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Nanomaterials –
Electronic Properties
Keya Dharamvir
Modifications due to :
•Quantum confinement
•Quantum size effect
•Energy bands and electronic transition
•Charge quantization
Nanostructures
STRUCTURE
SPATIAL
DIMENSION
CONFINEMENT
DIMENSION
Bulk
3
0
Surface/ Film
(Quantum Well)
2
1
Nanotubes, -wires
(Quantum wire)
1
2
Nano-particles,
clusters (Quantum
dots)
0
3
Microstructure vs. Nanostructure
Microstructure
/ Bulk
• Physics
Semi-classical
• Electron’s nature Particle-like
• E or k-space
Continuous
• Current
Continuous
• Decision
Deterministic
• Fabrication
Micro-fabrication
•Surface:volume Small
• Packing
Low
Nanostructure
Q. mechanical
Wave-like
Discrete
Quantized
Probabilistic
Nano-fabrication
Very large
Very high
Electrons’ Behaviour in Smaller Sizes
• Energy quantization
d ~ Fermi wave length of electron in a metal (lF)
or exciton diameter in a semionductor
• Charge quantization
Charging energy (Ec) >> Thermal energy (kT)
• Ballistic
d<mean free path (l)
Free electron case (3D box):
Y = exp(ikr) where k =2pn/L;
E= ħ2k2/2m
N = 2x (4pkF3/3)/(2p/L)3 = VkF3/3p2
electron concentration N = N/V
EF= (ħ2/2m) kF2 = (ħ2/2m) (3p2 N) 2/3; kF = (3p2 N)1/3
lF= 2p/kF= 2p (3p2 N)-1/3
Exciton : e-h pair bounded by attractive
electrostatic interaction (H atom-like)
E
Conduction
band
Eg
Valence
band
E
Exciton
levels
k
Eg
Eg-Eex
n =2
n=1
Exciton binding
energy: Eex
0
•Binding energy: Eex =me4/2eħ2n2
•Bohr (exciton) radius: r = n2eħ2/me2; 1/m=1/me +1/mh
Si
Ge
GaAs CdSe KCl
Eex (meV)
14.7 3.8-4.1 4.2
15
400
r (nm)
4.3 11.5
12.4
Quantum Confinement
r
dot
R
Exciton
radius Energy for the lowest excited state
relative to Egap
E(R) = h2p2/2mR2 – 1.8e2/2eR …
Particle in a box problem
•R<< r: Strong Confinement
- 1st term (localization) dominant
- Electron and hole are quantized
- Energy gap ~1/R2
eg) Si<4.3 nm, Ge<11.5 nm, GaAs<12.4
•R>> r: Weak confinement
- 2nd term (coulomb attraction) dominant
- Exciton confinement character
L.E. Brus, J. Chem. Phys. 80, 4403(1984)
Density of State: # of states per unit energy range
1D
k=2pn/L
E = ħk2/2m
k =(2mE)-1/2/ħ
N = 2xn/L= k/p
= (1/pħ)(2mE) 1/2
dN /dE = ((2m)1/2/2pħ)(E)-1/2
3D
DOS
dN /dE ~ E-1/2
DOS
DOS
N = 2n/L
2D
N =2pn2/L2
dN /dE = const
N =8pn3/3L3
dN /dE ~ E 1/2
E = ħk2/2m
= ħ/2m(kx2+ky2+kz2)
• k is discreet in confinement
directions only
Size Effect: Energy Levels and DOS
Semiconductor
Bulk
CB
Nano
atom
particle
2d
LUMO
DOS
Band E
F
gap
VB
3d
HOMO
Size controlled band gap tuning
Discrete Energy levels
A.P. Alivisatos, Science 271, 933 (1996)
Energy
1d
0d
Size Effect:1D-Quantum well states
F.J. Himpsel et al, Adv. Phys. 47, 511 (1998)
Size Effect: Optical Spectra
•
•
•
Shift to higher energy in smaller size
Discrete structure of spectra
Increased absorption intensity
A.P.Alivisatos, J. Phys. Chem. 100, 13227 (1996)
Size effect: Tunable Band Gap
Bulk Si = 1.14 eV
GaAs =1.5 eV
Optical excitation is significantly enhanced, both, in
frequency and intensity, in smaller sizes.
S. Ogut et al, Phys. Rev. Lett. 79, 1770 (1997)
Energy Bands
Go to P. 7 – 10 of Doc2
Energy Band Structure: Energy vs. k
a
2
0
1
E
Y = Cnfn
V = Cn V n
(h2/2m)2Y + V Y = E Y
Ej= a +2b cos 2pj/N
index j = 0, 1,  2 …
Define a new index k = 2pj/Na: wave vector
E(k) = a +2b coska,
Yk = eiknafn : Bloch wave function
(symmetry adapted LCAO)
….
….
a -2b
…. l= 2 p/k = 2a
a
a +2b
p/a
k=0
p/a
…. l= ∞
Electronic Transition
Electric Transition dipole moment mif = <ff |er| fi>
ff
mif
fi
E
p/a
k=0
p/a
ff
mif
fi
•Direct transition (Dk=0)
•In phase
•Added transition dipole
•Electronically
allowed transition
•Indirect transition (Dk ≠ 0)
•Out of phase
•Cancelled transition dipole
•Electronically forbidden
but vibronically allowed
•Band width: overap of wave functions
•Slope dE/hdk = hk/m = vg: group velocity of electron
Absorption spectra: Direct and Indirect Transition
Electronic absorption spectra for
three sizes of CdSe nanocrystals, in
the wurtzite (direct) and rock salt
(indirect) structures. In each
instance the direct gap spectrum is
structured and intense, while the
indirect gap one is featureless and
relatively weaker. The relative
absorption efficiencies do not change,
despite the concentration of
oscillator strength due to quantum
confinement.
Size Effect: Enhanced Absorption
For quantum dot,
•Energy levels: discrete
•DOS: delta function
N(E)
E
k
• D xD p ~ h
• x: well defined
• p=hk: Not well-defined
• k is not an exact quantum
number for QD
E
•Envelope functions sample larger k-space
•Overlap of wave functions
- Increased absorption intensity
M.S. Hybersten, Phys. Rev. Lett. 72, 1514 (1994)
Photon absorption: Direct vs. Indirect Transition
E
phonon
q
Eg
hv
k
Selection rule
Energy relationship
Interaction
Transition rate
Radiative efficiency
Example
k’ = k (Dk = 0)
k’ = k + q (Dk ≠ 0)
hv = Eg
hv = Eg + hv(q)
electronic: two body vibronic: three body
fast ~ 10 -7 sec
slow ~ 10-2 sec
high
low
GaAs (Eg (dir.) =1.4 eV) Si (Eg (ind.) = 1.1 eV)
(Eg (dir.) = 3.37 eV)
• P. 26, 27 of doc2
(Optical properties of semiconductor n
anoparticles)
• P. 18 of doc2
• (optical properties of metal nanoparti
cles)
Charge Quantization
e
e
3
2
1
N=0
d
•Charging energy: Ec = e2/2C >> kT
At T =300K
kT = 26 meV
C<< 3.1x10-19 F
C = 4pe d
4pe = 1.1x10-10 J -1 C2m-1
•For charge quantization,
the diameter of dot (d) must be << 28 nm
Tunneling Spectroscopy of InAs QD
STM
T=4.2K
Optical
d = 32A
S-like
P-like
Ec=0.11 eV: single electron charging energy
Eg=1.02 eV: nanocrystal band gap
U. Banin et al, Nature,
400, 926 (2000)
• P. 21, 22, 23 of doc2 for
Conduction through metal nanoparticle
s.
• P. 30 for
Comparison table
Property: Melting Temperature of Nanocrystal
A.P.Alivisatos, J. Phys. Chem. 100, 13227 (1996)
Property: Thermodynamic Behaviors of Metal Clusters
• As the cluster size decreases, the melting
temperature (Tm) monotonically decreases,
However, when the cluster size is small
enough, Tm does not vary monotonically
with cluster size.
• The absence of a premelting peak in heat
capacity curves for some clusers.
• Premelting: surface melting, partial
melting, orientational melting, and
isomerization
Y.J. Lee et al, J. Comp. Chem 21, 380 (2000), Phys. Rev. Lett. 86, 999 (2001)
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