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
AcknowledgementsAcknowledgements
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
OUTLINEOUTLINE
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 NT-Array Thin Film TransistorsFilm Transistors
USC, May 28, 2009 Slava V Rotkin
Courtesy Prof. John Rogers (UIUC)Courtesy Prof. John Rogers (UIUC)
NT aligned array : Novel type TFTNT aligned array : Novel type TFT
Novel fabrication technique (Left) allows fabrication of Thin-Film Transistors of parallel NT arrays.
Courtesy Prof. John Rogers (UIUC)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 ElectronicsAligned 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 GrowthAligned NT Growth
Courtesy J.A. RogersNT alignment is not independent of the gas flow direction:
competition of gas flow and surfacealignment = 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 Vg=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 FETSingle NT FET
insulatorinsulatorgate @ Vgate @ Vgg
source @ groundsource @ ground drain @ Vdrain @ Vdd
1D channel1D channel
• Gate voltage (charge of the gate electrode and "its vicinity") controls the transport
USC, May 28, 2009 Slava V Rotkin
integratedintegratedintegratedintegrated
Physics of NT Devices on SiO2Physics of NT Devices on SiO2
• weak interaction • electr. transport• thermal coupling• alignment
empty spaceempty spaceempty spaceempty 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 CapacitanceNanotube Quantum Capacitance
USC, May 28, 2009 Slava V Rotkin
Classical Capacitance: 1D caseClassical Capacitance: 1D caseClassical 1D capacitance: line charge has = 2 log r + const
therefore: Cg-1 = 2 log z/R
where z = min(d, L, lg)
Distance to metal leads around/nearby1D channel defines the charge density
(z) is different for different screeningof 1D, 2D and 3D electrodes.
RR
dd
LL
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: ~ - d2/dx2 ~ - d/dx
Atomistic Capacitance of 1D FETAtomistic Capacitance of 1D FET
USC, May 28, 2009 Slava V Rotkin
which is to be compared with 3D and 2D: ~ - d2/dx2 ~ - 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 FETAtomistic 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 capacitanceQuantum 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 ArrayCapacitance of the NT ArrayMethod 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/2d/
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
1 m
1 m
1 m
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-healingRandom 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 Cg
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. RandomnessCorrelation vs. Randomness
C, %
d, nm
25 50 75 100 125 150
2.42.62.8
3.23.4
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.
In a single tube FET total capacitance has 2 terms:
geometric capacitance
and quantum capacitance
for NT array geometrical capacitance further decreases:
10 20 50 100 200 5000.5
0.6
0.7
0.8
0.9
1
d, nm
C/Cclass
Quantum Capacitance in NT-Array TFTQuantum Capacitance in NT-Array TFT
Charge Scattering:Short IntroductionCharge 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
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 singularitiesConductivity: 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 ScatteringRemote impurity Scattering
Scattering in 1D systems is weak due to restricted phase space available for electron: k -> -k
Coulomb Center ScatteringCoulomb Center Scattering
on average the Coulomb potential
where e* and nS 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: ResultsCoulomb 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 averagingRIS Details: Statistical averaging
starting with the Coulomb potential
then, the scattering rate is
here we used notations:
and
on average isproportional to
Statistical averaging over a random impurity distribution of
scattering form-factor
DoSstrength of potential
Saturation Regime andHeat Dissipation ProblemSaturation Regime andHeat 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 GenerationSaturation 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 GenerationSaturation 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.
Vdq j
q~area~nm2
channel heating due to Joule losses and low thermal coupling to leads
q
jHeat Generation (2)Heat Generation (2)
Surface Phonon PolaritonSurface Phonon Polariton
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 <0
Surface Polariton in SiO2Surface Polariton in SiO2
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: Digression: A tutorial on SPP A tutorial on SPP
Digression: Digression: A tutorial on SPP A tutorial on SPP
USC, May 28, 2009 Slava V Rotkin
Maxwell equations in free space
Digression: Digression: A tutorial on SPP A tutorial on SPP
Digression: Digression: A tutorial on SPP A tutorial on SPP
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: Digression: A tutorial on SPP A tutorial on SPP
Digression: Digression: A tutorial on SPP A tutorial on SPP
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: Digression: A tutorial on SPP A tutorial on SPP
Digression: Digression: A tutorial on SPP A tutorial on SPP
Remote Polariton ScatteringRemote Polariton Scattering
Estimates for SiO2-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 ~107 V/m
• finally the scattering time
for vF~108 cm/s and SO~150meV :for vF~108 cm/s and SO~150meV : ~ 105 V/cm~ 105 V/cm
Physics of SPP scattering in SiO2Physics of SPP scattering in SiO2
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 SiO2Details of SPP scattering in SiO2
USC, May 28, 2009 Slava V Rotkin
Conductivity: van Hove singularitiesConductivity: 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.
rem
inder
Surface Polariton ScatteringSurface 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)Surface Polariton Scattering (2)
for vF~108 cm/s and SO~140meV : ~40 nm
2ki ~ 2/a ~ 1/nm
Forward scattering dominates:
q~1/ : forward scatteringq~2ki : backward scattering
USC, May 28, 2009 Slava V Rotkin
• for the SiO2 (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 RateSurface Polariton Scattering Rate
USC, May 28, 2009 Slava V Rotkin
ConclusionsConclusions
• Theory of NT scattering is not complete yet
• Physics of interactions in NTs at the hetero-interface with Si/SiO2 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
overheating of the channel : neglecting the thermal sink in the leads (~nm2)
Remote SPP ScatteringRemote 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 - IsF are for the total energy loss; while NT phonons take only a small fraction of that
where
j
qC
qph
QSPP
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 to= 77 and 300K
• inset: data collapse for (linear) dependence on the electron concentration (0.1 and 0.2 e/nm)
Remote SPP ScatteringRemote SPP Scattering
• NT transport in saturation regime is determined by both channels
USC, May 28, 2009 Slava V Rotkin
ConclusionsConclusions
• Theory of NT scattering is not complete yet
• Physics of interactions in NTs at the hetero-interface with Si/SiO2 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
USC, May 28, 2009 Slava V Rotkin