Brillouin zone, total charge can be obtained. The (Efs – E1) is derived by solving (5.2)
and (5.4), simultaneously. Finally, substituting the obtained (Efs – E1) into (5.1), we
can evaluate the Id.
Figure 5.2 shows an example of calculated Id-Vd characteristics of a [100] SiNW
with w of 2.69 nm, for various gate overdrives (Vg – Vt = 0.1, 0.4, 0.7 and 1.0 V) at a
room temperature (T = 300 K). A SiO2 with a thickness tox of 1 nm is adopted as the
gate insulator. Note that Id denotes the drain current normalized by the wire periphery.
5.3 Results and Discussion
Firstly, we adopted [100] Si NWs with circular cross sections as shown in Figure 5.3.
The curvature variation is no longer negligible in substantially small diameter. In
diameter smaller than 2 nm, we adopted every possible circular Si NWs with the
center of a Si atom. Figure 5.4 shows that effective mass of the lowest unprimed
subband widely fluctuates.
To try to avoid the fluctuation of the effective mass, we adopted square
61
cross-sectional shape with (110)-orientated surfaces as schematically shown in
Figure 5.5. The figure shows cross sections of the modeled Si NWs with width, w, of
0.77 and 2.69 nm, which were also called 5×5 and 15×15 in [37], from the number
of cross-sectional atoms. Figure 5.6 shows subband structures of Si NWs with w from
0.77 to 2.69 nm. While the band structure of bulk silicon has 6-fold degenerate
d ~ 2 nm
Figure 5.3: Cross-sectional atomic array of the 2-nm-diameter [100] Si NW with circular cross section. The curvature variation is no longer negligible in substantially small diameter. Large atoms are siliconand circumferential small atoms are hydrogen. 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6
Effe
ctiv
e m
ass
(m0)
Diameter (nm)
Nonparabolic EMA
First-principles calculation
Figure 5.4: Effective mass of the lowest unprimed subband as a function diameter. In diameterssmaller than 2 nm, we adopted every possible circular Si NWs with the center of a Si atom. The resultfrom the nonparabolic EMA has been calculated in the previous chapter. 3
62
conduction band minima (CBM) at ∆-valleys along the line from Γ to X, subband
minima of the [100] Si NW consist of four unprimed subband minima at Γ-valley and
two primed subbands at ∆-valleys. In subband structures of quantum-confined SiNW,
unprimed subband minima are lower than primed subband minima because the
quantization mass of the unprimed subband is larger than that of the primed subband.
Splitting in unprimed subband group at Γ point is observed in sufficiently thin wires,
whereas the subband structure in a sufficiently large wire is in 4-fold degeneracy. For
example, the Si NW with w of 0.77 nm has two 1-fold and one 2-fold degenerate
lowest unprimed subbands, whereas the Si NW with w of 2.69 nm has closer 1-fold
and 3-fold degenerate lowest unprimed subbands. The split increases in smaller wires,
and the upper unprimed subbands also have intense splitting.
w = 2.69 nmw = 0.77 nm
(110)
(110)(1
10)
(110
)
Figure 5.5: Cross-sectional atomic arrays of square Si NWs with [100]-directed channel and (110)-oriented surface. Large atoms are silicon and circumferential small atoms are hydrogen.
63
Ene
rgy
(eV
)
0
0.5
w = 0.77 nm
ZΓ Wave number
(a)
Ene
rgy
(eV
)
0
0.5
w = 1.15 nm
ZΓ Wave vector
(b)
w = 1.54 nm w = 1.92 nm
Ene
rgy
(eV
)
0
0.5
ZΓ Wave vector
(c)
Ene
rgy
(eV
)
0
0.5
ZΓ Wave vector
(d)
w = 2.30 nm w = 2.69 nm
Ene
rgy
(eV
)
0
0.5
ZΓ Wave vector
(e)
Ene
rgy
(eV
)
0
0.5
ZΓ Wave vector
(f)
Figure 5.6: Calculated band structures of Si NWs with various widths. In small width, valley splittingof four-fold degenerate unprimed subabnds is observed.
64
Figure 5.7 shows width dependence of band gap. Band gaps evaluated in the DFT
using the LDA are usually underestimated compared with the experimental values
[112], [113]. However, the modulation of the value predicted by the calculation for
the varied structure is generally believed to be reliable [114]. The bandgap of the Si
NW becomes wide as width decreases because of quantum confinement. Therefore,
with the increase in width, the bandgaps approach close to the bulk silicon. Figure 5.8
shows the width dependence at the electron effective masses for the lowest unprimed
and the lowest primed subbands in [100] Si NWs. An electron effective masse m* is
estimated using an approximation of eigenvalues E at band edges as described by
2
2
2*11
kE
m ∂∂
≈h
, (5.5)
where k and h are the wave vector and the reduced Planck’s constant, respectively.
For sufficiently large diameter of wires, one can expect that the effective mass moves
to 0.23 m0 for the unprimed subband and to 0.95 m0 for the primed subband, and they
0
0.5
1
1.5
2
2.5
3
0 1 2 3
Ban
d ga
p (e
V)
Width (nm)
Bulk band gap
Figure 5.7: Band gap as a function of width. Strong quantum confinement broadens band gap. Dottedline is the bulk band gap from the first-principles calculation. Although the result from the DFT withthe LDA does not give valid value of the band gap, tendency could be a good guide.
65
correspond to the transverse and the longitudinal effective masses at the band
minimum of bulk Si by the first-principles calculation, respectively. Here, m0 is the
electron mass. The first-principles calculation overestimates the effective masses of
the bulk Si, where known values of transverse and longitudinal effective masses are
0.19 and 0.916 m0, respectively. The effective masses consistently increase as
shrinking width both in the lowest unprimed and primed subbands. Size-dependence
of the subband structure basically agrees with those in previous works [31], [32], [33],
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4
Effe
ctiv
e m
ass
(m0)
Width (nm)
Symbols: [100] Si NW
Lowest primed subband
Lowest unprimed subband
Dotted lines: Bulk Si
Figure 5.8: Calculated effective masses of the lowest unprimed subband and the lowest primedsubband. The bulk effective masses from the first-principles calculation are overestimated from the known values, which are 0.19 and 0.916 m0 with transverse and longitudinal effective masses in bulk Si, respectively.
66
[34], [35], [36], [37]. Note that the wide fluctuation of the effective mass is
suppressed. We adopted square NWs to estimate size-dependent electrical
characteristics.
When a sufficiently large Vd is applied, the MOSFET enters into saturation region,
where total current consists of only forward current. From the Landauer’s formula,
ballistic Id in the saturation region can be approximately estimated by the sum of
(Efs – Ei) of each subband as described in (5.1), where Ei denotes i-th subband
minimum. Thus, size dependence of Ei will be discussed henceforth. Conventionally,
the saturation current is also expressed as
)( tgGinjd VVCI −=υ , (5.6)
where υinj denotes an average injection velocity, and CG denotes the effective total
gate capacitance per unit wire surface. From (5.2), the CG can be derived by
)(11)( 11
⎥⎦
⎤⎢⎣
⎡ −−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=−
−
qEE
VVCC
VVC fstg
centroidoxtgG
, (5.7)
where E1 denotes the lowest subband minimum. According to (5.7), (Efs – E1) and
Ccentroid degrade CG relative to the gate oxide capacitance Cox. These effects can be
regarded as the quantum effects. To investigate origin of the size-dependent
performance of SiNW-FETs, we analyze size dependence of the CG and the υinj, which
govern the Id. By analysis of the (Efs – Ei), we can also examine the subbands’ effect
on capacitance and injection velocity. Finally, in the following, all the size-dependent
parameters were calculated with the bias condition (Vg – Vt = 0.35 V) for on-current
derivation and Vd values large enough for saturation. The 1-nm-thick SiO2 was
adopted as a gate insulator. Temperature was set to 300 K.
Figure 5.9 shows width dependences of drain current, effective gate capacitance,
and injection velocity. Results of the nonparabolic EMA correspond to the cylindrical
67
Si NW MOSFETs with the same periphery, where the electrostatic capacitance per
68
0
0.2
0.4
0.6
0.8
1
0 2 4 6
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6
0
0.2
0.40.6
0.8
1
1.2
1.4
0 2 4 6
Nonparabolic EMA
First-principles calculation
Nonparabolic EMA
First-principles calculation
Nonparabolic EMA
First-principles calculation
WIdth (nm)
WIdth (nm)
WIdth (nm)
Drain current, Id(mA/m)
Effective gate capacitance,
CG(F/cm
2)
Injection velocity, inj(10
7cm/s)
(a)
(b)
(c)
Figure 5.9: Width dependences of (a) drain current, (b) effective gate capacitance, and (c) injectionvelocity. Results of the nonparabolic EMA correspond to the cylindrical Si NW MOSFETs with thesame periphery, where the electrostatic capacitance per unit wire surface holds. (Cox = 3.45 µF/cm2, Vg – Vt = 0.35 V, Vd = 0.5 V).
69
unit wire surface holds. The compact model overestimates effective gate capacitance
because threshold voltage was overestimated by about 0.8 V. If atomic array is
symmetric, the nonparabolic EMA could be good approximation:
･ Although the values are smaller than nonparabolic EMA, even bulk effective
mass is overestimated with the first-principles calculation;
･ Injection velocity degrades as in nonparabolic EMA.
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3WIdth (nm)
Subbandlevel, E–Efs(eV)
Lowest two primed subband
Lowest four unprimed subbands
Solid symbols: Two-foldOpen symbols: One-fold
Figure 5.10: Width dependences of subband minima of the lowest four unprimed subband and thelowest two primed subband based on the source Fermi level. (Cox = 3.45 µF/cm2, Vg – Vt = 0.35 V, Vd= 0.5 V).
70
Size-dependences of subband minima are shown in Figure 5.10. Carrier degeneracy of
most subbands degrade as shrinking diameter. Therefore, the injection velocity is
degraded.
5.4 Conclusions
We estimated electrical characteristics of ballistic NW MOSFETs by a combination of
the first-principles band calculation and a compact model for electrostatics. In
substantially small [100] Si NW nMOSFETs, we should carefully take into account
the electronic structure because, in small cylindrical Si NWs, effective mass widely
fluctuated with curvature variation. Finally, if the cross section of the NW is
rectangular, the wide fluctuation of the effective mass can be suppressed, where the
injection velocity was degraded as in nonparabolic EMA even in substantially small
thickness.
71
Chapter 6
Size and Corner Effects on Electron
Mobility of Rectangular Silicon
Nanowire MOSFETs
6.1 Introduction
As a useful metric to estimate the performance of MOSFETs, the low-field mobility
has been actively investigated with fabricated Si NW MOSFETs [8], [10], [12], [15],
[16], [39], [40], [41], [42], [43]. A number of computational studies have also
investigated the low-field mobility of NW MOSFETs [44], [45], [46], [47], [48], [49],
[50]. Sakaki [44] reported that the GaAs NW shows high mobility at low temperature
owing to the suppression of Coulomb scattering by reduced density of states. In small
Si NW MOSFETs, unfortunately, the benefit of the reduced density of states is
eliminated by the increase in electron-phonon wave function overlap for the phonon
scattering [45]. Nevertheless, the mobility was enhanced in several fabricated Si NW
72
MOSFETs [8], [10], [39], [40], [42]. Koo et al. [39] and Sekaric et al. [42] suggested
that the mobility enhancement in small Si NW MOSFETs would be due to
stack-induced stress. As well as the size-dependent mobility, the cross-sectional
electrostatics in Si NW MOSFETs has been focused on [40], [115], [116], [117],
[118]. Fossum et al. [116] and Poljak et al. [118] tried to suppress the corner
components of current, increasing off-state current. On the other hand, Moselund et al.
[40] suggested that the local volume inversion at the corner would cause the mobility
enhancement under on-state.
In this chapter, to elucidate the size and corner effects on the mobility, we
investigated the cross-sectional distribution of the low-field phonon-limited electron
mobility in rectangular Si NW MOSFETs with various orientations. To calculate the
phonon-limited mobility, we employed the Kubo-Greenwood formula [94], [95].To
derive the spatially resolved mobility, we took into account the subband composition
for local electrons. Using the subband composition, we could extract the contribution
of each subband to the local mobility. The obtained spatially resolved mobility
showed that the corner mobility was lower than the side mobility; therefore, the
corner component was not advantageous for the phonon-limited mobility unless we
consider other effects such as the strain effect.
6.2 Simulation Methods
Adopting a parabolic effective mass approximation (EMA), we calculated the
subband structure and electrostatics in the rectangular Si NW MOSFET by the
self-consistent solution of two-dimensional Schrödinger and Poisson equations. The
differential equations were solved by the finite difference method. For the calculations
of the low-field mobility and the momentum relaxation times due to phonon scattering
73
mechanisms, we referred to papers of Kotlyar et al. [45] and Jin et al. [52]. Here, we
describe several equations to derive and discuss our results. The total mobility, µtot,
which is associated with all subbands, is described in (6.1),
∑∑
=
µµ
µµµµ
µn
n
tot , (6.1)
where the subband index, µ, consists of valley index and principle quantum number,
µµ is the mobility for subband µ, and nµ is the number of electrons occupying subband
µ. The nµ is described in (6.2),
∫∞
=µ
µµ ρE
dEEfEn )()( , (6.2)
where f(E) is the Fermi-Dirac distribution function, ρµ(E) is the density of states for
subband µ, and Eµ is subband energy level, which denotes the minimum energy of
subband µ. As shown in subsection 2.3.2, the low-field mobility for subband µ is
derived by Kubo-Greenwood formula [94], [95]:
∫∞
−=µ
µµµµ
µ ρυτµE
B
dEEfEfEEETkn
q )](1)[()()()( 002 , (6.3)
where q, kB, and T are the elementary charge, the Boltzmann’s constant, and
temperature, respectively, and τµ(E) and υµ(E) are momentum relaxation time and
group velocity for subband µ [45], [52]. In the EMA, density of states and group
velocity are described in (2.59) and (2.60).
The total momentum relaxation time, τµ(E), due to phonon scattering mechanisms
is described in (6.4),
∑=
+=6
1ac )(
1)(
1)(
1j
j EEE µµµ τττ, (6.4)
where τµac(E) and τµ
j(E) are the momentum relaxation times due to elastic acoustic
74
phonon scattering and six-type inelastic phonon scattering mechanisms, respectively.
Here, the six-type inelastic phonon scattering mechanisms involve g- and f-type
transitions of transverse acoustic, longitudinal acoustic, and longitudinal optical
phonon modes [66]. The acµτ and j
µτ are described in (6.5) and (6.6),
∑′
′′=µ
µµµηηυµ
ρδρ
πτ i
l
B FEugTkΞ
E ,,2Si
2
ac )()(
1h
, (6.5)
∑′
′′′ ⎟⎠⎞
⎜⎝⎛ +
−±−
±=µ
µµµηηυµ
ωωρ
ωρπ
τ 21
21
)(1)(1
)(2
)()(
1,,
Si
2
mh
h jj
jj
j
jtj N
EfEf
FEgg
KDE
, (6.6)
where gυ is the valley degeneracy, ρSi is the density of Si, ul is the sound velocity, Ξ
and (DtK)j are the deformation potentials, Nj and jωh are the phonon number and
energy, η is the valley index, δη,η’ is the Kronecker delta, and jg ηη ′, is δη,η’ for g-type
process and 2(1 − δη,η’) for f-type process [45], [52]. Fµ ,µ’ is the overlap factor
described in (6.7),
∫∫ ′′ = dxdyyxyxF 22, |),(||),(| µµµµ σσ , (6.7)
where σµ(x, y) is the transverse part of the envelope wave function for subband µ.
Cross-sectional spatially resolved local mobility, µlocal(x, y), can be calculated by
(6.8),
),(
|),(|),(
2
local yxn
nyxyx
∑= µ
µµµ µσµ , (6.8)
where n(x, y) is the local electron density described in (6.9),
∑=µ
µµσ nyxyxn2
),(),( . (6.9)
Using the concept of the local density of states, which is denoted by |σµ(x, y)|2ρµ(E),
and assuming spatially independent τµ(E), we could derive the spatially resolved
mobility as described in (6.8). The spatially independent τµ(E) is acceptable because
75
the matrix elements of scattering potentials are already integrated over the space. In
other words, the µµ is spatially independent. Therefore, we can interpret the spatially
resolved mobility by taking into account spatially different contribution of each
subband.
6.3 Results and Discussion
Figure 6.1 shows that a schematic model of a rectangular NW MOSFET. To calculate
the subband structures of Si NWs with the EMA, we considered three two-fold
degenerate valleys with anisotropic effective masses. We adopted intrinsic body and
the gate insulator of 1-nm-thick SiO2. The band offset of the Si and SiO2 conduction
bands corresponded to 3.15 eV, and the electron effective mass in the insulator was set
to 0.5 m0. We assumed the Si and SiO2 permittivities as 11.8 ε0 and 3.9 ε0, respectively.
As parameters to calculate the mobility, we used a gate voltage of 1 V and the room
temperature of 300 K. The mobility was compared with the constant electron density
of 0.8×1013 /cm2. The parameters for the intravalley acoustic phonon scattering rate in
(6.5) are shown in Table 6.1, where the Ξ in (6.5) was set to 14.6 eV [52]. The
parameters for the intervalley phonon scattering rate in (6.6) are shown in Table 6.2
[66].
76
x
y
z
GateSiO2
Si NW
w
h
Figure 6.1: Schematic model of a rectangular NW MOSFET.
TABLE 6.1
PARAMETERS FOR INTRAVALLEY ACOUSTIC PHONON SCATTERING
ρ ul Ξ
2.33 g/cm3 9×105 cm/s 14.6 eV
TABLE 6.2
PARAMETERS FOR INTERVALLEY PHONON SCATTERING
j Mode (DtK)j (108 eV/cm) ħωj (meV) Selection rule
1 TA 0.5 12 g
2 LA 0.8 19 g
3 LO 11 62 g
4 TA 0.3 19 f
5 LA 2 47 f
6 TO 2 59 f
77
6.3.1 Size and Orientation Effects
We evaluated size dependence of phonon-scattering-limited electron mobility, where
the side surface height, h, was fixed as 10 nm and the top surface width, w, was
adjusted from 14 nm to 1 nm. Figure 6.2 shows schematic models of adopted Si NWs
with various directions and orientations; where [100] and [110] are NW directions,
and (100) and (110) are orientations of wafer. Note that the direction and the
orientation are represented by the general notations according to symmetry. To take
into account arbitrary direction, effective mass tensor needs to be re-expressed in the
device coordinate system (x, y, z) where the transport direction is along z axis, and the
confinement occurs in x-y plane [119]. According to Bescond et al. [119], the 3D
Schrödinger equation can be reduced to
),(),(),(22 trans
222
2
2
2
22
yxEyxyxUm
kyxyx
xxyyyxx σσωωω =⎥
⎦
⎤⎢⎣
⎡+−⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
+∂∂
−hh . (6.10)
The 3D envelope wave function, χ, can be expressed by the transverse part of the
envelope wave function σ as described in
)(),(),,( zyxikzeyxzyx ++= βασχ , (6.11)
where
2xyyyxx
zyxyyyzx
ωωωωωωω
α−
+−= , (6.12)
2xyyyxx
zxxyzyxx
ωωωωωωω
β−
+−= . (6.13)
ωij is the reciprocal effective mass tensor, and mtrans is the transport effective mass given
by
78
lt
xyyyxxmωω
ωωω2
2
trans
−= , (6.14)
where ωt and ωl are 1/mt and 1/ml , respectively. The ωij and mtrans for the [100] and
[110] Si NWs are shown in Table 6.3 [119].
Figure 6.3 shows cross-sectional local electron density, which gives that volume
inversion becomes strong as shrinking width. Size dependences in electron density are
similar among all sets of channel directions and surface orientations. Figure 6.4 shows
size dependences of the mobility. In [110]/(100) Si NWs, the decrease in the mobility,
consistent with the facet-driven expectation, was shown in width larger than 8 nm
(100)[10
0]
[100]/(100) Si NW
(100)
[110]
[110]/(100) Si NW
(110)
[110]
[110]/(110) Si NW
Figure 6.2: Schematic models of adopted Si NWs with various directions and orientations. [100] and[110] are NW directions, and (100) and (110) are orientations of wafer.
TABLE 6.3
EFFECTIVE MASS TENSOR
Wire orientationa
Wire surface
Minimum type ωxx ωyy ωxy mtrans Degeneracy
[100] (010)/(001) ∆1 1/mt 1/ml 0 mt 2
[100] (010)/(001) ∆2 1/ml 1/mt 0 mt 2
[100] (010)/(001) ∆3 1/mt 1/mt 0 ml 2
[110] (1−
10)/(001) ∆1 1/ml 1/mt 0 mt 2
[110] (1−
10)/(001) ∆2 (mt + ml)/(2mtml) 1/ mt 0 (mt + ml)/2 2
[110] (1−
10)/(001) ∆3 (mt + ml)/(2mtml) 1/ mt 0 (mt + ml)/2 2
aCarriers traverse along wire orientation.
79
because of dominant surface inversion. Electron mobility drastically decreases in
w < 6 nm because of the high scattering rate due to the large form factor of the
80
Electron density (/cm3)
h= 10 nm
h = 10 nm
w = 14 nm 6 nm 3 nm
h= 10 nm
(110)
(100)
(100)
(100)
(110)
(100)
(a) [100]/(100) Si NW
(b) [110]/(100) Si NW
(c) [110]/(110) Si NW
Figure 6.3: Width dependence of cross-sectional local lectron density in (a) [100]/(100) Si NWs, (b) [110]/(100) Si NWs, and (c) [110]/(110) Si NWs. (h = 10 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2). 4
0
200
400
600
800
0 2 4 6 8 10 12 14 16
Mob
ility
(cm
2 /V･s
)
Width (nm)
[100]/(100)[110]/(100)[110]/(110)
Figure 6.4: Width dependence of phonon-scattering-limited mobility in [100]/(100), [110]/(100), and [110]/(110) Si NWs. (h = 10 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2). 3
81
electron-phonon wave function [45], [46]. Figure 6.5 shows width dependences of
cross-sectional specially resolved mobility. We can distinguish orientation and corner
effects based on the specially resolved mobility analysis. Because corner mobility is
always lower than the (100)-surface mobility, the corner effect without a strain effect
would not affect the drastic mobility increase from experimental results. In the next
subsection, we discuss the corner effect.
6.3.2 Corner Effect
We adopted 4- and 12-nm-width rectangular Si NW MOSFETs with [100]-directed
channel and (100)-orientated surfaces. The same orientation of surface is adopted to
Mobility (cm2/V?s)
h= 10 nm
w = 14 nm 6 nm 5 nm 4 nm 3 nm
(100)(100)
(100)
(100)(110)
(110)
h= 10 nm
h= 10 nm
(a) [100]/(100) Si NW
(b) [110]/(100) Si NW
(c) [110]/(110) Si NW
Figure 6.5: Width dependence of cross-sectional specially resolved mobility in (a) [100]/(100) Si NWs,(b) [110]/(100) Si NWs, and (c) [110]/(110) Si NWs. We can distinguish orientation and corner effects based on the specially resolved mobility analysis. Corner mobility is always lower than the(100)-surface mobility (h = 10 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).
82
focus the corner effect. The low-field phonon-limited electron mobilities of 4- and
12-nm-width Si NW MOSFETs were 528 and 693 cm2/V･s, respectively. The lower
mobility of the smaller Si NW MOSFET is due to the fact that the overlap factor
described in (6.7) increases in the substantially confined wire [45], [46].
Figure 6.6 shows the cross-sectional local electron density described in (6.9). In the
12-nm-width Si NW MOSFET, most electrons distribute near the surface, and the
corner electron density is approximately twice as high as the side electron density
02
46
810
12
02
46
810
120
2
4
6
8
10
x 1019
x (nm)y (nm)
Electron density, n(x,y) (/cm3)
01
23
4
01
23
40
5
10
15
x 1019
x (nm)y (nm)
Electron density, n(x,y) (/cm3)
(b) w = 12 nm
(a) w = 4 nm
Corner
Side
Figure 6.6: Cross-sectional local electron density in (a) 4- and (b) 12-nm-width SiNW MOSFETs. The corner electron density is approximately twice as high as the side electron density in the 12-nm-width Si NW MOSFET. (h = w, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).
83
because of denser electric field lines, whereas the volume inversion occurs in the
4-nm-width Si NW MOSFET. Figure 6.7 shows the cross-sectional distributions of
the local phonon-limited electron mobility described in (6.8). In the 4-nm-width Si
NW MOSFET, electrons with the volume inversion correspond to spatially hardly
varied mobility because most electrons occupy the same subband group, i.e., the
lowest subband group. On the other hand, in the 12-nm-width Si NW MOSFET, the
surface mobility is higher than the center mobility. In the surface mobility, the corner
mobility is lower than the side mobility despite the local volume inversion at the
02
46
810
12
02
46
810
12500
600
700
800
x (nm)y (nm)
Local mobility, µlocal(x,y) (cm2/V
⋅ s)
01
23
4
01
23
4300
350
400
450
500
550
x (nm)y (nm)
Local mobility, µlocal(x,y) (cm2/V
⋅ s)
(a) w = 4 nm
(b) w = 12 nm
Corner Side
Figure 6.7: Cross-sectional spatially resolved phonon-scattering-limited mobility in (a) 4- and (b) 12-nm-width SiNW MOSFETs. The corner is lower than the side mobility in the 12-nm-width Si NW MOSFET. (h = w, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).
84
corner. Since the fluctuation of the electron density distribution is more drastic than
that of the mobility distribution, the cross-sectional conductivity distribution would be
dominated by the electron density distribution.
Although the difference between corner and side mobility is relatively small, the
difference should be discussed to interpret spatially resolved carrier transport. To
discuss the cross-sectional distribution of the mobility, we took into account the
subband composition of the electrons at the corner and side in the 12-nm-width Si NW
MOSFET. Figure 6.8 shows that, at the corner and side, almost the same number of
electrons belong to the 1st lowest subband group, and, only at the corner, a number of
electrons occupy the 2nd lowest subband group. The 1st subband group corresponds to
four-fold unprimed subbands, and the 2nd subband group corresponds to two-fold
primed subbands. Here, the unprimed subband denotes the four-fold degenerate
subband whose minimum is located at Γ. We can explain difference of the subband
0
0.2
0.4
0.6
0.8
1N
umbe
r of e
lect
rons
(/cm
3 )
Corner Side
Others
1st subband group
2nd subband group
x 1020
Figure 6.8: The number of electrons occupying each subband group at the corner and side. At thecorner, the large rate of electrons belongs to the 2nd subband group. (h = w = 12 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).
85
02
46
810
12
02
46
810
120
0.5
1
1.5
2
x 1013
x (nm)y (nm)
Probability density (/cm2)
02
46
810
12
02
46
810
120
0.5
1
1.5
2
x 1013
x (nm)y (nm)
Probability density (/cm2)
(b) 2nd subband group
(a) 1st subband group
Figure 6.9: Sum of probability densities for (a) the 1st subband group and (b) the 2nd subband group. The most electrons of the 2nd subband group distributes near the corners. (h = w = 12 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).
TABLE 6.4
MOBILITY FOR EACH SUBBAND GROUP
Subband Type µµ (cm2/V･s)
1st subband group 693
2nd subband group 528
86
compositions at the corner and side by taking into account electron probability density,
|σµ(x, y)|2, for each subband group. Figure 6.9 shows the sum of the probability
densities for each subband group. The probability densities for the 1st subband group
are almost evenly distributed near the surface; on the other hand, the probability
densities for the 2nd subband group concentrate at the four corners. Since wave function
within rapidly varying potential at the corner could be represented by high energy plane
wave function basis, electrons of the higher subbands are localized at corner. Because
of the probability density, the large rate of the corner electrons belongs to the 2nd
subband group.
87
The corner mobility is lower than the side mobility because the mobility of the 2nd
subband group, which is occupied by corner electrons, is lower than the mobility of the
1LUSG as shown in Table 6.4. According to (6.3), the mobility for each subband is
determined by the density of states, the group velocity, and the momentum relaxation
time. The product of f(E)[1 – f(E)] in the (6.3) implies that the electrons with an energy
close to the Fermi level mainly contribute conductivity under low electric field. Hence,
the rate of the conductivity-contributing electrons for the 1st subband group is smaller
88
than that for the 2nd subband group as deduced from Figure 6.10(a). In terms of
contributing electrons, the 1st subband group is more disadvantageous for the mobility
because non-contributing electrons decrease the average mobility. Figure 6.10(a) also
shows that the subband minima of the 1st subband group and 2nd subband group are
lower than the Fermi level, whereas other subbands such as the primed subbands were
higher than the Fermi level. Figure 6.10(b) shows that the group velocity for the 1st
subband group is higher than that for the 2nd subband group. Figure 6.10(c) shows that
89
the scattering rate for the 1st subband group is lower than that for the 2nd subband group
around the Fermi level, where the scattering rate is the reciprocal of the momentum
relaxation time. The total phonon scattering is mainly governed by the intravalley
acoustic phonon scattering mechanisms as shown in Figure 6.10(d) because the
intervalley scattering rate due to phonon emission around the Fermi level becomes
small in highly degenerate semiconductor as deduced from the term of
[1 – f(E – ħωj)] /[1 – f(E)] in (6.6). For the 1st subband group, the intravalley acoustic
Figure 6.10: (a) Density of states, (b) group velocity, and (c) total phonon scattering rate (d) intravalley acoustic phonon scattering rate for subband µ. The horizontal axes are based on the quasi Fermi level, Efn. (h = w = 12 nm, tox = 1 nm, Ninv = 0.8 × 1013 /cm2).
90
phonon scattering rate around the Fermi level is determined by the sum of intrasubband
scattering and intersubband scattering to the 2nd lowest unprimed subband. In contrast,
for the 2nd subband group, the scattering rate is determined by the intrasubband
scattering. Although the scattering rate for the 1st subband group is determined by two
types of scattering events, the scattering rate for the 1st subband group is lower than that
for the 2nd subband group because of the difference between the overlap factors. Since
the electron probability density for the 2nd subband group were more concentrate than
that for the 1st subband group as shown in Figure 6.9, the overlap factor for the
intrasubband scattering event from the 2nd lowest unprimed subband to itself was larger
than that for the intersubband scattering event from the 1st lowest unprimed subband to
the 2nd lowest unprimed subband: the former and latter overlap factors were 2.1 and
1.1 × 1012 /cm2, respectively. Consequently, the total phonon scattering rate for the 2nd
subband group is higher than that for the 1st subband group. Although, in terms of the
rate of conductivity-contributing electrons, the 1LUSG is disadvantageous for the
mobility, the higher group velocity and the lower scattering rate for the 1st subband
group yield higher mobility for the 1st subband group than for the 2nd subband group.
The subband composition at the corner and side also helps interpret the
cross-sectional distribution of local injection velocity under a ballistic transport
regime. In the ballistic transport regime, the injection velocity is proportional to the
drain current. The injection velocity is determined only by the average velocity at the
top of the barrier at the source end [108], [102]. The average velocity for the 2nd
subband group is lower than that for the 1st subband group as shown in Figure 6.10(b),
and the large rate of the corner electrons belong to the 2nd subband group as shown in
Figure 6.8; therefore, corner injection velocity would be lower than the side injection
velocity.
91
6.4 Conclusions
We derived the cross-sectional distribution of the phonon-limited electron mobility in
rectangular Si NW MOSFETs with the Kubo-Greenwood formula. Taking into account
the subband composition of the local electrons, we discussed the corner effects based
on a spatially resolved mobility analysis. When w < 6 nm, the mobility drastically
modulated. The 4-nm-width Si NW MOSFET showed a small spatial fluctuation in the
cross-sectional mobility distribution with volume inversion. On the other hand, the
12-nm-width Si NW MOSFET showed a large spatial fluctuation of the cross-sectional
mobility distribution, where most electrons were distributed near the surface. We also
revealed that the corner mobility was lower than the side mobility because the large rate
of the corner electrons belongs to the 2nd subband group with the lower group velocity
and higher scattering rate. Therefore, our results could imply that the corner component
did not improve the phonon-limited mobility in terms of the subband composition
without other effects such as the strain effect. Finally, analogous to the spatial mobility
distribution, the cross-sectional distribution of the local injection velocity under
ballistic transport regime could also be interpreted, and the corner injection velocity
would be lower than the side injection velocity. The size effect without considering
strain does not cause a drastic mobility increase in experimental results because the
electronic structure hardly changes and the corner mobility is lower than the side
mobility.
92
Chapter 7
Modeling of Quasi-Ballistic Transport
in Nanowire MOSFETs
7.1 Introduction
In subsection 7.1.1, we describe the request for the interpretation of the quasi-ballistic
transport with downscaling. In subsection 7.1.2, we briefly introduce Natori’s model
for high-field transport [60], [61], which is referred to develop a comprehensive
model of the quasi-ballistic transport here.
7.1.1 Quasi-Ballistic Transport
Downscaling of the state-of-the-art MOSFETs has required the interpretation of the
quasi-ballistic transport. We can describe the low-field transport in a long channel
device by the macroscopic variables, such as average electron density and electron
temperature, with near-equilibrium. The high field transport has been studied with the
93
velocity saturation [120], [121] and the velocity overshoot [122], [123]. M.
Lundstrom [51], [52] has developed a scattering theory with the kT layer for the
high-field transport. According to the kT-layer theory, the backscattering coefficient,
R, of the carriers injected from the source can be described in (7.1),
kT0
kT
LLR+
=λ
, (7.1)
where λ0 is the mean free path for the backscattering under near equilibrium, and LkT
is the critical distance with the kT layer as described in Figure 7.1. This theory can
describe the quasi-ballistic transport in an ultra-short channel device with taking into
account critical scattering events within the kT layer. Although the kT-layer theory has
been empirically validated with a Monte-Carlo simulation [51], it has not been clearly
postulated:
Equation (7.1), derived with assuming near-equilibrium analogous to diffusive
regime, is not valid under high field even around the top of the barrier where the
longitudinal electric field is small [53];
The critical distance even in the framework of the kT-layer theory was discrepant
from the LkT [54], [55], [56], [57].
kBT
Source
Drain
Channel
kT layer
Figure 7.1: We describe the kT layer within a schematic potential profile. The critical distance LkT is the distance between the top of the barrier and the position of the potential drop by kBT. The effect of the backscattering beyond the critical distance is neglected.
94
With renouncing the crude assumptions of the kT-layer theory, E. Gnani et al. [58]
and K. Natori [60] developed a quasi-ballistic transport model for Si NW MOSFETs
by directly solving the BTE with constraint of dominant elastic scattering due to
acoustic phonon. Further work by K. Natori [60], [61] took into account the inelastic
scattering due to the optical phonon emission, where he assumed endless drain for
simplicity. The assumption of the endless drain interrupts interpretation of the
quasi-ballistic transport because the carriers end up relaxing their energy.
In this chapter, we develop a quasi-ballistic transport model for NW MOSFETs
based on Natori’s model [60], [61]. Our model takes into account the finite drain
length and the distribution function of the carriers injected from source. Our model is
also validated by a numerical simulation based on the deterministic solution of the 1D
MSBTE.
7.1.2 Natori’s Model for Quasi-Balistic Transport
In Natori’s model [60], [61], the channel is divided into two zones as shown in Figure
hππ
0z
ze
Elastic zone
Relaxation zone
Figure 7.2: Concept of Natori’s quasi-ballistic transport model. Contrary to the kT-layer theory, the critical distance is set to the length between the top of the barrier and the position where carriers can emit the optical phonon energy, ħω. He took into account the effect of the backscattering beyond the critical distance.
95
7.2: elastic zone and relaxation zone. Contrary to the kT-layer theory, the critical
distance is set to the distance between the top of the barrier and the position where
carriers can emit the optical phonon energy, ħω. This critical zone is called the elastic
zone, where the carriers traverse with suffering elastic scattering only. Furthermore,
the effect of the backscattering beyond the critical zone is taken into account. In the
relaxation zone, the carriers traverse with energy relaxation for the optical phonon
emission as well as the elastic scattering.
As shown in Figure 7.3, a one-flux scattering matrix for a slab between z1 and z2
can be described in (7.2),
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
+
−
+
),(),(
)()()()(
),(),(
2
1
21
21
1
2
EzfEzf
ETERERET
EzfEzf ,
(7.2)
R1,2 and T1,2 are backscattering coefficient and transmission coefficient, where the
subscripts of 1 and 2 correspond to the carriers injected from the left and the right side,
respectively. f+(z,E) and f–(z,E) are the carrier distribution functions of a position z and
a total energy level E with positive and negative velocity. The expression by the
f+(z1,E)R1
T1
T2
R2
z1 z2
f–(z2,E)
f+(z2,E)f–(z1,E)
Figure 7.3: f+(z1,E), f–(z1,E), f+(z2,E), and f–(z2,E) can be described by a one-flux scattering matrix for a slab between z1 and z2. R1 and T1 correspond to the carriers injected from the left side, and R2 and T2correspond to the carriers injected from the right side.
96
one-flux scattering matrix of the (7.2) can changed to (7.3),
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
+
−
+
),(),(
1)()()()()()(1
),(),(
1
1
1
22121
22
2
EzfEzf
ERERERERETET
TEzfEzf .
(7.3)
Multiplying the scattering matrix in (7.3) for each slab, we can describe the scattering
matrix of continuous slabs.
Now, we briefly introduce Natori’s model with taking into account only the lowest
subband. The one-flux scattering matrix for the elastic zone between 0 and ze can be
described in (7.4),
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
+
−
+
),0(),0(
1)()()(21
)(11
),(),(
1
11
1e
e
εε
εεε
εεε
ff
RRR
Rzfzf ,
(7.4)
which has been reduced by T1,2 = 1 – R1,2 and R1 = R2 with considering elastic
scattering only. Here, f+ and f– newly denote functions of ε that is an energy level from
the top of the barrier as described in ε ≡ E – E1(0), where E1(z) is the subband
minimum at z.
From (7.4), the backscattering coefficient of the carriers injected from source is
described in (7.5),
),(),()(1
),(),()](21[)(
),0(),0()(
e
e1
e
e11
εε
ε
εε
εε
εεε
zfzfR
zfzfRR
ffR
+
−
+
−
+
−
−
−+== .
(7.5)
The backscattering coefficient for the elastic zone, R1, is derived by solving the BTE
with elastic scattering only [58], [60]:
∫
∫
+=
e
e
0ac
0ac
1
),(11
),(1
)(z
z
dzz
dzz
R
ελ
ελε ,
(7.6)
where the λac is the mean free path for the backscattering. The backscattering
97
coefficient at the end of the elastic zone, f–(ze,ε) /f+(ze,ε), is also derived by solving the
BTE with out-scattering to the lower energy level within the relaxation zone.
f–(ze,ε) /f+(ze,ε) is given by (7.7)
2
ac
op
ac
op
e
e 1),(),(
⎟⎟⎠
⎞⎜⎜⎝
⎛−+≈+
−
SS
SS
zfzf
εε
, (7.7)
which is approximated with assuming endless drain [60], [61]. Here, Sac and Sop are
transition coefficients associated with the elastic backscattering for the acoustic
phonon and transition coefficients associated with the energy relaxation for the optical
phonon emission, respectively.
There are few limitations in Natori’s model. To calculate the drain current, we
should model the distribution function of the carriers from source. There is no
guarantee that the backscattering coefficient of the carriers injected from source is the
same as that of the carriers injected from drain. The assumption of the endless drain
interrupts interpretation of the quasi-ballistic transport. The model developed in this
chapter could handle these limitations.
7.2 Modeling of Quasi-Ballisic Transport
In subsection 7.2.1, we describe carrier distribution functions at the top of the barrier
by one-flux scattering matrices, where the device is divided into five zones. In
subsection 7.2.2, we derive the backscattering and transmission coefficients for each
zone, which are the elements in the scattering matrices, by solving the 1D BTE
modified under appropriate approximations.
98
7.2.1 Expression by one-flux scattering matrices
In our concept, the device is divided into five zones as shown in Figure 7.4: source
zone, barrier zone, elastic zone, relaxation zone, and drain zone. The ideal source and
drain are located at each end of the device. According to Natori’s model, the elastic
zone is set to the region between the top of the barrier and the position where carriers
can relax the optical phonon energy, ħω. The region between the ideal source and the
top of the barrier is divided into two zones, where potential profile in the barrier zone
hardly depends on the length of the source zone and the longitudinal potential in the
source zone hardly varies. In the barrier zone and the elastic zone, we assumed that
the scattering event is constrained with the dominant elastic scattering due to acoustic
phonon. In relaxation zone, we considered the out-scattering to the lower energy level
with optical phonon emission as well as elastic scattering. In the source zone and the
drain zone, we considered both the in-scattering from the lower energy level with
optical phonon absorption and the out-scattering to the lower energy level with optical
h66
0 z0 ze zr zd
Ideal drain
Ideal source
Source zone
Drain zone
Elastic zone
Relaxation zone
Barrier zone
zsFigure 7.4: A device is divided into five zones in the model developed here. The ideal source and drainare located at each end of the device.
99
phonon emission as well as elastic scattering. Considering the in-scatteirng, we can
avoid excessive energy relaxation in substantially long length of the source and drain
zones. According to voltage conditions, the other regions can be regarded as the
elastic zone without sufficient energy to relax the optical phonon energy. Finally, we
should note that this model is for the 1D transport without intersubband scattering.
The current for the lowest subband is calculated by the carrier distribution function
at the top of the barrier as described by (7.8),
∫∞ −+ −=
0 00 )],(),([ εεεπ
dzfzfqIh
, (7.8)
where ε is an energy level from the top of the barrier as described in ε ≡ E – 1E (z0).
In (7.8), we do not consider any degeneracy except for the spin degeneracy. To derive
the carrier distribution function at the top of the barrier, we divide the scattering
matrices into that from 0 to z0 and those from z0 to zd as described in (7.9) and (7.10),
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
+
)0(111
)()( S
1s
2s2s1s2s1s
1b
2b2b1b2b1b
2b2s0
0
ff
RRRRTT
RRRRTT
TTzfzf ,
(7.9)
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
++
)()(
1111)z(
0
0
1e
2e2e1e2e1e
1r
2r2r1r2r1r
d1
d2d2d1d2d1
d2r2e2D
d
zfzf
RRRRTT
RRRRTT
RRRRTT
TTTff ,
(7.10)
where the variable ε is omitted for simple expression. The subscripts of backscattering
and transmission coefficients are associated with the initial of the zone name and the
direction of the injected carriers. fS, and fD are the source and drain Fermi-Dirac
distributions as described in (7.11) and (7.12),
⎟⎟⎠
⎞⎜⎜⎝
⎛ −++
≡
TkEzE
f
B
fs01S )(exp1
1)(ε
ε , (7.11)
⎟⎟⎠
⎞⎜⎜⎝
⎛ −++
≡
TkEzE
f
B
fd01D )(exp1
1)(ε
ε , (7.12)
100
which give the distribution function of the carriers injected from the ideal source and
drain. Simultaneously solving (7.9) and (7.10), we can derive the f+(z0,ε) and f–(z0,ε)
as described by
)()( 0SSS0 zfRfTzf −+ += , (7.13)
DS
SS
S0 11)( f
RRTf
RRRTzf
−+
−=− ,
(7.14)
where
b1s2
b2b1b2b1s2b2S 1
)(RR
RRTTRRR−
−−≡ ,
(7.15)
b1s2
b1s1S 1 RR
TTT−
≡ . (7.16)
)(1))(()()1(
r2r1r2r1e1d1r2r1e1
r2r1r2r1e2e1e2e1e2e1e2e1r1d1r2e1
RRTTRRRRRRRTTRRTTRRTTRRRRR
−−−−−−+−+−
≡ , (7.17)
)(1 r2r1r2r1e1d1r2r1e1
d2r2e2
RRTTRRRRRTTTT
−−−−≡ .
(7.18)
The variable ε is omitted for simple expression. Here, R is the backscattering
coefficient of the carrier injected from the barrier zone, and T is the transmission
coefficient of the carrier injected from the drain. The backscattering and transmission
coefficients for each zone can be derived by analytically or semi-analytically solving
the BTE.
7.2.2 Solution of the Boltzmann Transport Equation
As shown in the subsection 2.3.4, the 1D MSBTE has been rearranged as:
),(),(),(
|),(| outinµµµµ
µµµµ εε
εευ zCzC
zzf
z ±±±
−=∂
∂± , (7.19)
where υµ, ±µf , ±in
µC , and ±outµC are newly defined by the εµ that is an energy level
101
from the top of the barrier for subband µ as described in
)0(µµε EE −≡ , (7.20)
where E is the total energy. Since the scattering integrals of (7.19) are coupled by other
energy levels as described in (2.73) and (2.74), the BTE of (7.19) is nonlinear equation.
Therefore, we need to numerically solve (7.19). With several assumptions, however,
we analytically or semi-analytically solve the BTE for each zone as described in the
following subsections. Hereafter, we will consider only the lowest subband and not
represent the subband index µ.
7.2.2.1 Barrier and Elastic Zones
In the barrier zone and the elastic zone, we take into account elastic scattering only.
Figure 7.5 shows validation of this approximation in the elastic zone. Because the
barrier zone is usually substantially short, we can neglect the inelastic scattering. The
1D BTE with only elastic acoustic phonon scattering can be described in
),(),(
1),(),(
1),(
acac
εελ
εελ
ε zfz
zfzdz
zdf −++
−=− , (7.21)
),(),(
1),(),(
1),(
acac
εελ
εελ
ε zfz
zfzdz
zdf +−−
−= , (7.22)
where the λac is the mean free path for the backscattering as described in
)]()([4),(),(21
),(1
101
ac
acac zEzEmS
zzz −+=≡
επετευελ h.
(7.23)
Velocity υ(z,ε) and momentum relaxation time τac(z,ε) are described in
102
mzEzEz )]()([2),( 101 −+
≡ε
ευ , (7.24)
)]()([2),(1
101
ac
ac zEzEmS
z −+≡
επετ h.
(7.25)
By solving (7.22) and (7.23) simultaneously, we derive distribution functions, f+(z,ε)
and f–(z,ε), as described in
),(
),(11
),(1
),(
),(11
),(11
),( e
ac
ac0
ac
ac
e
0
0
e
0
e
ε
ελ
ελε
ελ
ελε zfdz
z
dzzzf
dzz
dzzzf
z
z
z
z
z
z
z
z−++
∫
∫
∫
∫
++
+
+= ,
(7.26)
),(
),(11
),(11
),(
),(11
),(1
),( e
ac
ac0
ac
ac
e
0
0
0
e
ε
ελ
ελε
ελ
ελε zfdz
z
dzzzf
dzz
dzzzf
z
z
z
z
z
z
z
z
e
−+−
∫
∫
∫
∫
+
++
+= . (7.27)
Based on the one-flux scattering matrix of (7.2) between z0 and ze, we can derive the
backscattering and transmission coefficients for the elastic zone as described in
Figure 7.5: Scattering rate under nondegenerate equilibrium in a cylindrical Si NW MOSFET. Solidline is the result considering elastic acoustic phonon scattering only, and dotted line is that considering both elastic acoustic phonon scattering and inelastic optical phonon scattering. The inelastic optical phonon scattering could be neglected below ħω = 63 meV. (d = 3 nm, tox = 1 nm, Vg = 0.6 V, Vd = 0 V).
103
∫
∫
+==
e
0
e
0
),(11
),(1
)()(
ac
ace2e1 z
z
z
z
dzz
dzzRR
ελ
ελεε , (7.28)
)(1)()( e1e2e1 εεε RTT −== . (7.29)
The barrier zone is substantially short enough to neglect inelastic scattering events.
Therefore, through the same process, we can derive the backscattering and
transmission coefficients for the barrier zone as described in
∫
∫
+==
0
s
0
s
),(11
),(1
)()(
ac
acb2b1 z
z
z
z
dzz
dzzRR
ελ
ελεε , (7.30)
)(1)()( b1b2b1 εεε RTT −== . (7.31)
7.2.2.2 Relaxation Zone
In relaxation zone, we considered the out-scattering to the lower energy level as well
as elastic scattering. The 1D BTE with elastic acoustic phonon scattering and inelastic
optical phonon out-scattering with energy relaxation can be described in
),(),(
2/)],(),([1
),(),(
1),(),(
1),(
op
acac
εελ
ωεωε
εελ
εελ
ε
zfz
zfzf
zfz
zfzdz
zdf
+−+
−++
−+−−+
−=−
hh,
(7.32)
),(),(
2/)],(),([1
),(),(
1),(),(
1),(
op
acac
εελ
ωεωε
εελ
εελ
ε
zfz
zfzf
zfz
zfzdz
zdf
−−+
+−−
−+−−+
−=
hh,
(7.33)
where the λop is the mean free path for the out-scattering to lower energy level as
described in
104
])()()][()([2
),(),(1
),(1
101101
op
opop
ωεεπ
ετεελ
hh −−+−+=
≡
zEzEzEzEmS
zzvz.
(7.34)
To analytically solve the simultaneous non-homoegenous differential BTE with (7.32)
and (7.33), we approximate
)]()([(2),(1
101
op
op zEzEmS
z −+≈
επελ h,
(7.35)
which is reasonable approximation in high-field [60]. To avoid excessive energy
relaxation in degenerate system, we also approximate
)(),( ωεωε hh −≈−+Dfzf , (7.36)
)(),( ωεωε hh −≈−−Dfzf . (7.37)
By solving (7.32) and (7.33) simultaneously with above approximations, we derive
general solution given by
⎥⎦
⎤⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛
−++++
⎥⎦
⎤⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛
++−+=⎟⎟
⎠
⎞⎜⎜⎝
⎛
∫
∫−
+
g
e
),(12exp
22
),(12exp
22
),(),(
ac
22
2
ac
22
2
z
z
z
z
dzz
XXXXXXXX
dzz
XXXXXXXX
zfzf
ελβ
ελα
εε
, (7.38)
where
)](1[ Dac
op ωε h−−≡ fSS
X . (7.39)
Based on the one-flux scattering matrix of (7.2) between ze and zr, we can derive the
backscattering and transmission coefficients for the relaxation zone as described in
⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−
==
∫
∫
r
e
r
e
),(122exp
221
221
),(122exp1
)()(
ac
2
22
2
r2r1z
z
z
zac
dzz
XXXXXX
dzz
XXRR
ελ
ελεε ,
(7.40)
105
⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−+
==
∫
∫
r
e
r
e
),(122exp
221
221
),(122exp22
)()(
ac
2
22ac
22
r2r1z
z
z
z
dzz
XXXXXX
dzz
XXXXTT
ελ
ελεε .
(7.41)
7.2.2.3 Source and Drain Zones
In the source zone and the drain zone, we considered both the in-scattering from the
lower energy level and the out-scattering to the lower energy level as well as elastic
scattering. Considering the in-scatteirng in the drain zone, we can avoid excessive
energy relaxation in the substantially long drain zone. The 1D BTE with elastic
acoustic phonon scattering and inelastic optical phonon in- and out-scatterings with
energy relaxation and excitation can be described in
)],(1[/),(
2/)],(),([
),(),(
2/)],(),([1
),(),(
1),(),(
1),(
opopop
op
acac
εελ
ωεωε
εελ
ωεωε
εελ
εελ
ε
zfSSz
zfzf
zfz
zfzf
zfz
zfzdz
zdf
+−+
+−+
−++
−′−+−
−
−+−−+
−=−
hh
hh , (7.42)
)],(1[/),(
2/)],(),([
),(),(
2/)],(),([1
),(),(
1),(),(
1),(
opopop
op
acac
εελ
ωεωε
εελ
ωεωε
εελ
εελ
ε
zfSSz
zfzf
zfz
zfzf
zfz
zfzdz
zdf
−−+
−−+
+−−
−′−+−
−
−+−−+
−=
hh
hh , (7.43)
where
])([4)(1
),(1
s101
acsacac EzE
mSz −+
≡≈επελελ h
, (7.44)
])(][)([2)(1
),(1
s101
s101
opsopop ωεεπελελ hh −−+−+
≡≈EzEEzE
mSz
. (7.45)
106
Here, Sop’ is a transition coefficient associated with the energy excitation for the
optical phonon absorption. From detailed balance condition, we can describe the
relationship between Sop and Sop’ as in
)](1)[(),()](1)[(),( SSopSSop εωεεωεεε ffzSffzS −−=−−′hh . (7.46)
To avoid excessive energy relaxation in degenerate system as in the previous
subsection, we can approximate
)(),( S ωεωε hh −≈−+ fzf , (7.47)
)(),( S ωεωε hh −≈−− fzf . (7.48)
We can reduce the BTE of (7.42) and (7.43) to
)(1),(
)(1),(
)(1
)(1),(
sin
sac
sout
sac ελ
εελ
εελελ
ε−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=− −+
+
zfzfdz
zdf , (7.49)
)(1),(
)(1),(
)(1
)(1),(
sin
sac
sout
sac ελ
εελ
εελελ
ε−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+= +−
−
zfzfdz
zdf . (7.50)
where soutλ and s
inλ are mean free paths for net out- and in-scatterings in the source
region. The soutλ and s
inλ are described in
)](1)[()(1
)(1
Ssop
Ssout εελ
ωεελ f
f−−−
≡h
, (7.51)
)](1)[()()](1[
)(1
Ssop
SSsin εελ
εωεελ f
ff−
−−≡
h.
(7.52)
By solving (7.49) and (7.50) simultaneously with those approximations, we derive
general solution as given by
)])((exp[
)(1)(
)(1)(
])(exp[
)(1)(
)(1)(
),(),(
ss
sout
s
sout
s
sout
s
sout
s
zzYY
YzY
Y
Y
zfzf
s −
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−
++
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
+
ε
ελε
ελε
βε
ελε
ελε
αεε ,
(7.53)
where
107
2sout
sout
sac
s )]([1
)()(2)(
ελελελε +≡Y .
(7.54)
Based on the one-flux scattering matrix of (7.2) between 0 and zs, we can derive the
backscattering and transmission coefficients for the source zone as described in
( )
( )ss
sout
s
sout
s
sout
s
sout
s
sss2
)(2exp
)(1)(
)(1)(
)(1)(
)(1)(
)(2exp1)(
LYY
Y
Y
Y
LYR
ε
ελε
ελε
ελε
ελε
εε
−+
−−
−
+
−−= ,
(7.55)
)(1)( s2s1 εε RT −= . (7.56)
In the drain zone, the calculation process is basically the same as that in the source
zone. The mean free paths are described in
])([4)(1
),(1
d101
acdacac EzE
mSz −+
≡≈επελελ h
, (7.57)
])(][)([2)(1
),(1
d101
d101
opdopop ωεεπελελ hh −−+−+
≡≈EzEEzE
mSz
. (7.58)
From detailed balance condition, we can describe the relationship between Sop and Sop’
as in
)](1)[(),()](1)[(),( DDopDDop εωεεωεεε ffzSffzS −−=−−′hh . (7.59)
To avoid excessive energy relaxation in degenerate system, we can approximate
)(),( D ωεωε hh −≈−+ fzf , (7.60)
)(),( D ωεωε hh −≈−− fzf . (7.61)
We can reduce the BTE of (7.42) and (7.43) to
)(1),(
)(1),(
)(1
)(1),(
din
dac
dout
dac ελ
εελ
εελελ
ε−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=− −+
+
zfzfdz
zdf , (7.62)
108
)(1),(
)(1),(
)(1
)(1),(
din
dac
dout
dac ελ
εελ
εελελ
ε−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+= +−
−
zfzfdz
zdf , (7.63)
where doutλ and d
inλ are mean free paths for net out- and in-scatterings in sdrain
region. The doutλ and d
inλ are described in
)](1)[()(1
)(1
Ddop
Ddout εελ
ωεελ f
f−
−−≡
h,
(7.64)
)](1)[()()](1[
)(1
Ddop
DDdin εελ
εωεελ f
ff−
−−≡
h.
(7.65)
By solving (7.62) and (7.63) simultaneously with those approximations, we derive
general solution as given by
)])((exp[
)(1)(
)(1)(
)])((exp[
)(1)(
)(1)(
),(),(
dd
dout
d
dout
d
rd
dout
d
dout
d
zzYY
YzzY
Y
Y
zfzf
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−
++−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
+
ε
ελε
ελε
βε
ελε
ελε
αεε ,
(7.66)
where
2dout
dout
dac
d )]([1
)()(2)(
ελελελε +≡Y .
(7.67)
Based on the one-flux scattering matrix of (7.2) between zr and zd, we can derive the
backscattering and transmission coefficients for the drain zone as described in
( )
( )dd
dout
d
dout
d
dout
d
dout
d
ddd1
)(2exp
)(1)(
)(1)(
)(1)(
)(1)(
)(2exp1)(
LYY
Y
Y
Y
LYR
ε
ελε
ελε
ελε
ελε
εε
−+
−−
−
+
−−= ,
(7.68)
)(1)( d1d2 εε RT −= . (7.69)
109
7.3 Validation by Numerical Simulation
The model developed in this chapter is validated a numerical simulation based on the
deterministic solution of the 1D MSBTE [92], [93]. To validate the model, we needed
to import the self-consistent longitudinal potential profile extracted from the
numerical simulation.
A 3-nm-diameter [100] Si NW MOSFET was used to achieve 1D-like transport,
which can neglect the intersubband scattering events. The difference of the energy
level between the first and the second subbands is 0.21 eV, which is large enough to
neglect the influence of the second subband. The diameter of 3 nm is also small
enough to keep the form factor constant because the wave function hardly varies
owing to the strong volume inversion as shown in Figure 3.6. The lowest subband
group of the [100] Si NW is four-fold degenerate. Hence, we can describe the current
ID:
∫∞ −+ −=
0 00D )],(),([2 εεεπυ dzfzfqgIh
, (7.70)
where gυ is the valley degeneracy, which is two in Si. Because the lowest subband is
unprimed subband, the transport effective mass m is mt. Transition coefficients of
elastic acoustic phonon scattering and inelastic optical phonon out-scattering with
energy relaxation are
FugTkΞS
l
B2
Si
2
ac ρπ
υh= ,
(7.71)
]1)([2
)(]1)([
2)(
op3Si
23
op3Si
23
op +++= ωωρ
πω
ωρπ
υυ
hh NFgg
KDNFg
gKD
S ggt
fft ,
(7.72)
where we assume the dominant optical phonon scatterings are of f3 and g3 types, and
the optical phonon energy ħω for both types of f3 and g3 is set to 63 meV [92]. F
110
denotes the form factor. The number of phonons, Nop, is given by the Bose-Einstein
factor as
11)( /op −
= TkBeN ωω
hh . (7.73)
According to the selection rule, gf3 is 2, and gg3 is 1. In the strong volume inversion,
the wave function can be approximated by the Bessel function. Because strong
volume inversion is accompanied with the diameter of 3 nm, the form factor from the
self-consistent calculation could approximate that with the Bessel function as
described in
∫≈R
drAkrJ
AkrJrF
0
20
20 )()(2π ,
(7.74)
where the normalizing constant A is given by
∫=R
drkrJrA0
20 |)(|2π , (7.75)
and the wave vector of the ground state is given by
RRj
k 405.21,0 ≈= . (7.76)
Figure 7.6 shows comparison of the backscattering coefficient R(ε) between this
model and the numerical simulation. The model is in good agreement with the
numerical simulation especially below 63 meV. Figure 7.6(c) shows that modeling for
the drain zone is available even in substantially long drain. Figure 7.7 shows
comparison of the carrier distribution functions at the top of the barrier and the drain
current, which is calculated from the distribution functions, would be well modeled.
Figure 7.7(b) shows that modeling for the source zone is available even in
substantially long source. Figure 7.8 shows that our model is in quantitatively good
agreement with the numerical simulation in I-V characteristics.
111
Figure 7.6: Backscattering coefficient, R(ε), for (a) saturation region with Ld = 10 nm, (b) subthreshold region with Ld = 10 nm, and (c) saturation region with long drain, Ld = 100 nm. Open symbols are results from the numerical simulation, solid lines are those from this model, and dotted lines are thoseconsidering elastic acoustic phonon scattering only. The modeling of the drain zone is available even in substantially long drain. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Lg = 30 nm, Nd
s = Ndd = 2 × 1020 /cm3).
112
113
Figure 7.7: Distribution functions at the top of the barrier in (a) saturation region with short source (Ls = 10 nm) and (b) subthreshold region with long source (Ls = 100 nm). Open symbols are the results from the numerical simulation, solid lines are those form this model, and dotted lines are Fermi-Diracdistribution function within the ideal source. The modeling of the source zone is available even in substantially long source. (d = 3 nm, tox = 1 nm, Lg = 30 nm, Ld = 10 nm, Vd = 0.3 V, Nd
s = Ndd = 2 × 1020 /cm3).
114
7.4 Discussion
In this section, we investigate the quasi-ballistic transport and compare this model
with other models, such as Natori’s model [60], [61] and the AP-only model [58], [59].
Figure 7.9 shows the adopted potential profile, U(z), for discussion of quasi-ballistic
sport. The channel potential is linear. The barrier zone is neglected for simplicity. Us
and Ud are determined by source and drain donor impurity densities, Nds and Nd
d,
under equilibrium. Here, all the impurities are ionized. With this potential, the
backscattering coefficient R(ε) can be analytically derived, but, to calculate drain
current, we need numerical integration over energy.
Figure 7.10 shows that the ballisticity as functions of gate length and drain length,
where the ballisticity B is calculated by:
012345678
0 0.1 0.2 0.3 0.4 0.5
V g– V t
= 0.3
V
Vg – Vt = 0.0 V
Drain voltage (V)
Dra
in c
urre
nt (µ
A)
Numerical simulation
This model
Figure 7.8: Id-Vd characteristics from the numerical simulation and the drain current from this model.This model is in quantitatively good agreement with the numerical simulation. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Lg = 30 nm, Ld = 10 nm, Nd
s = Ndd = 2 × 1020 /cm3).
115
∫∫
∞ +
∞ −+
−
−≡
0 0
0 00
)](),([
)],(),([
εεε
εεε
dfzf
dzfzfB
D
, (7.77)
which gives the current gain compared with that of the ballistic limit. When we
neglect the injection from drain with a large Vd, we can approximate the B as
∫∫
∞ +
∞ + −≈
0 0
0 0
),(
)](1)[,(
εε
εεε
dzf
dRzfB . (7.78)
Figure 7.10 shows that the ballisticity of unity could not be obtained by taking into
account the realistic 1D drain. If the Ld is longer than 50 nm, about 20% of carriers
injected to drain zone is backscattered. Figure 7.11 shows drain current as a function
of source length. When Ls is shorter than 10 nm, elastic source approximation is
available. In a shorter 1D source, the resistivity seems to be smaller.
We compare between this model and other models. Figure 7.12(a) shows drain
current as a function of gate length. When Lg is longer than 50 nm, Natori’s model is
in good agreement with this model. When Lg is shorter than 10 nm, elastic
approximation is in good agreement with this model. Figure 7.12(b) shows drain
Linear potential
Lg LdLs
z
E
Source
Drain
Channel
0
Us
Ud
z0 zr zd
U(z0)
Figure 7.9: Adopted potential profile, U(z), for discussion of quasi-ballistic transport. There is no barrier zone for simplicity. Us and Ud are determined by source and drain donor impurity densities, Nd
s
and Ndd, under equilibrium, where Ud = Us – qVd when Nd
s = Ndd.
116
current as a function of drain length. In Natori’s model, the drain current does not
depend on drain length because endless channel is assumed. Elastic approximation
117
0
0.2
0.4
0.6
0.8
1
0 50 100 150
0
0.2
0.4
0.6
0.8
1
0 50 100 150
0
0.2
0.4
0.6
0.8
1
0 50 100 150
Gate length (nm)
Drain length (nm)
Bal
listic
ityB
allis
ticity
(a) Ld = 1 nm
Gate length (nm)
Bal
listic
ity
(b) Ld = 20 nm
(c) Lg = 1 nm
Figure 7.10: Ballisticity as functions of (a) gate length with Ld = 1 nm, (b) gate length with Ld = 20 nm, and (c) drain length with Lg = 1 nm. Although the Ld and Lg of 1 nm are not realistic, it makes sense as eliminating their influences. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Vd = 0.3 V, U(z0) = Efs, Nd
s = Ndd = 2 × 1020 /cm3).
118
119
0
1
2
3
4
5
0 50 100 150Source length (nm)
Draincurrent(τA)
This modelElastic only
Figure 7.11: Drain current as a function of source length. Close circles are results from this model, ,and open squares are those considering elastic acoustic phonon scattering only. (d = 3 nm, tox = 1 nm, Lg = 10 nm, Ld = 10 nm, Vd = 0.3 V, U(z0) = Efs, Nd
s = Ndd = 2 × 1020 /cm3). 12
00.5
11.5
22.5
33.5
44.5
0 50 100 150
0
1
2
3
4
5
0 20 40 60
Drain length (nm)
Draincurrent(.A)
Gate length (nm)
This modelNatori’s model
Elastic onlyDraincurrent(.A)
This modelNatori’s modelElastic only
(a) Ld = 10 nm
(b) Lg = 10 nm
Figure 7.12: Drain current as functions of (a) gate length with Ld = 10 nm and (b) drain length with Lg = 10 nm. Close circles are results from this model, open squares are those from Natori’s model, and open triangles are those considering elastic acoustic phonon scattering only. (d = 3 nm, tox = 1 nm, Ls = 10 nm, Vd = 0.3 V, U(z0) = Efs, Nd
s = Ndd = 2 × 1020 /cm3). 11
120
could not describe backscattering in long drain. When Lg is longer than 100 nm,
Natori’s model is in good agreement with this model.
Finally, we describe the advantages of this model. This model can describe
distribution function of the carriers injected from realistic source and transmission
coefficient for the carriers injected from drain. It can also take into account finite
length of drain and source.
7.5 Conclusions
We successfully developed the quasi-ballistic transport model by the one-flux
scattering matrices and the 1D MSBTE with dividing the device into five zones. The
backscattering coefficient, carrier distribution function, and drain current from this
model were in good agreement with the results from the numerical simulation. We
also found out that realistic 1D drain length was important to adjust the backscattering
coefficient, and realistic 1D source length was important to represent the distribution
of the injected carriers.
121
Chapter 8
Conclusions
8.1 Summary of Conclusions
Through modeling of gate capacitance and quasi-ballistic transport, we could interpret
device physics of ultra-scaled NW MOSFETs. In chapter 3, we successfully
developed a comprehensive gate capacitance model to distinguish the contributions of
the quantum effects with respect to the finite inversion layer centroid and the finite
density of states. The finite inversion layer centroid caused the increase in the total
capacitance for small NW MOSFETs. In chapter 7, we successfully developed the
semi-analytical quasi-ballistic transport model, where the model has been validated by
the device simulation based on the deterministic numerical solution of the 1D MSBTE.
We also found out that the 1D drain length was the parameter to adjust the
backscattering coefficient and that the 1D source length was the parameter to
represent the distribution function of the carriers injected from realistic source. In
122
conclusion, from the developed models and simulation results, we could interpret
device physics and find out what parameters control the performance in ultra-scaled
NW MOSFETs.
Through numerical simulations, we could find out what parameters control
performance of ultra-scaled NW MSOFETs. In chapter 4, based on the developed gate
capacitance model, we could interpret the size-dependent performance in Si and InAs
NW MOSFETs. Performance of a large-effective-mass NW like the Si NW depends
on the injection velocity, which generally increases as shrinking diameter; on the
other hand, that of a small-effective-mass NW like the InAs NW is insensitive to the
injection velocity. We also found out that the desirable diameter could be around 5
nm for Si NWs and 10 nm for InAs NWs. In chapter 5, we carefully took into account
the band structure effect in substantially small diameter since effective mass of Si
NWs widely fluctuated with curvature variation. If the NW cross section is
rectangular, the wide fluctuation of the effective mass was suppressed, where the
injection velocity was degraded as in nonparabolic EMA. In chapter 6, we revealed
that the size effect on the band structure and the increase in the corner component
would not cause the drastic increase in mobility from the experimental results because
the corner mobility is lower than the side mobility and the electronic structure hardly
changes. We also found out that the mobility drastically modulated in width smaller
than 6 nm.
8.2 Future Work
Finally, we introduce some further studies:
In chapter 6, we discussed the phonon-scattering-limited low-field mobility of the
123
rectangular NW MOSFETs. Unfortunately, we cannot interpret the drastic
mobility increase around the width of 10 nm. In our study, only limited case
without strain has been investigated. Hence, as the future work, we can investigate
strain effect and take into account other scattering mechanisms, e.g. surface
roughness scattering;
In chapter 7, a comprehensive quasi-ballistic transport model has been developed.
By using that, we can interpret more various aspects regarding the quasi-ballistic
transport in NW MOSFETs. Because we should carefully take into account the
longitudinal potential profile to obtain the exact drain current, the potential profile
needs to be additionally modeled. The modeling of the potential profile can also
be a future work. The quasi-ballistic transport model without considering the
intersuband scattering can be extended to that with considering the intersuband
scattering [61]. The compact model can also be developed based on the
quasi-ballistic transport model.
[124], [125], [126]
124
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Appendix A: Self-Consistent Calculation of the
Top-of-the-Barrier Semiclassical Ballistic Transport
Model
We introduce the self-consistent calculation flow in the cylindrical NW MOSFETs, in
detail. Since we can neglect the longitudinal electric field at the top of the barrier or
under the low-field limit, the 3D differential equations can be reduced to a 2D
problem. We adoped the cylindrical coordinate system for the cylindrical NW
MOSFETs [47], [124], [125]. The subband structure and electrostatics are calculated
as following three steps:
(1) By substituting arbitrary potential into Schrödinger equation, we can obtain the
subband structure and local electron density. In the cylindrical coordinate system,
the radial Schrödinger equation with spatially varying effective mass mµ(r) is
described in
)()()()(2)(
112 2
22
c
2
rRErRrUrl
rmdrd
rmr
drd
r µµµµ
=⎥⎥⎦
⎤
⎢⎢⎣
⎡++⎟⎟
⎠
⎞⎜⎜⎝
⎛−
hh , (A.1)
where Rµ(r) and Eµ are the radial envelope wave function and the energy
eigenvalue associated with the subband µ. The subband index, µ, includes the
valley index, principal quantum number, and angular quantum number [47]. l is
the angular quantum number, which is integer. Boundary conditions to solve (A.1)
are: dRµ / dr = 0 at r = 0 for l = 0; Rµ(0) = 0 for l ≠ 0; and Rµ(d / 2 + tox) = 0
regardless of the l. According to the top-of-the-barrier ballistic transport model, we
can calculate the radial electron density, n(r), at the top of the barrier as described
in
134
[ ]∑ ∫
∑∞
+=
=
µ
µµ
µµµ
µ
ρE
dEEfEfE
rR
nrRrn
)()(2
)()(
)()(
DS2
2
, (A.2)
where nµ and ρµ(E) are the number of electron and density of states, respectively.
According to the top-of-the-barrier semiclassical ballistic transport model [35],
[90], source and drain equilibrium distribution functions, fS and fD, are defined as
(2.66) and (2.67);
(2) By substituting the calculated local electron density into the Poisson equation, we
can obtain new potential at the top of the barrier. The radial Poisson equation with
spatially varying dielectric constant, εr(r), in terms of cylindrical coordinate
system is described in
0
a2
r])([)()(1
εε Nrnq
drrdUrr
drd
r+
−=⎥⎦⎤
⎢⎣⎡ , (A.3)
where Nd+ is donor concentration. Boundary conditions to solve (A.3) are dU / dr = 0
at r = 0 and U(d / 2 + tox) = − qVg; After correcting old potential with new potential,
we substitute the modified potential into the Schrödinger equation at the first step.
Repeating the first and second steps until the old and new potentials converge, we
obtain the exact potential and electronic structure.
135
Appendix B: Solution of the Poisson Equation in a
Cylindrical Coordinate System
We neglect the charge within the oxide layer and assume a uniformly doped channel.
If the dielectric constant within NW region, εnw, is uniform, Poisson equation in the
cylindrical coordinate system is given by
0nw
a2 ])([)(1
εεNrnq
drrdUr
drd
r+
−=⎟⎠⎞
⎜⎝⎛ , (B.1)
or
0nw
a ])([)(1εε
ψ Nrnqdr
rdrdrd
r+
=⎟⎠⎞
⎜⎝⎛ , (B.2)
where
)()( fb rUUrq −=ψ . (B.3)
By double integral of both side of (B.2), the surface potential is described in
)0(1)(1)0()]0()([
)(
d da
0nw0 0
0nw
oxfbg
ψεεεε
ψψψ
ψφ
++=
+−=
=−−
∫ ∫∫ ∫R
w
r
w
R rdrrN
rdrqdrrrn
rdrq
R
RVV
. (B.4)
When we eliminate terms with respect to Na in full depletion or non-doped bodies,
(B.4) is reduced to
( )0nw
ieff
00nw
)(ln)0()(
εε
εεψψ
Qx
drrnrRrqR
R
−=
⎟⎠⎞
⎜⎝⎛=− ∫
. (B.5)
where
136
R
rndrqQ
R
∫−≡ 0
i , (B.6)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−≡
∫∫
R
R
rndr
rndrrRx
0
0eff
)ln(exp . (B.7)
137
Appendix C: Gate Capacitance Modeling of Planar and
Double-Gate MOSFETs
In this appendix, we describe the inversion layer capacitance Cinv = d(–Qi) / dψs for
conventional planar and DG MOSFETs with a uniformly doped p-type substrate.
Since we neglect the charge within the oxide layer, we can adopt uniform dielectric
constant within NW region, εnw. Firstly, we consider the DG MOSFET. Electrostatics
of planar and DG MOSFETs can be described by the 1D Poisson equation:
0nw
a2
2 ])([)(εε
ψ Nxnqdx
xd += , (C.1)
where Na is the acceptor concentration. By solving the Poisson equation as in [18], we
can derive the ψs as described in (C.2),
c0nw
2da
0r
avgis 2
)(ψ
εεεεψ ++
−=
wqNxQ, (C.2)
where ψc is the central potential, Qi is the electron inversion charge density per unit
surface, and xavg is the average distance of the inversion electrons from the surface.
Note that we do not need to consider the effective inversion layer centroid in planar
symmetry. Here, wd is the depletion width. The second term of the right hand side in
(C.2) is neglected in full depletion, whereas the third term of the right hand side in
(C.2) is neglected in partial depletion.
Differentiating (C.2) with respect to ψs, we derive the Cinv as described in (C.3) and
(C.4):
)()()(1
i
c
i
avg
0nw
i
0nw
avg
inv Qdd
QddxQx
C −+
−−
+=ψ
εεεε, (C.3)
in full depletion and
138
)()()(1
i
d
i
avg
0nw
i
0nw
avg
inv Qdd
QddxQx
C −+
−−
+=ψ
εεεε,
(C.4)
in partial depletion, where ψd = qNawd2 / 2εnwε0, which gives the potential drop by the
depletion charge. The first and second terms of the right hand side in (C.3) and (C.4)
compose the electrostatic capacitance Ce as. When we define the quantum capacitance
as the capacitance due to the additional potential drop to charge inversion carriers, the
third term of the right hand side in (C.3) and (C.4) corresponds to the quantum
capacitance. Note that the addition potential drops, ψc and ψd, are independent of the
electric field caused by the inversion charge. If the DOS is substantially large, then
the additional potential drop becomes small; thus, the quantum capacitance reflects
the quantum capacitance due to the finite DOS. The Cinv in planar MOSFETs is the
same as (C.4). In weak inversion, the term dψd / d(– Qi) could be approximated by
1 / Cdos as reported in [97].
139
Appendix D: Carrier Degeneracy and Injection Velocity
In this appendix, we derive the relation between the carrier degeneracy, (Efs – E1), and
the injection velocity, υinj, where we take into account only one subband. According to
the top-of-the-barrier ballistic transport model [109], the injection velocity is
uni-directional velocity associated with source Fermi level. υinj can be described in
∫
∫∞
∞
=
1
1
)(2
)(
)(
S
S
inj
E
E
dEEfEq
dEEfq
ρπυ h , (D.1)
where E1 is the minimum of the subband, and source Fermi-Dirac distribution, fS, is
given by
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
≡
TkEE
Ef
B
fsS
exp1
1)( . (D.2)
The 1D density of states ρ (ε) is given by
)(22)(
1
t
EEmE−
=hπ
ρ , (D.3)
where mt is the transport effective mass of the unprimed subband. By substituting
(D.2) and (D.3) into (D.1), we can derive the υinj as described in
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
=
− TkEEF
TkEEF
mTk
B
B
t
B
1fs2/1
1fs0
inj2υ , (D.4)
where Fj(η) is the Fermi-Dirac integral of order j [126],
∫∞
−+=
0 )exp(1)(
ηεεεη dF
j
j . (D.5)
140
Figure D.1 shows that υinj increases with increasing carrier degeneracy (Efs – E1).
When carriers are nondegenerate with Efs several kBT below E1, the Fermi-Dirac
distribution function approximates to the Maxwell-Boltzmann distribution function,
and the υinj approximates to the uni-directional thermal velocity, υth, as described in
t
B
mTk
πυ 2
th = . (D.6)
0
0 . 5
1
1 . 5
2
2 . 5
-0 . 1 -0 . 05 0 0 . 05 0 . 1
Efs – E1 (eV)
⊥inj(107cm
2/V?s)
⊥th
Figure D.1: Injection velocity as a function of carrier degeneracy (solid line). The dashed line indicatesuni-directional thermal velocity.