1. 18. Nanostructures Imaging Techniques for nanostructures
Electron Microscopy
Optical Microscopy
Scanning Tunneling Microscopy
Atomic Force Microscopy
Electronic Structure of 1-D Systems
1-D Subbands
Spectroscopy of Van Hove Singularities
1-D Metals – Coulomb Interaction & Lattice Couplings
Electrical Transport in 1-D
Conductance Quantization & the Landauer Formula
Two Barriers in Series-Resonant Tunneling
Incoherent Addition & Ohm’s Law
Localization
Voltage Probes & the Buttiker-Landauer Formulism
Electronic Structure of 0-D Systems
Quantized Energy Levels
Semiconductor Nanocrystals
Metallic Dots
Discrete Charge States
Electrical Transport in 0-D
Coulomb Oscillations
Spin, Mott Insulators, & the Kondo Effects
Cooper Pairing in Superconducting Dots
Vibrational & Thermal Properties
Quantized Vibrational Modes
Transverse Vibrations
Heat Capacity & Thermal Transport

2. carbon nanotubes, quantum wires, conducting polymers, … .
1-D nanostructures:
carbon nanotubes, quantum wires, conducting polymers, … .
0-D nanostructures:
semiconductor nanocrystals, metal nanoparticles,
lithographically patterned quantum dots, … .
We’ll deal only with crystalline nanostructures.
SEM image
Gate electrode pattern of a quantum dot.

3. Model of CdSe nanocrystal
TEM image
AFM image of crossed C-nanotubes (2nm wide) contacted by Au electrodes (100nm wide) patterned by e beam lithography
Model of the crossed C-nanotubes & graphene sheets.

4. 2 categories of nanostructure creation:
Lithographic patterns on macroscopic materials (top-down approach).
Can’t create structures < 50 μm.
Self-assembly from atomic / molecular precusors (bottom-up approach).
Can’t create structures > 50 μm.
Challenge: develop reliable method to make structure of all scales.
Rationale for studying nanostructures:
Physical, magnetic, electrical, & optical properties can be drastically altered when the extent of the solid is reduced in 1 or more dimensions.
1. Large ratios of surface to bulk number of atoms.
For a spherical nanoparticle of radius R & lattice constant a:
R = 6 a ~ 1 nm →
Applications: Gas storage, catalysis, reduction of cohesive energy, …
2. Quantization of electronic & vibrational properties.

5. Imaging Techniques for nanostructures
Reciprocal space (diffraction) measurements are of limited value for nanostructures:
small sample size → blurred diffraction peaks & small scattered signal.
2 major classes of real space measurements : focal & scanned probes.
Focal microscope: probe beam focused on sample by lenses.
Resolution
β = numerical aperture
Scanning microscopy: probe scans over sample.
Resolution determined by effective range of interaction between probe & sample.
Besides imaging, these probes also provide info on
electrical, vibrational, optical, & magnetic properties.
focal microscope
DOS

6. Electron Microscopy Transmission Electron Microscope (TEM):
100keV e beam travels thru sample & focussed on detector.
Resolution d ~ 0.1 nm (kept wel aboved theoretical limit by lens imperfection).
Major limitation: only thin samples without substrates can be used.
Scanning Electron Microscope (SEM):
100~100k eV tight e beam scans sample while backscattered / secondary e’s are measured.
Can be used on any sample.
Lower resolution: d > 1 nm.
SEM can be used as electron beam lithography.
Resolution < 10 nm.
Process extremely slow
→ used mainly for prototypying & optical mask fabrication.

7. Optical Microscopy
For visible light & high numerical aperture ( β  1 ), d ~ nm.
→ Direct optical imaging not useful in nanostructure studies.
Useful indirect methods include
Rayleigh sacttering, absortion, luminescence, Raman scattering, …
Fermi’s golden rule for dipole approximation for light absorption:
Emission rate (α = e2 /  c ):
Real part of conductivity ( total absorbed power = σ E 2 ):
Absorption & emission measurements → electronic spectra.

8. Optical focal system are often used in microfabrication.
Spectra of Fluorescence of individual nanocrystals.
Mean peak: CB → VB
Other peaks involves LO phonon emission.
Fluorescence from CdSe nanocrystals at T = 10K
Optical focal system are often used in microfabrication.
i.e., projection photolithography.
For smaller scales, UV, or X-ray lithographies are used.

9. Scanning Tunneling Microscopy
Carbon nanotube
STM:
Metal tip with single atom end is controlled by piezoelectrics to pm precision.
Voltage V is applied to sample & tunneling current I between sample & tip is measured.
 = tunneling barrier
z = distance between tip & sample
Typical setup: Δz = 0.1 nm → Δ I / I = 1.
Feedback mode: I maintained constant by changing z.
→ Δz ~ 1 pm can be detected.

10. STM can be used to manipulate individual surface atoms.
“Quantum coral” of r  7.1 nm formed by moving 48 Fe atoms on Cu (111) surface.
Rings = DOS of e in 3 quantum states near ε F.
( weighted eDOS at E = εF + eV )

11. Atomic Force Microscopy
Laser
photodiode array
AFM:
Works on both conductor & insulator.
Poorer resolution than STM.
C ~ 1 N/m
F ~ pN – fN
Δz ~ pm
mm sized cantilever
Contact mode:
tip in constant contact with sample; may cause damage.
Tapping mode:
cantilever oscillates near resonant frequency & taps sample at nearest approach.
Q = quality factor
ω0 & Q are sensitive to type & strength of forces between tip & sample.
Their values are used to construct an image of the sample.

12. Magnetic Force Microscopy
MFM = AFM with magnetic tip
Other scanned probe techniques:
Near-field Scanning Optical Microscopy (NSOM)
Uses photon tunneling to create optical images with resolution below diffraction limit.
Scanning Capacitance Microscopy (SCM)
AFM which measures capcitance between tip & sample.

13. Electronic Structure of 1-D Systems
Bulk: Independent electron, effective mass model with plane wave wavefunctions.
Consider a wire of nanoscale cross section.
i, j = quantum numbers in the cross section
1-D subbands
Van Hove singularities at ε = εi, j

14. Spectroscopy of Van Hove Singularities
Carbon nanotube
Prob. 1
photoluminescence of a collection of nanotubes
STM

15. 1-D Metals – Coulomb Interaction & Lattice Couplings
Let there be n1D carriers per unit length, then
Fermi surface consists of 2 points at k = kF .
Coulomb interactions cause e scattering near εF .
For 3-D metals, this is strongly suppressed due to E, p conservation & Pauli exclusion principle.
τ0 = classical scattering rate
Caution: our Δε = Kittel’s ε. →
 quasiparticles near εF are well defined →
For 1-D metals,
Let →
for |k|  kF
 E & p conservation are satisfied simultaneously.

16. 1 + 2 → 3 + 4
Pauli exclusion favors
E, p conservation:
 For a given Δε1 , there always exist some Δε2 & Δε4 to satisfy the conservation laws provided Δε1 > Δε3 . → →
 quasiparticles near εF not well defined
Fermi liquid (quasiparticle) model breaks down.
Ground state is a Luttinger liquid with no single-particle-like low energy excitations.
→ Tunneling into a 1-D metal is suppressed at low energies.
Independent particle model is still useful for higher excitations (we’ll discuss only such cases).

17. 1-D metals are unstable to perturbations at k = 2kF .
E.g., Peierls instability: lattice distortion at k = 2kF turning the metal into an insulator.
Polyacetylene: double bonds due to Peierls instability. Eg  1.5eV.
Semiconducting polymers can be made into FETs, LEDs, … .
Proper doping turns them into metals with mechanical flexibility & low T processing.
→ flexible plastic electronics.
Nanotubes & wires are less susceptible to Peierls instability.

18. Electrical Transport in 1-D
Conductance Quantization & the Landauer Formula
1-D channel with 1 occupied subband connecting 2 large reservoir.
Barrier model for imperfect 1-D channel
Let Δn be the excess right-moving carrier density, DR(ε) be the corresponding DOS.
q = e
→ The conductance quantum
depends only on fundamental constants.
Likewise the resistance quantum

19. If channel is not perfectly conducting,
Landauer formula
T = transmission coefficient.
For multi-channel quasi-1-D systems
i, j label transverse eigenstates.
For finite T,
R = reflection coefficient.
Channel fully depleted of carriers at Vg = –2.1 V.

20. Two Barriers in Series-Resonant Tunneling
tj, rj = transmission, reflection amplitudes.
For wave of unit amplitude incident from the left
At left barrier
At right barrier → →
Resonance condition :
n  Integers

21. At resonance
Resonant tunneling
For t1 = t2 = t : →
For very opaque barriers, r  –1 ( φ  n π )
→ resonance condition becomes particle in box condition
while the off resonance case gives
Using
&
one gets (see Prob 3) the Breit-Wigner form of resonance
where

22. Incoherent Addition & Ohm’s Law
Classical treatment: no phase coherence. →  → →
(Prob. 4 )
= Sum of quantized contact resistance & intrinsic resistance at each barrier.
Let the resistance be due to back-scattering process of rate 1/τb .
For propagation over distance dL, →
(Prob. 4 )
Incoherence addition of each segment gives

23. Localization →  …  = average over φ* = average over k or ε .
larger than incoherent limit
Consider a long conductor consisting of a series of elastic scatterers of scattering length le .
Let R >>1, i.e., R  1 & T << 1, ( R + T = 1 ) .
For an additional length dL,
Setting →

24. → 
where
C.f. Ohm’s law R  L
For a 1-D system with disorder, all states become localized to some length ξ .
Absence of extended states → R  exp( a L / ξ ) , a = some constant.
For quasi-1-D systems, one finds ξ ~ N le , where N = number of occupied subbands.
For T > 0, interactions with phonons or other e’s reduce phase coherence to length lφ = A T −α . 
for each coherent segment.
Overall R  incoherent addition of L / lφ such segments.
For sufficiently high T, lφ  le , coherence is effectively destroyed & ohmic law is recovered.
All states in disordered 2-D systems are also localized.
Only some states (near band edges) in disordered 3-D systems are localized.

25. Voltage Probes & the Buttiker-Landauer Formulism
T(n,m) = total transmission probability for an e to go from m to n contact.
1,2 are current probes; 3 is voltage probe.
For a current probe n with N channels, µ of contact is fixed by V.
Net current thru contact is
Setting →
For the voltage probe n, Vn adjusts itself so that In = 0. →

26. In , Vn depend on T(n,m) → their values are path dependent.
Voltage probe can disturb existent paths.
Let every e leaving 1 always arrive either at 2 or 3 with no back scattering. if
Current out of 1:
no probe

28. Mesoscopic regime: le < L < lφ .
Semi-classical picture:
App. G
Aharonov-Bohm effect

29. Electronic Structure of 0-D Systems
Quantum dots: Quantized energy levels.
e in spherical potential well:
For an infinite well with V = 0 for r < R :
for r < R
βn, l = nth root of jl (x).
β0,0 = π (1S), β0,1 = 4.5 (1P), β0,2 = 5.8 (1D)
β1,0 = 2π (2S), β1,1 = 7.7 (2P)

30. Semiconductor Nanocrystals
CdSe nanocrystals
For CdSe:
For R = 2 nm,
For e, ε 0,0 increases as R decreases.
For h, ε 0,0 decreases as R decreases.
→ Eg increases as R decreases.
Optical spectra of nanocrystals can be tuned continuously in visible region.
Applications: fluorescent labeling, LED.
Kramers-Kronig relation:
same as bulk
For ω →  →
Strong transition at some ω in quantum dots → laser ?

31. Metallic Dots Mass spectroscopy (abundance spectra):
Large abundance at cluster size of magic numbers
( 8, 20, 40, 58, … )
→ enhanced stability for filled e-shells.
Average level spacing at εF :
For Au nanoparticles with R = 2 nm, Δε  2 meV.
whereas semiC CdSe gives Δε  0.76 eV.
→ ε quantization more influential in semiC.
Small spherical alkali metallic cluster Na
mass spectroscopy

32. Optical properties of metallic dots dominated by surface plasmon resonance.
If retardation effects are negligible, →
Surface plasma mode at singularity:
indep of R.
For Au or Ag, ωp ~ UV, ωsp ~ Visible.
→ liquid / glass containing metallic nanoparticles are brilliantly colored.
Large E just outside nanoparticles near resonance enhances weak optical processes.
This is made use of in Surface Enhanced Raman Scattering (SERS), & Second Harmonic Generation (SHG).

33. Discrete Charge States
Thomas-Fermi approximation:
U = interaction between 2 e’s on the dot = charging energy.
α = rate at which a nearby gate voltage Vg shifts φ of the dot.
Neglecting its dependence on state,
C = capacitance of dot.
Cg = capacitance between gate & dot
If dot is in weak contact with reservoir, e’s will tunnel into it until the μ’s are equalized.
Change in Vg required to add an e is

34. U depends on size &shape of dot & its local environment.
For a spherical dot of radius R surrounded by a spherical metal shell of radius R + d,
Prob. 5
For R = 2 nm, d = 1 nm & ε = 1, we have
U = 0.24 eV >> kBT = 0.026eV at T = 300K
→ Thermal fluctuation strongly supressed.
For metallic dots of 2nm radius, Δε  2meV → ΔVg due mostly to U.
For semiC dots, e.g., CdSe, Δε  0.76 eV → ΔVg due both to Δε & U.
Charging effect is destroyed if tunneling rate is too great.
Charge resides in dot for time δt  RC. ( R = resistance ) →
Quantum fluctuation smears out charging effect when δε  U, i.e., when R ~ h / e2 .
Conditions for well-defined charge states are
&

35. Electrical Transport in 0-D
For T < ( U + Δε ) / kB , U & Δε control e flow thru dot.
Transport thru dot is suppressed when µL & µR of leads lie between µN & µN+1 (Coulomb blockade)
Transport is possible only when µN+1 lies between µL & µR .
→ Coulomb oscillations of G( Vg ).

36. Coulomb Oscillations
GaAs/AlGaAs
T = 0.1K
Coulomb oscillation occurs whenever U > kBT, irregardless of Δε .
For Δε >> kBT, c.f. resonant tunneling:
Thermal broadening
Breit-Wigner lineshape

37. Single Electron Transistor (SET):
Based on Coulomb oscillations ( turns on / off depending on N of dot ).
→ Ultra-sensitive electrometer ( counterpart of SQUID for B ).
→ Single e turnstiles & pumps:
single e thru device per cycle of oscillation.
quantized current I = e ω / 2 π.
2-D circular dot
dI/dV: Line → tunneling thru given state.
White diamonds (dI/dV = 0 ) :
Coulomb blockades of fixed charge states
( filled shells for large ones ) N 1 2 3 … 7
(i, j)
(0,0)
(0,1) or (1,0)
(1,1) , (0,2), or (2,0)
Δ Vg
U /α e
(U + ) /α e
(U + ) /α e
Height of diamonds:

38. Spin, Mott Insulators, & the Kondo Effects
Consider quantum dot with odd number of e’s in blockade region.
~ Mott insulator with a half-filled band.
No external leads:
degenerated
Kondo effect : with external leads & below TK :
Ground state = linear combinations of  &  states with virtual transitions between them.
(intermediate states involve pairing with an e from leads to form a singlet state
→ Transmission even in blockade region.
For symm barriers & T << TK , T 1.
Singlets states in 3-D Kondo effect enhances ρ.

39. Cooper Pairing in Superconducting Dots
Competition between Coulomb charging & Cooper pairing.
For dots with odd number of e’s , there must be an unpaired e.
Let 2Δ = binding energy of Cooper pairs.
For 2Δ > U,
e’s will be added to dot in pairs.
Coulombe oscillations 2e – periodic.

40. Vibrational & Thermal Properties
Continuum approximation:
ω = vs K → ωj upon confinement.
Quantized vibrations around circumference of thin cylinder of radius R & thickness t << R.
j = 1,2, …
Longitudinal compressional mode →
Radial breathing mode
j = 1,2, …
Transverse mode

41. Measuring ωRBM gives good guess of R.
Raman spectrum of individual carbon nanotubes
( 160 cm–1 = 20 meV )
vL = 21 km/s →
Measuring ωRBM gives good guess of R.

42. Transverse Vibrations
Transverse mode is not a shearing as in 3-D, but a flexural wave which involves different longitudinal compression between outer & inner arcs of the bend.
Transverse standing wave on rectangular beam of thickness h, width w, & length L : →
C.f.
Torsion / shear mode
Si nanoscale beans: f  L–2
Micro / Nano ElectroMechanical systems ( M/N EM)

43. Heat Capacity & Thermal Transport
Quantized vibrational mode energies are much smaller than kBTroom .
→ Modes in confined directions are excited at Troom.
 Lattice thermal properties of nanostructure are similar to those in bulk.
For low T <  ω / kB , modes in confined directions are freezed out.
→ system exhibits lower-dimensional characteristics.
E.g.
(Prob.6)
Gth depends only on fundamental constants if T = 1.