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Silicon-Interface Scattering in Carbon Nanotube Transistors

Silicon-Interface Scattering in Carbon Nanotube Transistors. Slava V. Rotkin. Physics Department & Center for Advanced Materials and Nanotechnology Lehigh University. Acknowledgements. Dr. A.G. Petrov (Ioffe) Prof. J.A. Rogers (UIUC) Dr. V. Perebeinos and Dr. Ph. Avouris (IBM)

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Silicon-Interface Scattering in Carbon Nanotube Transistors

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  1. Silicon-Interface Scattering in Carbon Nanotube Transistors Slava V. Rotkin Physics Department & Center for Advanced Materials and Nanotechnology Lehigh University NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  2. Acknowledgements Dr. A.G. Petrov (Ioffe) Prof. J.A. Rogers (UIUC) Dr. V. Perebeinos and Dr. Ph. Avouris (IBM) Prof. K. Hess (UIUC) and Prof. P. Vogl (UVienna) NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  3. OUTLINE Introduction: - NT Transistors with "non-monolithic" channel The old "new" Surface Scattering - Remote Coulomb Impurity scattering - Remote Polariton Scattering Physics of Surface Phonon Polariton (SPP) SPP and heat dissipation in NT devices Conclusions NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  4. NT Transistors _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  5. Quantum physics of TFT capacitance • Fabrication of NT-Array TFTsrevealed new "old" physics. • very large gate coupling – too strong if not taking into account intertube coupling • non-uniformity of the channel – self-screening and "defect healing" • multi-layer dielectrics and surface E/M modes • interface scattering Most of the tubes are parallel, but the distance between neighbor tubes may vary. For TFT applications only semiconductor tubes are needed. Thus one needs to destroy (burn out) metallic tubes. Which randomizes the channel. self-consistent modeling (Poisson+Schroedinger eqs) including e/m response NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  6. Physics of NT Devices on SiO2 Weak van der Waals interactions... For a polar substrate -- such as quartz, sapphire, calcite -- new physics due to evanescent Electro-Magnetic (EM) modes, aka Surface Phonon-Polariton modes • weak interaction • electr. transport • thermal coupling • alignment empty space integrated NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  7. Charge Scattering:Short Introduction _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  8. Transport Theory: What to Forget and What to Remember Equilibrium distribution function is Fermi-Dirac function: e.d.f. is symmetric and thus j = 0 The asymmetric non-e.d.f. provides j > 0 (both in ballistic and diffusive model) Quantum-mechanical calculation of the conductivity may be reduced to the Drude formula electron velocity which enters the formula can be related to m.f.p. vttr=L NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  9. Conductivity: van Hove singularities Scattering rate is proportional to electron velocity which diverges at the subband edge. Thus, the Drude conductivity has "zeroes" at vHs. Which holds for both metallic and semiconductor tubes. after Prof. T. Ando NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  10. Remote impurity Scattering _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  11. Coulomb Center Scattering Scattering in 1D systems is weak due to restricted phase space available for electron: k -> -k the Coulomb impurities are on the substrate, not within the NT lattice – the remote impurity scattering on average the Coulomb potential where e* and nS are the charge and density of impurities NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  12. Coulomb scattering: Results Scattering in 1D systems is weak due to restricted phase space available for electron: k -> -k Within this model a universal expression for conductance was found Modeling uses the nonequilibrium solution of the Boltzmann transport equation where a quantum mechanical scattering rate is calculated in the Born Approximation and parameterized by the strength of the Coulomb centers' potential and DoS NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  13. RIS Details: Statistical averaging Statistical averaging over a random impurity distribution of starting with the Coulomb potential on average is proportional to strength of potential DoS scattering form-factor then, the scattering rate is here we used notations: and NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  14. Surface Phonon Polariton _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  15. Digression: A tutorial on SPP NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  16. Surface Polariton in SiO2 • Surface phonons exist in polar dielectrics: • due to the dielectric function difference between the substrate and the air, a surface EM wave could exist • dielectric function of the polar insulator has a zero at wLO, atthe LO phonon frequency • surface wave can be obtained by solving Maxwell equations with proper boundary conditions • Specifics of surface polaritons: • electric field is not normal to the surface (at 45o) • electric field decays exponentially from the surface (not a uniform solution of Maxwell equations) • existence of a surface mode essentially depends on existence of the anomalous dispersion region e<0 E q H NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  17. Remote Polariton Scattering _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  18. Physics of SPP scattering in SiO2 • Estimates for SiO2-quartz: • electric field in the air is proportional to decay constant, determined from Mxw.Eq+B.C., and F-factor • relevant l is proportional to the wavelength of hot electron • electric field ~107V/m • finally the scattering time for vF~108cm/s and wSO~150meV : e ~ 105V/cm NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  19. Conductivity: van Hove singularities Scattering rate is proportional to the velocity which diverges at the subband edge. Thus, the Drude conductivity has peculiarities at vHs. reminder Prof. T. Ando NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  20. Surface Polariton Scattering • RPS rate varies for intra-subband andinter-subband scattering • RPS has maximum at the van Hove singularities (for semiconductor-SWNT) inter-subband transitions are negligible due to non-zero angular momentum transfer JETP Letters, 2006 At vHs our Born approximation fails which manifests itself as diverging scattering rate NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  21. Surface Polariton Scattering (2) Correct many-body picture includes phonon renormalization of the electron spectrum. Within iterative Quantum Mechanical calculation (aka SCBA) new scattering rate obtained: - averaged near the vHs - still faster than other channels JETP Letters, 2006 Forward scattering dominates: q~1/l : forward scattering q~2ki : backward scattering for vF~108cm/s and wSO~140meV : l~40 nm 2ki ~ 2p/a ~ 1/nm NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  22. Remote SPP Scattering Rate T=77; 150; 210; 300; 370; 450 K lattice T • scattering rate increases with the electric field strength because of stronger warming of the electron distribution function • similarly it increases with the temperature • concentration dependence is weak and can be attributed to the tails of distribution function NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  23. SPP Scattering Rate and Mobility • for the SiO2 substrate the SPP channel is likely prevailing over inelastic scattering, such as due to NT (own) optical phonons for the small distance to the polar substrate < ~ 4 nm; JETP Letters, 2006 (3V,300K) Nano Letters, 2009 • low-field mobility at 100+K is totally dominated by SPP • the effect is even stronger for high-k dielectrics due to increase of the Froehlich constant : x20 and more; • RPS has a weak dependence on the NT radius, thus for narrow NTs it will dominate over the other 1/R mechanisms SPP NT NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  24. SPP Scattering Rate and Mobility • for the SiO2 substrate the SPP channel is likely prevailing over inelastic scattering, such as due to NT (own) optical phonons for the small distance to the polar substrate < l ~ 4 nm; JETP Letters, 2006 • SPP low-field mobility for a large number of various chirality NTs allows to infer empirical scaling on the NT radius • comparison with other mechanisms: R2 for NT acoustic phonons • lattice temperature is taken as given lattice T Nano Letters, 2009 NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  25. Saturation Regime _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  26. Saturation Regime: Optical Phonons Scattering in 1D systems is weak due to restricted phase space available for the electron: k -> -k. However, the strong scattering at high drift electric field is inevitable: saturation regime. The scattering mechanism is an optical phonon emission which results in fast relaxation rates for the hot electrons and holes. Inelastic scattering rates have been calculated for SWNTs earlier: NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  27. Saturation Regime: Heat Generation What was known so far? Inelastic optical phonon relaxation scattering is likely a factor determining the saturation current in SWNTs : The hot electron energy is transferred to the SWNT phonon subsystem. The energy dissipation depends on the environment (thermal coupling). NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  28. SPP and Saturation Regime Kane, PRL, 2000 Deviation from Ohm's law: first nonvanishing term in R(Vd)=Ro +Vd/Io Inverse drain current vs. inverse applied electric field low-F and high-F Is are essentially different, being determined by different scattering mechanisms [17,0] NT at the doping level 0.1 e/nm NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  29. SPP and Saturation Regime Kane, PRL, 2000 Deviation from Ohm's law: first nonvanishing term in R(Vd)=Ro +Vd/Io Inverse drain current vs. inverse applied electric field low-F and high-F Is are essentially different, being determined by different scattering mechanisms low-F scattering is due to all phonons (including NT intrinsic phonon modes) and high-F scattering is due to SPP mechanism NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  30. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  31. Modern Electronics andHeat Dissipation Problem _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  32. ITRS Grand Challenges: The Heat ? S. Borkar, “Design challenges of technology scaling,” IEEE Micro, vol. 19 (4), 23–29, Jul.–Aug. 1999. "Energy in Nature and Society: General Energetics of Complex Systems" by V. Smil (2008) Among main evaluation parameters for novel semiconductor electronics technologies the power consumption, and in particular the power dissipation become more and more important NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  33. SPP Heat Dissipation _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  34. Joule Heat Generation Vd j q q j q~area~nm2 channel heating due to Joule losses and low thermal coupling to leads It exists, however, a relaxation mechanism which transfers the energy directly to the substrate without intermediate exchange with the SWNT lattice (phonons) which is an inelastic remote optical phonon scattering Pioneering work by K. Hess and P. Vogl – back to 1972 – RIP scattering in Si. The mechanism appeared to be ineffective for Si MOS-FETs and was almost forgotten for decades... NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  35. SPP and Overheating j overheating of the channel : we neglect the thermal sink in the leads (area~nm2), then only substrate contributes via thermal coupling: qC qph where QSPP • two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : • NT phonons warm the NT lattice but • the Kapitsa • resistance is high NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  36. SPP and Overheating j overheating of the channel : we neglect the thermal sink in the leads (area~nm2), then only substrate contributes via thermal coupling: qph where • two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : • NT phonons warm the NT lattice but • the Kapitsa • resistance is high QSPP • assume for a moment that SPP channel is absent • Joule losses are NOT the same as the total dissipation: NT phonons take only a small fraction of IdF NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  37. SPP and Overheating 2 4 6 8 10 12 F (V/mm) PSPP/PNT 200 100 50 20 10 5 2 1 j overheating of the channel : we neglect the thermal sink in the leads (area~nm2), then only substrate contributes via thermal coupling: qph where • two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : • NT phonons warm the NT lattice but • the Kapitsa • resistance is high QSPP substrate T • assume for a moment that SPP channel is absent • Joule losses are NOT the same as the total dissipation: NT phonons take only a small fraction of IdF NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  38. SPP and Overheating (2) • ratio of "real"-to-expected losses for two tubes (R~0.5 and 1.0 nm) at two to= 77 and 300K • inset: data collapse for (linear) dependence on the electron concentration (0.1 and 0.2 e/nm) • SPP scattering is higher in smaller diameter tubes: simply the SPP field is stronger • opposite R-dependence for two scattering mechanisms NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  39. SPP and Overheating (2) • even in case of no other thermal coupling to substrate, SPP channel releases the heat (R~0.5 nm, T=300K) • inset: same data vs. Joule loss • NT transport in saturation regime is determined by both channels • different temperature dependence for two scattering mechanisms NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  40. Conclusions • Theory of NT scattering after 10 years still has new uncovered physics • Physics of interactions in NTs at the hetero-interface with Si/SiO2 is rich for fundamental research • Hot electron scattering due to SPP modes is by orders of magnitude faster channel for non-suspended NT • Remote SPP scattering provides a new and very effective thermo-conductivity mechanism NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  41. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  42. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  43. Nanotube Quantum Capacitance _____________ NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  44. L R d Classical Capacitance: 1D case Classical 1D capacitance: line charge has f = r 2 log r + const therefore: Cg-1 = 2 log z/R where z = min(d, L, lg) Distance to metal leads around/nearby 1D channel defines the charge density r(z) is different for different screening of 1D, 2D and 3D electrodes. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  45. Atomistic Capacitance of 1D FET The transverse size a of nanowires and nanotubes is less than the Debye screening length and other microscopic lengths of the material. Classic view: Linear connection between electric potential and charge Q=C V , in a 1D device: r ~ - C jext which is to be compared with 3D and 2D: r ~ - d2j/dx2r ~ - dj/dx Quantum Mechanical view: Selfconsistent calculation of the charge density Rotkin et.al. JETP-Letters, 2002 NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  46. Atomistic Capacitance of 1D FET The transverse size a of nanowires and nanotubes is less than the Debye screening length and other microscopic lengths of the material. Classic view: Linear connection between electric potential and charge Q=C V , in a 1D device: r ~ - C jext which is to be compared with 3D and 2D: r ~ - d2j/dx2r ~ - dj/dx Quantum Mechanical view: Selfconsistent calculation of the charge density Rotkin et.al. JETP-Letters, 2002 NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  47. 1 mm 1 mm 1 mm Capacitance of the NT Array Method of potential coefficients (or EE circuit analysis): Screening by neighbor NTs in the array – total capacitance is of a bridge circuit Fig. : Gate coupling in array-TFT as a function of the screening by neighbor NTs (top to bottom): same SiO2 thickness = 1.5 um, NT densities = 0.2, 0.4 and 2 NT/um 2d/L Screening depends on single parameter: 2d/Lo which has a physical meaning of the number of NTs electrostatically coupled in the array. The tubes that are further apart do not "know" about each other NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  48. Random Array Coupling: Self-healing DC/C -0.15 -0.25 -0.35 Current nonuniformity is a deficiency for device production. ConsiderDr due to non-uniform screening. d=40 nm d=600 nm Three sample distributions of the tubes in the random-tube array (d=160 nm, 80% variance). One may expect a severe variance in device characteristics because of non-uniform Cg NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  49. Correlation vs. Randomness DC, % 3.4 d, nm 3.2 3.0 25 50 75 100 125 150 2.8 2.6 2.4 The capacitance of a random TFT array (a single given realization) as a function of the external screening (insulator thickness). The low density TFT array is within a single tube limit... ...in the high density TFT array the inter-NT coupling is very strong and stabilizes the overall device response. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

  50. 1 C/Cclass 0.9 0.8 0.7 0.6 d, nm 0.5 10 20 50 100 200 500 Quantum Capacitance in NT-Array TFT In a single tube FET total capacitance has 2 terms: geometric capacitance and quantum capacitance for NT array geometrical capacitance further decreases: L NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University

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