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18. NanostructuresImaging Techniques for nanostructures

Electron MicroscopyOptical MicroscopyScanning Tunneling MicroscopyAtomic Force Microscopy

Electronic Structure of 1-D Systems1-D SubbandsSpectroscopy of Van Hove Singularities1-D Metals – Coulomb Interaction & Lattice Couplings

Electrical Transport in 1-DConductance Quantization & the Landauer FormulaTwo Barriers in Series-Resonant TunnelingIncoherent Addition & Ohm’s LawLocalizationVoltage Probes & the Buttiker-Landauer Formulism

Electronic Structure of 0-D SystemsQuantized Energy LevelsSemiconductor NanocrystalsMetallic DotsDiscrete Charge States

Electrical Transport in 0-DCoulomb OscillationsSpin, Mott Insulators, & the Kondo EffectsCooper Pairing in Superconducting Dots

Vibrational & Thermal PropertiesQuantized Vibrational ModesTransverse VibrationsHeat Capacity & Thermal Transport

1-D nanostructures: carbon nanotubes, quantum wires, conducting polymers, … .

0-D nanostructures: semiconductor nanocrystals, metal nanoparticles, lithographically patterned quantum dots, … .

Gate electrode pattern of a quantum dot.

SEM image

We’ll deal only with crystalline nanostructures.

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.

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:

2

2

3

3

4

43

surf

bulk

NR

a

aN R

3a

R

R = 6 a ~ 1 nm → 1

2surf

bulk

N

N

Applications: Gas storage, catalysis, reduction of cohesive energy, …

2. Quantization of electronic & vibrational properties.

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

Focal microscope: probe beam focused on sample by lenses.

2 2sind

β = numerical aperture Resolution

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.

jj

D DOS

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.

Optical MicroscopyFor visible light & high numerical aperture ( β 1 ), d ~ 200-400 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:

22i j j iw j e i

E r

Emission rate (α = e2 / c ):

3

2 2

3 2

42 j i

j i i jw j e i j ic

E r r

Real part of conductivity ( total absorbed power = σ E 2 ):

2

2

,

2ˆ i j j i

i j

ej i f f

V

n r ˆEE n

Absorption & emission measurements → electronic spectra.

Fluorescence from CdSe nanocrystals at T = 10K

Spectra of Fluorescence of individual nanocrystals.Mean peak: CB → VBOther peaks involves LO phonon emission.

Optical focal system are often used in microfabrication.i.e., projection photolithography.For smaller scales, UV, or X-ray lithographies are used.

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.

2

2exp 2I

mz

= tunneling barrierz = 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.

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.

2

j j F jj

d I

I dVeV r ( weighted eDOS at E = εF + eV )

Atomic Force MicroscopyLaser

mm sized cantilever

photodiode array

C ~ 1 N/mF ~ pN – fNΔz ~ pm

F C z

AFM: • Works on both conductor & insulator.• Poorer resolution than STM.

Contact mode: tip in constant contact with sample; may cause damage.

Tapping mode:cantilever oscillates near resonant frequency & taps sample at nearest approach.

20

222 2 0

0

Fz

C

Q

0

0

Fz Q

C

0

0

Fz

C

Q = quality factor per cyclestored

dissipated

E

E

ω0 & Q are sensitive to type & strength of forces between tip & sample.Their values are used to construct an image of the sample.

Magnetic Force Microscopy

MFM = AFM with magnetic tip

0

0 0

z z

F z z F zF

zz

0 0

2

2

z z z z

B Bz

z z

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.

Electronic Structure of 1-D Systems

Bulk: Independent electron, effective mass model with plane wave wavefunctions.Consider a wire of nanoscale cross section.

2 2

, 2i j

k

m

,, , , i k zi jx y z x y e i, j = quantum numbers in the cross section

,,

i ji j

DD ,,i j

i j

d N d k

dD

k d

2,

2 22 2 i j

L m

,,

,

4

0

i ji j

i j

L

hv

Van Hove singularities at ε = εi, j

1-D subbands

Spectroscopy of Van Hove Singularities

STM

photoluminescence of a collection of nanotubes

Carbon nanotube

Prob. 1

1-D Metals – Coulomb Interaction & Lattice Couplings

Let there be n1D carriers per unit length, then 1

2 22

2D F Fn k k

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. 2

0

11 F

Fee

τ0 = classical scattering rate

→ 3ee

D

2

0 F

quasiparticles near εF are well defined

For 1-D metals, 2

2 2

2 Fk km

2

2 F Fk k k km

2Fk

km

for |k| kF

E & p conservation are satisfied simultaneously.

LetFk k k →

F

Caution: our Δε = Kittel’s ε.

2

0

1

F

→ 31 as 0

D

2Fk

km

1 + 2 → 3 + 4

1 1, 0k

3 4, 0 Pauli exclusion favors

E, p conservation: 1 2 3 4k k k k

Fk k k F

2 2, 0k

3 3, 0k

4 4, 0k

0

For a given Δε1 , there always exist some Δε2 & Δε4 to satisfy the conservation laws provided Δε1 > Δε3 .

0

11

Fee

→ 1

ee

D

0 F

quasiparticles near εF not well defined→ 1

~D

const

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).

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.

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

I n qv

Let Δn be the excess right-moving carrier density, DR(ε) be the corresponding DOS.

RD q Vqv

L

22q V v

hv

22 eV

h q =

e

→ The conductance quantum 22

Q

eG

h depends only on fundamental constants.

Likewise the resistance quantum 2

1

2QQ

hR

G e

Channel fully depleted of carriers at Vg = –2.1 V.

If channel is not perfectly conducting,

22

F F

eG

h T Landauer formula

For multi-channel quasi-1-D systems

,,

F i j Fi j

T T

i, j label transverse eigenstates.

= transmission coefficient.

22

, ,F L R

eI V T d f eV f

h

T

For finite T,

22

hR

e

T

2

1

2

h

e

T T

T 2 22 2

h h

e e

RT

= reflection coefficient.

Two Barriers in Series-Resonant Tunneling

tj, rj = transmission, reflection amplitudes.

expj j t jt t i expj j r jr r i

For wave of unit amplitude incident from the left

1 1a t r b

2i k L i k Lb e r a e

2i k Lc t a e

At left barrier

At right barrier

2

2mk

→ 1 22

1 21

i k L

i k L

t t ec

r r e

2

1 1 2i k La t r r e a 1

21 21 i k L

t

r r e

→

1 2

1 2

2

2 1 2

2

1 21

t t

r r

i k L

i k L

t t ec

r r e

T

2 2

1 22 2

1 2 1 2 1 21 2 cos 2 r r

t t

r r kL r r

Resonance condition : 1 2* 2 2r rkL n n Integers

1 2

1 2

1 2

2

1 21

t t

r r

i k L

i k L

t t ec

r r e

1 2 *

1 2 1 20

t ti k L n n i n

n

t t e r r e

At resonance 1 2 1 20

n ni k L

n

c t t e r r

1 2

1 21i k Lt t

er r

For t1 = t2 = t :

2

21

i k Ltc e

r

i k Le → * 2 1n T Resonant

tunneling

For very opaque barriers, r –1 ( φ n π ) → resonance condition becomes particle in box condition

2 2

1 2, 1t t

2 2

1 2t tTwhile the off resonance case gives

k L n

Using

one gets (see Prob 3) the Breit-Wigner form of resonance

1 22 2

1 2

4

4 n

T where

2

2j jt

* 2 n &

Incoherent Addition & Ohm’s LawClassical treatment: no phase coherence.

2 2 2 2

1 1a t r b 1 1a t r b

2i k L i k Lb e r a e

2i k Lc t a e

→ 2 2 2

2b r a

2 2 2

2c t a

2 2 2 2 2

1 1 2a t r r a 2

12 2

1 21

t

r r

→

2 22 1 2

2 2

1 21

t tc

r r

T

→22

hR

e

T(Prob. 4 )

2 2

1 22 22

1 2

12

r rh

e t t

= 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,1

b F

dLd

vR

b

dL

l → 1D

d R

d L

2

1

2 b

h

e l

1 2

21 2

12

h

e

R R

T T

21D

m

n e

(Prob. 4 )Incoherence addition of each segment gives

22Qb

h LR R

e l

Localization

22

hR

e

T

2 2

1 22 2

1 2 1 21 2 cos *

t t

r r r r

T

2 2

1 2 1 22 22

1 2

1 2 cos *

2

r r r rh

e t t

→

2 2

1 2 1 22 22

1 2

1 2 cos *

2

r r r rhR

e t t

2 2

1 22 22

1 2

1

2

r rh

e t t

larger than

incoherent limit

… = average over φ* = average over k or ε .

Consider a long conductor consisting of a series of elastic scatterers of scattering length le .

Let R >>1, i.e., 1 & << 1, ( + = 1 ) .

2

1

2

h dR dR

e d

R RT T

For an additional length dL, e

d Ld

lR 1d d R T

Setting 2

1r R 2

2r d R2

1t T 2

2t d T

→ 2

1

2 1

h d

e d

R R

T R 21 1

2

hd d

e R R R

T

1 1R d R R 1 2R d R 1 2e

d LR

l

2 2

1 22 22

1 2

1

2

r rh

e t t

1 2e

d LR dR R

l

→

2

e

d LdR R

l

0

2ln

e

R L

R l

0 0LR R

where

22Q

hR

e

2

2exp

2 e

h LR

e l

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 .

2

2exp

2 e

lhR

e l

for each coherent segment.

For sufficiently high T, lφ le , coherence is effectively destroyed & ohmic law is recovered.

Overall R incoherent addition of L / lφ such segments.

All states in disordered 2-D systems are also localized.Only some states (near band edges) in disordered 3-D systems are localized.

Voltage Probes & the Buttiker-Landauer Formulism

1,2 are current probes; 3 is voltage probe.

(n,m) = total transmission probability for an e to go from m to n contact.

2

,2 n mn n n m

m

eI N V V

h

T

For a current probe n with N channels, µ of contact is fixed by V.

Net current thru contact is

Setting 0 ,n nI V V n → ,n m

nm

N T

For the voltage probe n, Vn adjusts itself so that In = 0.

,1 n mn m

mn

V VN

T→

,

,

n mm

mn m

m

V

T

T

,

,

n mm

mn n m

m

T

T

In , Vn depend on (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.

3,1

3 3,1 3,2

VV

T

T T 2

V 3,1 3,2T Tif

Current out of 1: 2

1,33

2eI V V

h T

21,32 1

12

eV

h

T22e

Vh

no probe

Mesoscopic regime: le < L < lφ .

Semi-classical picture: , exp

mm nj l

j

i et a d

c

p A l

App. G

, 2,n m m ntT

2

2 2

1 2 1 2 1 2

2exp 2 cos

/loop

iea a a a a a

c hd

c e

A l

Aharonov-Bohm effect

loop S

d dS AA l

0

hc

e

Electronic Structure of 0-D Systems

Quantum dots: Quantized energy levels.

e in spherical potential well: , , ,n l m n l , , , ,, , ,n l m n l l mr R r Y

For an infinite well with V = 0 for r < R :

2 2,

, 22 *n l

n l m R

,

,n l

n l l

rR r j

R

for r < R

, 0l n lj

β0,0 = π (1S), β0,1 = 4.5 (1P), β0,2 = 5.8 (1D)

β1,0 = 2π (2S), β1,1 = 7.7 (2P)

βn, l = nth root of jl (x).

Semiconductor NanocrystalsCdSe nanocrystals

For CdSe:

* 0.13cm m

2

,, 2

0,0

2.9n ln l

eV

R

For R = 2 nm, 0,1 0,0 0.76 eV

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.

2 2

0

2 sP ds

s

Kramers-Kronig relation:

For ω → 2ne

m

0

2s ds

2

0 2

n es ds

m

→same as bulk

Strong transition at some ω in quantum dots → laser ?

Metallic Dots

Small spherical alkali metallic cluster

Na mass spectroscopy

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 : 21

3F

FD N

For Au nanoparticles with R = 2 nm, Δε 2 meV.whereas semiC CdSe gives Δε 0.76 eV.→ ε quantization more influential in semiC.

Optical properties of metallic dots dominated by surface plasmon resonance.

41

3

extP E

If retardation effects are negligible,

2

2

n e

m

→ 2

2

1

43

extP Emn e

2

2

3

34 1

ext

p

E

Surface plasma mode at singularity: 3p

sp

For Au or Ag, ωp ~ UV, ωsp ~ Visible.

indep of R.

→ 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).

Discrete Charge States

Thomas-Fermi approximation: 1 1N N e 1N gNU e V

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,

2eU

C

2

1

1g N N

eV

e C

gC

C

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

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,

2e dU

R 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 )

→h

t

h

RC

2

2

1e h

C e R

Quantum fluctuation smears out charging effect when δε U, i.e., when R ~ h / e2 .

2

hR

eConditions for well-defined charge states are

2

B

ek T

C&

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 ).

Coulomb Oscillations

GaAs/AlGaAsT = 0.1K

Thermal broadening Breit-Wigner lineshape

2

1

1g N N

eV

e C

Coulomb oscillation occurs whenever U > kBT, irregardless of Δε .

1 22 2

1 2

4

4 n

T

For Δε >> kBT, c.f. resonant tunneling:

2 2

1 22 2

1 2 1 2 1 21 2 cos 2 r r

t t

r r kL r r

T

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 )

Height of diamonds: 2

max

eeV

C

1i j i j 2 2 21,

2U x y m x y

12g N i jV N U

e

N 1 2 3 … 7

(i, j) (0,0) (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

Spin, Mott Insulators, & the Kondo EffectsConsider quantum dot with odd number of e’s in blockade region.~ Mott insulator with a half-filled band.

ˆBμ zNo external leads:

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

degenerated

0 01exp

2K

UT U

U

→ Transmission even in blockade region.For symm barriers & T << TK , 1.

Singlets states in 3-D Kondo effect enhances ρ.

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.

Vibrational & Thermal Properties

Continuum approximation:ω = vs K → ωj upon confinement.

Quantized vibrations around circumference of thin cylinder of radius R & thickness t << R.

Longitudinal compressional mode

Radial breathing mode

Transverse mode

j

jK

R L

L j

vj

R j = 1,2, …

21

2tot VU Y e dV 2

22

Y Vr

R

re

R

→

2RBM

Y V

MR

1 Y

R Lv

R L

Yv

j

jK

R

2

12L

T j

v t j

R

j = 1,2, …

Raman spectrum of individual carbon nanotubes( 160 cm–1 = 20 meV )

vL = 21 km/s

14RBM

meV

R nm

→

Measuring ωRBM gives good guess of R.

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 :

0, cosy z t y K z t

2

2

ye t

z

2K y t

/2

22

0 /2

1

2

L h

tot

h

U Y K y t dt d z

4 2 21

24Y V K h y

→ 21

12T Lv h K

C.f.

twist K Torsion / shear modeSi nanoscale beans: f L–2

Micro / Nano ElectroMechanical systems ( M/N EM)

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.

2 21 2

3D B

V

L k TC

hv

E.g. (Prob.6)

2 21

3D B

th

k TG

h

T

Gth depends only on fundamental constants if = 1.

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