Transcript Scattering - Lehigh University

Heterogeneous carbon-based devices:
Towards integration with Si technology
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)
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
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)
Y-cut [01-10]
Z-cut [0001]
X-cut [2-1-10]
Courtesy Prof. John Rogers (UIUC)
SEM reconstruction ("fake" 3D view)
of NT-TFT and gold electrodes.
USC,
May 28,on
2009
SEM of NT
growth
different quartz facets
NTs can be transferred
on plastic
Slava V Rotkin
Aligned NT for Transparent Electronics
NTs are so small that absorption in a single layer of wellseparated tubes
is negligible
Adapted from Zhou (2008)
USC, May 28, 2009
Slava V Rotkin
Aligned NT Growth
NT alignment is not independent of the gas flow direction:
Courtesy J.A. Rogers
competition of gas flow and surface
alignment = serpentine growth
Criss-crossed
NT arrays
USC, May 28, 2009
NT transistor as an element of
a FM-radio
Slava V Rotkin
Single NT FET
Physics of NT FieldEffect Transistor (FET):
• NT channel is
conducting at Vg=0
(non-intentional pdoping)
source @ ground
drain @ Vd
1D channel
• 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
insulator
gate @ Vg
• Gate voltage (charge of the gate electrode
and "its vicinity") controls the transport
Physics of NT Devices on SiO2
• weak interaction
• electr. transport
• thermal coupling
• alignment
Weak van der Waals
interactions...
For a polar substrate
-- such as quartz,
sapphire, calcite -new physics due to
evanescent ElectroMagnetic (EM) modes,
aka Surface Phononintegrated
Polariton
modes
USC, May 28, 2009
empty space
Slava V Rotkin
Nanotube Quantum
Capacitance
USC, May 28, 2009
Slava V Rotkin
Classical Capacitance: 1D case
Classical 1D capacitance: line charge has f = r 2 log r + const
therefore: Cg-1 = 2 log z/R
L
R
where z = min(d, L, lg)
Distance to metal leads around/nearby
1D channel defines the charge density
d
r(z) is different for different screening
of 1D, 2D and 3D electrodes.
USC, May 28, 2009
Slava V Rotkin
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/dx2
r ~ - dj/dx
Quantum Mechanical view:
Selfconsistent calculation of
the charge density
USC, May 28, 2009
Rotkin et.al.Slava
JETP-Letters,
V Rotkin 2002
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/dx2
r ~ - dj/dx
Quantum Mechanical view:
Selfconsistent calculation of
the charge density
USC, May 28, 2009
Rotkin et.al.Slava
JETP-Letters,
V Rotkin 2002
Quantum physics of TFT capacitance
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"
Most of the tubes are parallel, but the
distance between neighbor tubes may vary.
• multi-layer dielectrics and
surface E/M modes
• interface scattering
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
USC, May 28, 2009
Slava V Rotkin
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
1 mm
1 mm
2d/L
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 mm
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
Random Array Coupling: Self-healing
Current nonuniformity is a deficiency for device production.
Consider Dr due to non-uniform screening.
DC/C
-0.15
-0.25
-0.35
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
USC, May 28, 2009
Slava V Rotkin
Correlation vs. Randomness
The capacitance of a random TFT
array (a single given realization)
as a function of the external
screening (insulator thickness).
DC, %
3.4
3.2
3.0
2.8
2.6
2.4
d, nm
25 50 75 100 125 150
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.
USC, May 28, 2009
Slava V Rotkin
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:
1
C/Cclass0.9
0.8
L
0.7
0.6
d, nm
0.5
10
20
50
100
200
500
Charge Scattering:
Short Introduction
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 enters the formula
USC, May 28, 2009
Slava V Rotkin
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
USC, May 28, 2009
Slava V Rotkin
Remote impurity
Scattering
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
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
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
Saturation Regime and
Heat Dissipation Problem
Saturation Regime: Heat Generation
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.
USC, May 28, 2009
Slava V Rotkin
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).
USC, May 28, 2009
Slava V Rotkin
Heat Generation (2)
q
j
j
q
Vd
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-S in Si.
The mechanism appeared to be ineffective for Si MOS-FETs and was
almost USC,
forgotten
for decades...
May 28, 2009
Slava V Rotkin
Surface Phonon Polariton
Surface 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
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)
• 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
• existence of a surface mode essentially depends
on existence of the anomalous dispersion region e<0
USC, May 28, 2009
Digression:
A tutorial
on SPP
Slava V Rotkin
Maxwell equations in free space
USC, May 28, 2009
Digression:
A tutorial
on SPP
Slava V Rotkin
Maxwell equations in free
space are solved by anzatz
additional materials connection:
Maxwell equations in free space
algebraic form of Maxwell
equations in free space
E
q
H
surface requires that:
USC, May 28, 2009
Digression:
A tutorial
on SPP
Slava V Rotkin
all field components (but
one) can be found from BC:
frequency of the SPP
provides consistency of BC:
"a" for air
E
q
H
"b" for bulk
USC, May 28, 2009
Digression:
A tutorial
on SPP
Slava V Rotkin
Remote Polariton
Scattering
Physics of SPP scattering in SiO2
Estimates for SiO2-quartz:
• electric field in the air is
proportional to decay
constant, determined from
MEq+BC, and F-factor
• relevant l is proportional
to the wavelength of hot
electron
for vF~108 cm/s
and wSO~150meV :
• electric field ~107 V/m
• finally the scattering time
e ~ 10
5 V/
cm
Details of SPP scattering in SiO2
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 wSO is the SPP frequency
USC, May 28, 2009
Slava V Rotkin
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.
Prof. T. Ando
USC, May 28, 2009
Slava V Rotkin
Surface Polariton Scattering
• RPS rate varies for intra-subband and inter-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
At vHs our Born approximation fails which
manifests itself as diverging scattering rate
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
Forward scattering dominates:
q~1/l : forward scattering
q~2ki : backward scattering
for vF~108 cm/s
and wSO~140meV : l~40 nm
2ki ~ 2p/a ~ 1/nm
Surface Polariton Scattering Rate
• 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 < l ~ 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
USC, May 28, 2009
Slava V Rotkin
Conclusions
• Theory of NT scattering is not complete yet
• Physics of interactions in NTs at the heterointerface with Si/SiO2 is rich
• Hot electron scattering due to SPP modes
provides a new and very effective thermoconductivity mechanism
• Graphenes – another example of nano-heterointerface where quantum effects may nicely
develop into effects useful for applications
USC, May 28, 2009
Slava V Rotkin
Remote SPP Scattering
overheating of the channel : neglecting
the thermal sink in the leads (~nm2)
j
qC
qph
where
• two scattering mechanisms :
• NT phonons warm the NT lattice
but are inefficient
QSPP
• 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
USC, May 28, 2009
Slava V Rotkin
Remote SPP Scattering
• 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)
• NT transport in saturation regime is
determined by both channels
• different temperature dependence
for two scattering mechanisms
USC, May 28, 2009
Slava V Rotkin
Conclusions
• Theory of NT scattering is not complete yet
• Physics of interactions in NTs at the heterointerface with Si/SiO2 is rich
• Hot electron scattering due to SPP modes
provides a new and very effective thermoconductivity mechanism
• Graphenes – another example of nano-heterointerface 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
USC, May 28, 2009
Slava V Rotkin