Transcript 投影片 1 - National Cheng Kung University

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
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.
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:
R = 6 a ~ 1 nm →
N surf
Nbulk

1
2
N surf
Nbulk
4 R 2
2
a

4 R3
3 a3
Applications: Gas storage, catalysis, reduction of cohesive energy, …
2. Quantization of electronic & vibrational properties.

3a
R
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
d



2 2sin 
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
D          j 
j
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 Microscopy
For 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:
wi  j 
2
j e Er i
2
  j   i   
Emission rate (α = e2 /  c ):
w j i 
2
j e Er i
2
  i   j    
4  j   i 
3
3 2
c
j r i
2
Real part of conductivity ( total absorbed power = σ E 2 ):
2e2
    
V

i, j
j nˆ  r i
2
 f  i   f  j    j   i   


Absorption & emission measurements → electronic spectra.
E  E nˆ
Fluorescence from CdSe
nanocrystals at T = 10K
Spectra of Fluorescence of individual nanocrystals.
Mean peak: CB → VB
Other 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.

I  exp  2

2m 
z 
2

 = 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.
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
dI
   j  rj    F  eV   j 
I dV
j
( weighted eDOS at E = εF + eV )
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
F  C z
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.
F
z  
C
02

2

2 2
0

Q = quality factor 
 0 


Q


2
z 0 
F 0
Q
Estored
per cycle
Edissipated
z0 
C
ω0 & Q are sensitive to type & strength of forces between tip & sample.
Their values are used to construct an image of the sample.
F0
C
Magnetic Force Microscopy
MFM = AFM with magnetic tip
F
F  z0  z   F  z0  
z
z
z  z0
B

z
z  z0
2 B

 z2
z
z  z0
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
k2
   i, j 
2m
  x, y, z   i, j  x, y  eik z
i, j = quantum numbers in the cross section
1-D subbands
D     Di , j  
i, j
d Ni , j d k
L
Di , j   
 2 2
d k d
2
Van Hove singularities at ε = εi, j
m
2 2    i, j 
 4L

  hvi , j
 0

   i, j
   i, j
Spectroscopy of Van Hove Singularities
Carbon nanotube
Prob. 1
photoluminescence of a
collection of nanotubes
STM
1-D Metals – Coulomb Interaction & Lattice Couplings
Let there be n1D carriers per unit length, then
n1D 
2
2
2k F  k F
2

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.
1
 ee
→
1   F 
 

0  F 
  3D  
For 1-D metals,
Let

 ee
2
  


0  F 
 
k  k  kF
1   
 

0  F 
→
τ0 = classical scattering rate
2
  3D 

→
k
     F
Caution: our Δε = Kittel’s ε.
2
2
k

2m
2
2
F

 

2
2m
1 as   0
 k  kF  k  kF 
2
kF
k
m
for |k|  kF
 E & p conservation are satisfied simultaneously.
 quasiparticles near
εF are well defined
2
kF
 
k
m
     F
k  k  kF
1+2→3+4
Pauli exclusion favors  3 ,  4  0
1 , k1  0
 3 , k3  0
 2 , k2  0
 4 , k4  0
E, p conservation:
k1  k2  k3  k4  0
 For a given Δε1 , there always exist some Δε2 & Δε4
to satisfy the conservation laws provided Δε1 > Δε3 .
1
 ee
→

1 
0 F
→
 1D 
~ const

 1D  
 
 ee


0 F
 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).
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
Let Δn be the excess right-moving carrier density, DR(ε) be the corresponding DOS.
2 2
D   q V

q Vv
I  n qv  R
qv
hv
L
→ The conductance quantum
2 e2
GQ 
h
Likewise the resistance quantum
RQ 
2 e2

V
h
q = e
depends only on fundamental constants.
1
h
 2
GQ 2 e
If channel is not perfectly conducting,
2 e2
G  F  
T  F 
h
Landauer formula
T = transmission coefficient.
For multi-channel quasi-1-D systems
T  F    Ti , j  F 
i, j
i, j label transverse eigenstates.
For finite T,
2 e2
I  F ,V , T  
h
R

 d  f   eV   f   T  
L
h
h T  1  T 

2 e2 T
2 e2
T
R = reflection coefficient.
Channel fully depleted of carriers at Vg = –2.1 V.
R


h
h R

2 e2 2 e2 T
Two Barriers in Series-Resonant Tunneling
tj, rj = transmission, reflection amplitudes.
t j  t j exp  i t j 
rj  rj exp  i r j 
For wave of unit amplitude incident from the left
a  t1  r1 b
At left barrier
ik L
 r2 a e i k L
At right barrier b e
c  t2 a e i k L
→
a  t1  r1 r2 e
Tc 
2
2i k L
t1 t2 e
t1

a
1  r1 r2 e2 i k L

i k L  t 1 t 2
1  r1 r2 e


i 2 k L  r 1  r 2
Resonance condition :
→
t1 t2 e i k L
c
1  r1 r2 e2 i k L
2


k
t1
2
t2
2
1  2 r1 r2 cos  2kL  r1  r 2   r1
*  2kL  r1  r 2  2 n
2
r2
2
n  Integers
2m
2
c
t1 t2 e
1  r1 r2 e
At resonance

i k L  t 1 t 2

i 2 k L  r 1  r 2
c  t1 t2 e
c
For t1 = t2 = t :

i k L
  t1 t2 e


n0
r1
n
r2
n

i k L  t 1 t 2


r
n0

t1 t2
1  r1 r2
2
t
1 r
2
e i k L  e  i k L
n
1
r2 ei n *
e i k L
T *  2 n  1
→
For very opaque barriers, r  –1 ( φ  n π )
→ resonance condition becomes particle in box condition
while the off resonance case gives
Using
2
t1 , t2
2
1
T  t1
n
2
t2
Resonant
tunneling
k L  n
2
*  2 n
&
one gets (see Prob 3) the Breit-Wigner form of resonance
T   
4 1 2
 1  2 
2
 4    n 
2
where
j 

tj
2
2
Incoherent Addition & Ohm’s Law
Classical treatment: no phase coherence.
a  t1  r1 b
→
b eik L  r2 a e i k L
b
c  t2 a e i k L

→
a
2
 t1
2
 r1
2
 t1
2
2
 r2
2
2
 t2
2
a
c
2
r2
2
a
r
h 
h
 2 1  1
R 2
2e 
2e T
t1
2
2
2

a
a
2
b
2
r2
2
2
2
2
2
t1
1  r1
 r1
T c
→

h  R1 R2


 2 1 


2
T 1 T 2 
t2  2e 
r2
2

t1
2
1  r1
t2
2
2
r2
2
2
(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, d R 
1 dL
dL

 b vF
lb
Incoherence addition of each segment gives
R  RQ 
→
1D 
h L
2e2 lb
dR
h 1
 2
dL
2e lb

m
n1D e 2 
(Prob. 4 )
Localization
2
T
t1 t2
h
h 1  2 r1 r2 cos  *  r1
R 2
 2
2
2
2 e T 2e
t1 t2
2
1  2 r1 r2 cos  *  r1
2
r2
→
2
2
r2
2
 …  = average over φ* = average over k or ε .
h 1  r1
R  2
2e
2
r2  2 r1 r2 cos  *
2
t1
2
t2
2
h 1  r1 r2
 2
2
2
2e
t1 t2
2
2
h 1  r1 r2
2e 2 t1 2 t2 2
2
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
1).
dL
For an additional length dL, d R 
le
Setting
→
r1
2
R
t1
2
T
+ T=
dR  dT  1
r2
2
 dR
t2
2
 dT
h  1 R dR 
h
h 1 R dR


1  R d R 1  d R  


2e2  T d T 
2e2 T
2e2 T 1  d R 


dL
 
 R 1   R  1 d R    R 1  2 d R    R  1  2
le


R  dR 
2

dL
R  dR  R 1  2

le  →


ln
R
2L

R 0
le
R 
where
 2L 
h
exp
 
2e2
 le 
dR  R
R
0
2d L
le
 R
L 0
 RQ 
h
2e 2
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 .

R 
 2l 
h
exp


2e 2
 le 
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.
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.
2e2 

 n,m
Net current thru contact is I n 
N
V

T
V

n n
m
h 
m

Setting
I n  0 , Vn  V n
Nn   T 
→
n , m
m
For the voltage probe n, Vn adjusts itself so that In = 0.
T  V

T 
n,m
→
1
Vn 
Nn
T
m
 n,m
Vm
m
n,m
m
T  

T 
n,m
m
n
m
n,m
m
m
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.
T 3,1 V
V3  3,1
3,2
T  T 
Current out of 1:


V
2
2e2
I
V  T 1,3 V3
h
3,1
3,2
if T    T  

2e2  1 1,3 

V 1  T 
h  2

2e2

V
h
no probe
Mesoscopic regime: le < L < lφ .
Semi-classical picture:
t
 m, n 
i
  a j exp 
j


m
l
e  

p

A   d l

c

 
T
 n, m
 t
 m, n 
2
App. G
Aharonov-Bohm effect

loop
 ie

a1  a2 exp 
A  d l

c

 loop
2
 a1
2
 a2
2
 2 a1
A  d l     A  dS  
 2  
a2 cos 

 hc / e 
S
0 
hc
e
Electronic Structure of 0-D Systems
Quantum dots:
Quantized energy levels.
 n,l ,m  r,  ,    Rn,l  r  Yl ,m  ,  
 n,l , m   n,l
e in spherical potential well:
For an infinite well with V = 0 for r < R :
2
 n,l 
 n2,l
2m * R
  n,l r 
Rn,l  r   jl 

R


2
βn, l = nth root of jl (x).
for r < R
jl   n ,l   0
β0,0 = π (1S),
β0,1 = 4.5 (1P),
β1,0 = 2π (2S),
β1,1 = 7.7 (2P)
β0,2 = 5.8 (1D)
Semiconductor Nanocrystals
CdSe nanocrystals
For CdSe:
2
mc *  0.13 m
For R = 2 nm,
 n,l
  n,l   2.9eV 

 0,0   R 2 


 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.
Kramers-Kronig relation:
For ω → 
     
ne2
2
    

m  
2




P
0
  s
ds
s2   2
   s  ds

→
0
Strong transition at some ω in quantum dots → laser ?

0
   s  ds 
 n e2
2m
same
as bulk
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 :  
1
D  F 

2 F
3N
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
Optical properties of metallic dots dominated by surface plasmon resonance.
If retardation effects are negligible,
n e2
    
m 2
→
P
1
m
4

n e2
3
2
Surface plasma mode at singularity:
P
1
Eext
sp 

4
3


Eext
3
 3 2

4  2  1
 

 p

p
3
Eext
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).
Discrete Charge States
Thomas-Fermi approximation:
N 1   N 1  e    N 1  NU   eVg
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,
e2
U
C

Cg
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
1 
e2 
Vg 
    
 e  N 1 N C 
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,
e2 d
U
 R Rd
Prob. 5
For R = 2 nm, d = 1 nm & ε = 1, we have
U = 0.24 eV >> kBT = 0.026eV
→ Thermal fluctuation strongly supressed.
at T = 300K
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 )
→
2
h
h
e
h 1

 

RC
t
C e2 R
Quantum fluctuation smears out charging effect when δε  U, i.e., when R ~ h / e2 .
Conditions for well-defined charge states are
R
h
e2
&
e2
C
k BT
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/AlGaAs
T = 0.1K
1 
e2 
Vg 
    
 e  N 1 N C 
Coulomb oscillation occurs
whenever U > kBT, irregardless of Δε .
For Δε >> kBT, c.f. resonant tunneling:
T
t1
Breit-Wigner lineshape
t2
2
1  2 r1 r2 cos  2kL  r1  r 2   r1
T   
Thermal broadening
2
4 1 2
 1  2 
2
 4    n 
2
2
r2
2
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
U  x, y  
1
m 2  x 2  y 2 
2
i j  i  j 1 
Vg N 
1
   N  2 U 
 e ij
N
1
2
3
…
7
(i, j)
(0,0)
(0,0)
(0,1) or (1,0)
…
(1,1) , (0,2), or (2,0)
U /α e
(U +  ) /α e
…
(U +  ) /α e
Δ Vg
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:
eVmax
e2
  
C
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:
μ   B zˆ
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
TK 
1
2
   0  0  U  
 U exp 


U


→ Transmission even in blockade region.
For symm barriers & T << TK , T 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.
Kj 
j
R
L j  j
vL
R
j = 1,2, …
Longitudinal compressional mode
e
r
R
RBM 
Radial breathing mode
j
Kj 
R
Transverse mode
U tot 
→
YV
MR 2

T j
1
R
Y


1
YV
2
2
Y
e
dV


r


2 V
2R2
vL
R
v t  j
 L  
12  R 
vL 
Y
2
j = 1,2, …

Raman spectrum of individual carbon nanotubes
( 160 cm–1 = 20 meV )
vL = 21 km/s
→
Measuring ωRBM gives good guess of R.
 RBM 
14  meV 
R  nm 
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 :
y  z, t   y0 cos  K z  t 
2 y
2
e 2 t K yt
z
Utot
1
 Y
2
L
  K
Si nanoscale beans: f 
C.f.
2
y t  dt d z 
2
0  h /2
→ T 
L–2
h /2
1
Y V K 4 h2 y 2
24
1
vL h K 2
12
twist  K
Torsion / shear mode
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.
E.g.
1D
V
C
1D
th
G
2 2 L kB2 T

3hv

 2 kB2 T
3h
(Prob.6)
T
Gth depends only on fundamental constants if T = 1.