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Ballistic and quantum transports in carbon nanotubes
Discrete energy levels in carbon nanotubes
Two atoms two energy levels
Three atoms three energy levels
atoms
CB
VB
Metallic(no band gap) Semiconductor
Small band gap (0.1-3eV) Insulator (large band gap > 5 eV)
N = infinite (metals)
EF
Ve-
Free-e-
VB
CB
Spacing between levels becomes too small to be distinguishedSo it can be regarded as a band structure
Fermi sea
Free-e-
As long as kinetic energy is sufficient free electron movementcan change from lane to lane
a. Underlying mechanism for ballistic transport
Bulk Cu
Corresponding band structure
CB
VB
Conduction electron paths in all directions within CB
EF
Nanowire (quantum wire)
Nanodot (quantum dot)
EF
EF
Discrete levels
Sub-bands
Spacing increase
Band gap
M
EF
temp
Science, 283, 52 (1999).
Quantum transport in carbon nanotubes
a. Metallic CNTs have two conduction bands (two conduction channels). b. one conduction channel of quantum unit = Go = 12.9 (K)-1 = 2e2/h.c. Two conduction channels = 2Go = 4e2/h = 6.5 (K)-1
If contact resistance is small and negligiblethen CNT resistance between contacts Should be 6.5K
In practice, resistance exceeds 6.5 k and underlyingMechanism comes from
A. Semiconducting tubesB. Huge contact resistanceC. Defects in metallic CNTs
Electrons are unable to enter into CNT (joule heating)
High contact resistance
Thick contact barrier
Thin contact barrier
CNT
CNT
+-Electric charges induce voltage at leads but no current flow in CNT (coulomb oscillation)
tunneling
Existing conduction electrons in CNT
Existing electrons in CNTs exclude incoming electrons
Coulomb blockage
Thin contact barrier
CNT
tunneling
Gate voltage
Gate voltage
Gate voltage: electric field modulatedchemical potential (energy levels)
gate
Two electrons occupying one level
gate
Modulation of energy levels by gate voltage
Go
Ballistic transport in carbon nanotubes
Electron conduction has no resistanceand no heat generation and structures are defect-free.
e-
Conductor (e.g. Cu)
e-
Transport delay by resistance
e-
Resistance point (scattering)
Resistance comes from thermal vibration of crystal lattice, electrons and impurities
forward scatteringbackscattering
mean free pathrelaxation time
mean free path: no resistance
Cu (mean free path) = 1 m
Size reduction to below 1 m in length
scattering pointscattering point
Cu nanowiree-
free path with no resistance
Nanowire on electrodes
electrodes
resistance at wire-electrode contact
e-
forward scatteringforward scattering
backscattering
Fabry-Perot interference
When defects exist the ballistic transport is absent
Fabry-perot interference
defect
No Fabry-perot interference
Defects in CNTs
EF
Scattering center
Blocking of two conduction channels此時碳管電阻值昇高
Mean free path in Cu is 1 m
1m
Mean free path is ca. 100-300 nm in CNT
Scattering centers
Conditions for ballistic transport
a. Tube length electron mean free paths (or no defects)
b. Low contact resistance (low capacitance)
+ -
High contact resistance (capacitor-like)
CNT
capacitor
leads
dielectric
c. Gate voltage is not needed (different from that of quantum wire)
gate
High contact resistance (tunneling) Low contact resistance
Science, 280, 1744
R
+
-
R
Distance
A
A
R
+
-
B
R
+
-
C
B C
A general case
Ballistic transport effects
a. No heat generation, because no electron-phonon interaction (i.e. no scattering by defects)
b. Stepwise I-V profile (or quantum conduction)
Conduction via individual atoms
a. Nano-contact
b. electro-sharpening of metal wire
Diffusive conduction Quantum conduction
c. Mechanical break junction
Single atom電極
電子
How to transmit through a single atom
Conduction through individual orbits
3
6
5
Ohmic conductor (linear I-V profile)
metals
Voltage
Current
Non-linear I-V profile (non-ohmic conductor)
voltage
current
Light bulb
semiconductors
voltage
current
Theory of coulomb blockage
source drain
nanotube
If one transfers the charge Q from the source to the grain the change in the energy of the system is
http://edu.ioffe.ru/register/?doc=galperin/l13pdf1.tex
the first item is the work by the source of the gate voltage while the second is the energy of Coulomb repulsion at the grain.
the effective capacitance C
the gate voltage VG
source drain
nanotubee-
-+Polarization of leads
Q = –CVG So Q can be tuned by the gate voltage VG
the charge is transferred by the electrons with the charge –e.
Then, the energy as a function of the number n of electrons at the drain is
the difference
at certain values of VG,
and the difference vanishes.
It means that only at that values of the gate voltage resonant transfer is possible.