Heterogeneous carbon-based devices: Towards integration with Si technology Slava V. Rotkin Physics Department & Center for Advanced Materials and Nanotechnology, Lehigh University

USC, May 28, 2009 Slava V Rotkin 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)

USC, May 28, 2009 Slava V Rotkin OUTLINE Motivation: NT array Thin Film Transistors (TFT) - Charge coupling: Classical and Quantum terms - When "dice" has only "6" face 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

USC, May 28, 2009 Slava V Rotkin NT-Array Thin Film Transistors

USC, May 28, 2009 Slava V Rotkin Courtesy Prof. John Rogers (UIUC) NT aligned array : Novel type TFT Novel fabrication technique (Left) allows fabrication of Thin-Film Transistors of parallel NT arrays. Courtesy Prof. John Rogers (UIUC) X-cut [2-1-10] Z-cut [0001]Y-cut [01-10] SEM reconstruction ("fake" 3D view) of NT-TFT and gold electrodes. SEM of NT growth on different quartz facets NTs can be transferred on plastic

USC, May 28, 2009 Slava V Rotkin Aligned NT for Transparent Electronics NTs are so small that absorption in a single layer of well- separated tubes is negligible Adapted from Zhou (2008)

USC, May 28, 2009 Slava V Rotkin Aligned NT Growth Courtesy J.A. Rogers NT alignment is not independent of the gas flow direction: competition of gas flow and surface alignment = serpentine growth Criss-crossed NT arrays NT transistor as an element of a FM-radio

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 V g ground V d 1D channel Gate voltage (charge of the gate electrode and "its vicinity") controls the transport

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin Nanotube Quantum Capacitance

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin 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

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

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/ 

USC, May 28, 2009 Slava V Rotkin 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  C/C One may expect a severe variance in device characteristics because of non-uniform C g

USC, May 28, 2009 Slava V Rotkin 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 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.

In a single tube FET total capacitance has 2 terms: geometric capacitance and quantum capacitance for NT array geometrical capacitance further decreases: d, nm C/C class  Quantum Capacitance in NT-Array TFT

Charge Scattering: Short Introduction

USC, May 28, 2009 Slava V Rotkin 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 enters the formula The asymmetric non-e.d.f. provides j > 0 (both in ballistic and diffusive model) Equilibrium distribution function is Fermi-Dirac function:

USC, May 28, 2009 Slava V Rotkin 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.

Remote impurity Scattering

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

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

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

Saturation Regime and Heat Dissipation Problem

USC, May 28, 2009 Slava V Rotkin 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: However, recent optics experiments indicated that the relaxation rates for hot electrons are even faster, which suggests a possibility for a new unknown scattering mechanism. Saturation Regime: Heat Generation

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin 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-S in Si. VdVd q j q~area~nm 2 channel heating due to Joule losses and low thermal coupling to leads q j Heat Generation (2)

Surface Phonon Polariton

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 in polar dielectrics: due to the dielectric function difference between the substrate and the air, a surface e.m.w. could exist dielectric function of the polar insulator has a singularity at the LO phonon frequency surface wave with a strong decay of the electric field in the air appears and interacts with the NT charges

USC, May 28, 2009 Slava V Rotkin Digression: A tutorial on SPP Digression:

USC, May 28, 2009 Slava V Rotkin Maxwell equations in free space Digression: A tutorial on SPP Digression:

USC, May 28, 2009 Slava V Rotkin 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: Digression: A tutorial on SPP Digression: Maxwell equations in free space

USC, May 28, 2009 Slava V Rotkin 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:

USC, May 28, 2009 Slava V Rotkin

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:

USC, May 28, 2009 Slava V Rotkin last component of the field can be found only with QM/QED Digression: A tutorial on SPP Digression:

USC, May 28, 2009 Slava V Rotkin

Remote Polariton Scattering

Estimates for SiO 2 -quartz: electric field in the air is proportional to decay constant, determined from MEq+BC, 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

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin Scattering rate = lifetime ~ 30 meV No sharp transition could happen Selfconsistent calculation of the lifetime of the... RP-polaron RP-polaron Remote Polariton Scattering E(k), eV k, 1/A

USC, May 28, 2009 Slava V Rotkin [19,0] NT + RPS E(k), eV k, 1/A Remote Phonon Polaron Energy

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin 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

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

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

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin Conclusions Theory of NT scattering is not complete yet Physics of interactions in NTs at the hetero- interface with Si/SiO 2 is rich Hot electron scattering due to SPP modes provides a new and very effective thermo- conductivity mechanism Graphenes – another example of nano-hetero- interface where quantum effects may nicely develop into effects useful for applications

USC, May 28, 2009 Slava V Rotkin scattering rate increases with the electric field strength because of stronger warming of the electron distribution function Remote SPP Scattering Rate

USC, May 28, 2009 Slava V Rotkin Remote SPP Scattering IVCs with and without taking into account SPP mechanism The saturation regime is clearly seen at larger bias (larger field) for SPP scattering Inset: mobility vs. field

USC, May 28, 2009 Slava V Rotkin overheating of the channel : neglecting the thermal sink in the leads (~nm 2 ) Remote SPP Scattering two scattering mechanisms : NT phonons warm the NT lattice but are inefficient SPP phonons take the heat directly into bulk substrate; Joule losses - I s F are for the total energy loss; while NT phonons take only a small fraction of that where j qCqC q ph Q SPP

USC, May 28, 2009 Slava V Rotkin different temperature 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) Remote SPP Scattering NT transport in saturation regime is determined by both channels

USC, May 28, 2009 Slava V Rotkin Conclusions Theory of NT scattering is not complete yet Physics of interactions in NTs at the hetero- interface with Si/SiO 2 is rich Hot electron scattering due to SPP modes provides a new and very effective thermo- conductivity mechanism Graphenes – another example of nano-hetero- interface where quantum effects may nicely develop into effects useful for applications

USC, May 28, 2009 Slava V Rotkin

decreasing scan rate Hysteresis in SWNT-array Transistors Robert-Peillard, Rotkin, 2005 Experiment: Laminated Device, CVD Tubes Courtesy J.Rogers

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 V g ground V d 1D channel

USC, May 28, 2009 Slava V Rotkin

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

USC, May 28, 2009 Slava V Rotkin 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

USC, May 28, 2009 Slava V Rotkin finally the relaxation time for the RiC scattering potential is conductivity depends on the Fermi level Remote impurity Scattering

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

USC, May 28, 2009 Slava V Rotkin