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

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

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

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

5. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Fabrication of NT-Array TFTs revealed 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. Quantum physics of TFT capacitance 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

6. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University integratedintegrated Physics of NT Devices on SiO 2 weak interaction electr. transport thermal coupling alignment empty space 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

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

8. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University e.d.f. is symmetric and thus j = 0 Transport Theory: What to Forget and What to Remember 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. v  tr  The asymmetric non-e.d.f. provides j > 0 (both in ballistic and diffusive model) Equilibrium distribution function is Fermi-Dirac function:

9. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Conductivity: van Hove singularities after Prof. T. Ando 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.

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

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

12. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Scattering in 1D systems is weak due to restricted phase space available for electron: k -> -k Coulomb scattering: Results 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

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

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

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

16. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Specifics of surface polaritons: electric field is not normal to the surface (at 45 o ) 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  <0 Surface Polariton in SiO 2 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  LO, at the LO phonon frequency surface wave can be obtained by solving Maxwell equations with proper boundary conditions q H E

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

18. Maxwell equations in free space Digression: A tutorial on SPP Digression:

19. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University E q Maxwell equations in free space are solved by anzatz algebraic form of Maxwell equations in free space surface requires that: H additional materials connection: Maxwell equations in free space Digression: A tutorial on SPP Digression:

20. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University E "b" for bulk "a" for air q all field components (but one) can be found from BC: frequency of the SPP provides consistency of BC: H Digression: A tutorial on SPP Digression:

21. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University E "b" for bulk "a" for air q all field components (but one) can be found from BC: frequency of the SPP provides consistency of BC: H E H J Digression: A tutorial on SPP Digression:

22. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University last component of the field can be found only with QM/QED Digression: A tutorial on SPP Digression:

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

24. Remote Polariton Scattering

25. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Estimates for SiO 2 -quartz: electric field in the air is proportional to decay constant, determined from Mxw.Eq+B.C., and F-factor relevant is proportional to the wavelength of hot electron electric field ~10 7 V / m finally the scattering time for v F ~10 8 cm / s and  SO ~150meV : for v F ~10 8 cm / s and  SO ~150meV :  ~ 10 5 V / cm Physics of SPP scattering in SiO 2

26. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Interaction potential (e-dipole) where the (dipole) polarization is calculated following Mahan et al. here q is the SPP wavenumber; x is normal to the surface F is related to Froehlich constant: and  SO is the SPP frequency Details of SPP scattering in SiO 2

27. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Scattering rate = lifetime ~ 30 meV No sharp transition could happen Selfconsistent calculation of the lifetime of the... RP-polaron RP-polaron Remote Polariton Scattering 0.020.040.060.080.1 0.65 0.75 0.8 0.85 E(k), eV k, 1/A

28. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University [19,0] NT + RPS 0.020.040.060.080.1 0.2 0.4 0.6 0.8 1 1.2 1.4 E(k), eV k, 1/A Remote Phonon Polaron Energy

29. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University T=0K - therefore, only SO-phonon emission is included  m=0 - intra-subband transitions  m=1 - inter-subband transitions (neglecting higher m's) q~1/ (forward) and q~2k (backward) scattering Remote Polariton Scattering

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

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

32. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University 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 Surface Polariton Scattering (2) for v F ~10 8 cm / s and  SO ~140meV : ~40 nm 2k i ~ 2  /a ~ 1/nm Forward scattering dominates: q~1/ : forward scattering q~2k i : backward scattering JETP Letters, 2006

33. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University for the SiO 2 (quartz) substrate the SPP scattering is likely prevailing over inelastic scattering by NT (own) optical phonons for the small distance to the polar substrate <  ~ 4 nm; the effect is even stronger for high-k dielectrics due to increase of the Froehlich constant : x20 and more; the effect is independent of the radius of the NT, thus for narrow NTs it will dominate over the other 1 / R mechanisms Surface Polariton Scattering Rate JETP Letters, 2006

34. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University 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 Remote SPP Scattering Rate lattice T T=77; 150; 210; 300; 370; 450 K

35. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University for the SiO 2 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; SPP Scattering Rate and Mobility JETP Letters, 2006 (3V,300K) 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 Nano Letters, 2009 SPP NT

36. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University for the SiO 2 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; SPP Scattering Rate and Mobility 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: R 2 for NT acoustic phonons lattice temperature is taken as given Nano Letters, 2009 lattice T

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

38. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University 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: Saturation Regime: Optical Phonons

39. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University 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). Saturation Regime: Heat Generation

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

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

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

43. Modern Electronics and Heat Dissipation Problem

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

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

46. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University 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 The mechanism appeared to be ineffective for Si MOS-FETs and was almost forgotten for decades... Pioneering work by K. Hess and P. Vogl – back to 1972 – RIP scattering in Si. VdVd q j q~area~nm 2 channel heating due to Joule losses and low thermal coupling to leads q j Joule Heat Generation

47. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : NT phonons warm the NT lattice but the Kapitsa resistance is high overheating of the channel : we neglect the thermal sink in the leads (area~nm 2 ), then only substrate contributes via thermal coupling: where j qCqC q ph Q SPP SPP and Overheating Material g=1/ , W/(m·K) Silica Aerogel0.004 - 0.04 Air0.025 Wood / wool0.04 - 0.4 Water (liquid)0.6 Thermal epoxy1 - 7 Glass1.1 Concrete, stone1.7 – 2.4 Stainless steel12.11 ~ 45.0 Aluminium200 Copper380 Silver429 Diamond900 - 2320

48. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University assume for a moment that SPP channel is absent overheating of the channel : we neglect the thermal sink in the leads (area~nm 2 ), then only substrate contributes via thermal coupling: Joule losses are NOT the same as the total dissipation: NT phonons take only a small fraction of I d F where j q ph Q SPP SPP and Overheating two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : NT phonons warm the NT lattice but the Kapitsa resistance is high

49. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University assume for a moment that SPP channel is absent overheating of the channel : we neglect the thermal sink in the leads (area~nm 2 ), then only substrate contributes via thermal coupling: Joule losses are NOT the same as the total dissipation: NT phonons take only a small fraction of I d F where j q ph Q SPP SPP and Overheating two scattering (NT and SPP) and two coupling (SPP and Kapitsa) mechanisms : NT phonons warm the NT lattice but the Kapitsa resistance is high 1 2 5 10 20 50 100 200 24681012 F (V/mm) P SPP /P NT substrate T

50. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University opposite R-dependence for two scattering mechanisms ratio of "real"-to-expected losses for two tubes (R~0.5 and 1.0 nm) at two t o = 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 SPP and Overheating (2)

51. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University different temperature dependence for two scattering mechanisms 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 SPP and Overheating (2)

52. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Conclusions Theory of NT scattering after 10 years still has new uncovered physics Physics of interactions in NTs at the hetero-interface with Si/SiO 2 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

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

55. ITRS Grand Challenges: The Heat Among main evaluation parameters for novel semiconductor electronics technologies the power consumption, and in particular the power dissipation become more and more important Main performance metrics to be compared with the existing Si-CMOS technology: Cost Speed Integration and... Power

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

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

58. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Quantum Mechanical view: Selfconsistent calculation of the charge density Rotkin et.al. JETP-Letters, 2002 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:  ~ - C  ext which is to be compared with 3D and 2D:  ~ - d 2  /dx 2  ~ - d  /dx Atomistic Capacitance of 1D FET

59. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University which is to be compared with 3D and 2D:  ~ - d 2  /dx 2  ~ - d  /dx 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:  ~ - C  ext Quantum Mechanical view: Selfconsistent calculation of the charge density Rotkin et.al. JETP-Letters, 2002 Atomistic Capacitance of 1D FET

60. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University 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 Screening depends on single parameter: 2d/  o 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 2d/  Fig. : Gate coupling in array-TFT as a function of the screening by neighbor NTs (top to bottom): same SiO 2 thickness = 1.5 um, NT densities = 0.2, 0.4 and 2 NT/um 1  m

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

62. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University The capacitance of a random TFT array (a single given realization) as a function of the external screening (insulator thickness). Correlation vs. Randomness  C, % d, nm 255075100125150 2.4 2.6 2.8 3.2 3.4 3.0 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.

63. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University In a single tube FET total capacitance has 2 terms: geometric capacitance and quantum capacitance for NT array geometrical capacitance further decreases: 102050100200500 0.5 0.6 0.7 0.8 0.9 1 d, nm C/C class  Quantum Capacitance in NT-Array TFT

64. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Capacitance and Gate Coupling In the limit of dense array (or far gate) the coupling is close to 1 - maximum allowed coupling. Performance of the array transistor is comparable to the flat channel one.  To compare effectiveness of the gate coupling for the NT TFT and a flat channel FET we introduce a coupling ratio  which is the amount of surface charge of the array transistor as compared to the planar device 2d/ 

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

69. Scattering rate = lifetime ~ 30 meV No sharp transition could happen Selfconsistent calculation of the lifetime of the... RP-polaron RP-polaron Remote Polariton Scattering 0.020.040.060.080.1 0.65 0.75 0.8 0.85 E(k), eV k, 1/A

70. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University [19,0] NT + RPS 0.020.040.060.080.1 0.2 0.4 0.6 0.8 1 1.2 1.4 E(k), eV k, 1/A Remote Phonon Polaron Energy

71. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University T=0K - therefore, only SO-phonon emission is included  m=0 - intra-subband transitions  m=1 - inter-subband transitions (neglecting higher m's) q~1/ (forward) and q~2k (backward) scattering Remote Polariton Scattering

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

74. Charge Trapping

75. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Physics of NT Field- Effect Transistor (FET): NT channel is conducting at V g =0 (non-intentional p- doping) long mean free path (due to 1D symmetry) optical phonons limit the high field current work function of the electrodes defines the height of the contact Schottky barrier Single NT FET insulator gate @ V g source @ ground drain @ V d 1D channel Gate voltage (charge of the gate electrode and "its vicinity") controls the transport

76. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University decreasing scan rate Hysteresis in SWNT-array Transistors Robert-Peillard, Rotkin, 2005 Experiment: Laminated Device, CVD Tubes Courtesy J.Rogers

77. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Physics of current hysteresis in NT FETs: Gate voltage controls the charge of the channel In addition to the charge stored in the gate (gate capacitance), strong electric field generates charges at the interfaces (add.capacitances) This shifts the threshold voltage (and changes mobility) The field is self- consistent with the charge NT FET: Rad-hardening problem insulator gate @ V g source @ ground drain @ V d 1D channel

78. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Field Damping Factor  SWNT @ low-  substrate, all lengths in nanometers

79. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University d Field Damping Factor  SWNT @ low-  substrate, all lengths in nanometers

80. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Gate Coupling Factor 

81. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Gate Coupling Factor d

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

83. Equilibrium d.function is known: it is symmetric and thus j = 0 Boltzmann Transport Equation allows to compute non-equilibrium distribution function collision integral full derivative of f (t, r, p) Newton's law Boltzmann Transport Equation (0)

84. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Boltzmann Transport Equation (1) for elastic scattering collision integral is needed to compute non-equilibrium distribution function from BTE : I (f) and finally collision integral is:

85. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Boltzmann Transport Equation (2) to solve equation we need transition probability for I(f) which we get from Fermi Golden Rule consider example of Coulomb impurity scattering: then matrix elements are Fourier components of V(r) impurities are equivalent

86. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Boltzmann Transport Equation (2) we substitute square of the matrix element in FGR : when averaging series over impurity positions only diagonal terms are to be kept (RPA) : we change summation in I(f) into integration and finally obtain :

87. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University BTE: Relaxation time approximation we assume non-equilibrium d.f. is "close" to equilibrium d.f. we note that collision integral must be 0 for f 0 because it's at equilibrium: relaxation time approximation : for example, for Coulomb impurity scattering rel.time is: then :

88. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Relaxation time approximation (2) to solve Boltzmann Transport Equation we assume then, we neglect f 1 in the l.h.s. and obtain the solution : which is to be used to compute current, next: (en passant: we assume ∂f/∂r=0) and simplify it :

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

90. For Fermi-Dirac degenerate e-gas f 0 is step function, thus : Relaxation time approximation (3) remember, in electromagnetism the current is where  is the conductivity we obtain thus QM expression for  where diffusivity is and conductivity reduces to the Drude formula

91. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Remote impurity Scattering BTE conductivity for particular case of RiS : conductivity contains DoS (same as velocity) the electron velocity near the band edge is constant for M- SWNT and f(EF) for S-SWNT in TBA one obtains

92. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University finally the relaxation time for the RiC scattering potential is conductivity depends on the Fermi level Remote impurity Scattering

93. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Scattering: Analytical results - universality semiconducting and metal NT Numerical results: scattering rate vs. EF Remote impurity Scattering (2) interband scattering depends on NT type to dependence of the scattering

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

95. Scattering amplitude relaxation time can be defined via total probability of scattering from some state |k> into all other states |k'> : QM scattering amplitude can be found using Fermi Golden Rule the conductivity requires DoS, distribution function, and  Our next goal is the scattering matrix element

96. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Using TB to find the scattering rate Scattering rate for the Coulomb potential calculation of the matrix element starts with TB-wavefunctions Fourier of 3D Coulomb potential In the envelope function approximation we end with the 1D Fourier of Coulomb potential

97. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Fourier Transform in 1D which is effectively 2D Coulomb potential 1D Fourier of Coulomb potential can be simplified at large distances: At small distances 1D Coulomb is diverging if not screened -212 -1.5 -0.5 0.5 z/R V after screening we obtain the effective triangular potential (i.e. an effective electric field, E )

98. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University NT Boltzmann Transport Equation using FGR in Born Approximation: we introduce mean free path then, partial conductivity (due to 1 subband / 1 channel): and total conductivity is as follows:

99. NCN Seminar, UIUC Mar 4 2009 Slava V Rotkin, Lehigh University Scattering matrix elements in NTs k  EFEF Backscattering is only process that results in finite resistance BTE reads as: K+ K-K'+ K'- Prof. T. Ando Backscattering probability is greatly reduced in NTs for a potential of non-vanishing radius

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

101. 1.Stacy E. Snyder, and Slava V. Rotkin, “Optical Identification of a DNA-Wrapped Carbon Nanotube: Signs of Helically Broken Symmetry", Small, accepted, 2008. 2.Seong Jun Kang, Coskun Kocabas, Taner Ozel, Moonsub Shim, Ninad Pimparkar, Muhammad A. Alam, Slava V. Rotkin, and John A. Rogers, “High performance electronics using dense, perfectly aligned arrays of single walled carbon nanotubes”, Nature Nanotechnology, vol. 2 (no.4) 230-236 (2007). 3.Vadim Puller, and Slava V. Rotkin, "Helicity and Broken Symmetry in DNA-Nanotube Hybrids", Europhysics Letters 77 (2), 27006--1-6 (Jan 2007). 4.Qing Cao, Ming-Gang Xia, Coskun Kocabas, Moonsub Shim, John A. Rogers, and Slava V. Rotkin, “Gate Capacitance Coupling of Single-walled Nanotube Thin-film Transistors”, Applied Physics Letters, vol. 90 (2), 023516 (2007). 5.Slava V. Rotkin, Narayan R. Aluru, and Karl Hess, ”Multiscale Theory and Modeling of Carbon Nanotube Nano-Electromechanical Systems”, in "Handbook of Nanoscience, Engineering and Technology (2nd Edition)", Eds.: W. Goddard, D. Brenner, S. Lyshevski, G.J. Iafrate; Taylor and Francis-CRC Press, Chapter 13, pp. 13.20-13.32 (2007). 6.Slava V. Rotkin, Alexander Shik, “Electrostatics of nanowires and nanotubes: Application for field-effect devices”, in the Special Issue Nanowires and Nanotubes, Editor: Peter Burke, Publ.: World Scientific, Singapore. International Journal of High Speed Electronics and Systems, vol. 16 (no.4), 937-958, (2006). 7.Stacy E. Snyder, and Slava V. Rotkin, “Polarization component of the cohesion energy in the complexes of a single-wall carbon nanotube and a DNA", JETP Lett 84, 348, (2006). 8.Alexey G. Petrov, Slava V. Rotkin, “Hot carrier energy relaxation in single-wall carbon nanotubes via surface optical phonons of the substrate” JETP Lett 84 (3), 156-160 (2006). 9.Yan Li, Umberto Ravaioli, and SV. Rotkin, "Metal-Semiconductor Transition and Fermi Velocity Renormalization in Metallic Carbon Nanotubes", Phys. Rev. B 73, 035415 (2006). 10.L. Rotkina, S. Oh, J.N. Eckstein, S.V. Rotkin, “Logarithmic behavior of the conductivity of electron-beam deposited granular Pt/C nanowires”, Phys. Rev. B 72, 233407 (2005). 11.Salvador Barraza-Lopez, Slava V. Rotkin, Yan Li, and Karl Hess, "Conductance Modulation of Metallic Nanotubes by Remote Charged Rings", Europhysics Lett 69, 1003 (2005). 12.Slava V. Rotkin, “From Quantum Models to Novel Effects to New Applications: Theory of Nanotube Devices”, in “Applied Physics of Nanotubes: Fundamentals of Theory, Optics and Transport Devices”, Nanoscience and Nanotechnology Series, Ser.Ed.: Ph. Avouris, Springer Verlag GmbH & Co. KG (2005). 13.Yan Li, Deyu Lu, Klaus Schulten, Umberto Ravaioli, and Slava V. Rotkin, “Screening of Water Dipoles Inside Finite-Length Armchair Carbon Nanotubes”, Journal of Computational Electronics, vol. 4, 161-165 (2005). 14.Arnaud Robert-Peillard, Slava V. Rotkin, “Modeling Hysteresis Phenomena in Nanotube Field-Effect Transistors”, IEEE Transactions on Nanotechnology, 4 (2), 284-288 (2005). 15.Deyu Lu, Yan Li, Slava V. Rotkin, Umberto Ravaioli, and Klaus Schulten, “Finite-Size Effect and Wall Polarization in a Carbon Nanotube Channel”, Nano Lett 4, 2383-2387 (2004). 16.Yan Li, Slava V. Rotkin, and Umberto Ravaioli, "Metal-Semiconductor Transition in Armchair Carbon Nanotubes by Symmetry Breaking", Applied Physics Lett 85, 4178 (2004). 17.Alexey G. Petrov, Slava V. Rotkin, "Transport in Nanotubes: Effect of Remote Impurity Scattering", Phys. Rev. B vol. 70 (3), 035408-1-10, 15 Jul 2004. 18.Slava V. Rotkin, and Karl Hess, "Possibility of a Metallic Field-Effect Transistor", Applied Physics Letters vol. 84 (16), p.3139-3141, 19 April 2004. 19.Slava V. Rotkin, Harry Ruda, Alexander Shik, "Field-effect transistor structures with a quasi-1D channel", International Journal of Nanoscience vol. 3 (1/2), 161-170, Feb 2004. 20.Kirill A. Bulashevich, Slava V. Rotkin, Robert A. Suris, "Excitons in Single Wall Carbon Nanotubes", International Journal of Nanoscience vol. 2 (6), pp. 521-526, Dec 2003. 21.Slava V. Rotkin, Harry Ruda, Alexander Shik, "Universal Description of Channel Conductivity for Nanotube and Nanowire Transistors", Applied Physics Letters 83, 1623, 2003. 22.Alexey G. Petrov, Slava V. Rotkin, "Breaking of Nanotube Symmetry by Substrate Polarization", Nano Letters vol. 3, No.6, 701-705, 2003. 23.Yan Li, Slava V. Rotkin, Umberto Ravaioli, "Electronic response and bandstructure modulation of carbon nanotubes in a transverse electrical field", Nano Letters 3, 183, 2003. 24.Slava V. Rotkin, "Theory of Nanotube Nanodevices", in Nanostructured Materials and Coatings for Biomedical and Sensor Applications. Editors: Y.G. Gogotsi and Irina V. Uvarova. Kluwer, pp. 257-277, 2003. 25.Slava V. Rotkin, Vaishali Shrivastava, Kirill A. Bulashevich, and Narayan R. Aluru, "Atomistic Capacitance of a Nanotube Electromechanical Device", International Journal of Nanoscience vol. 1, No. 3/4, 337-346, 2002. 26.Slava V. Rotkin, Ilya Zharov, "Nanotube Light-Controlled Electronic Switch", International Journal of Nanoscience vol. 1, No. 3/4, 347-355, 2002. 27.Kirill A. Bulashevich, Slava V. Rotkin, "Nanotube Devices: Microscopic Model", JETP Letters vol. 75 (4), 205-209, 2002. 28.Slava V. Rotkin, Yuri Gogotsi, "Analysis of non-planar graphitic structures: from arched edge planes of graphite crystals to nanotubes", Materi. Res. Innovations, 5, 191, 2002. 29.Marc Dequesnes, Slava V. Rotkin, Narayan R. Aluru, "Parameterization of continuum theories for single wall carbon nanotube switches by molecular dynamics simulations", Journal of Computational Electronics 1 (3), 313-316, 2002. 30.Slava V. Rotkin, Karl Hess, "Many-body terms in van der Waals cohesion energy of nanotubes", Journal of Computational Electronics 1 (3), 323-326, 2002. 31.Marc Dequesnes, Slava V. Rotkin, Narayan R. Aluru, "Calculation of pull-in voltages for carbon nanotube-based nanoelectromechanical switches", Nanotechnology 13, 120, 2002. List of publications used in this presentation: downloadable from http://theory.physics.lehigh.edu/rotkin/text/pub-list.html