Nanomaterials –Electronic Properties

Keya Dharamvir

•Quantum confinement

•Quantum size effect

•Energy bands and electronic transition

•Charge quantization

Modifications due to :

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 Nanostructure/ Bulk

• Physics Semi-classical Q. mechanical• Electron’s nature Particle-like Wave-like• E or k-space Continuous Discrete• Current Continuous Quantized• Decision Deterministic Probabilistic• Fabrication Micro-fabrication Nano-fabrication •Surface:volume Small Very large• Packing Low Very high

Electrons’ Behaviour in Smaller Sizes

• Energy quantization d ~ Fermi wave length of electron in a metal (F)

or exciton diameter in a semionductor • Charge quantization Charging energy (Ec) >> Thermal energy (kT)

• Ballistic d<mean free path ()

Free electron case (3D box):exp(ikr) where k =2n/L; E= ħ2k2/2m N = 2x (4kF

3/3)/(2/L)3 = VkF3/32

electron concentration N = N/VEF= (ħ2/2m) kF

2 = (ħ2/2m) (32 N) 2/3; kF = (32 N)1/3

F= 2kF= 2 (32 N)-1/3

Exciton : e-h pair bounded by attractive electrostatic interaction (H atom-like)

E

Eg

Conductionband

Valence band

Exciton levels

•Binding energy: Eex =e4/2ħ2n2

•Bohr (exciton) radius: r = n2ħ2/e21/me

+1/mh

Si Ge GaAs CdSe KClEex (meV) 14.7 3.8-4.1 4.2 15 400r (nm) 4.3 11.5 12.4

E

Eg

0

Exciton bindingenergy: EexEg-Eex

n=1n =2

k

Quantum Confinement

Exciton radius

R

•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)

Energy for the lowest excited state relative to Egap

E(R) = h22/2R2 – 1.8e2/2R …

r

dot

Particle in a box problem

Density of State: # of states per unit energy range

N =2n2/L2 N =8n3/3L3

dN /dE = const dN /dE ~ E 1/2

k=2n/LE = ħk2/2mk =(2mE)ħN = 2xn/L= k/ = (ħ)2mE)

dN /dE = ((2m)/2ħ)(E)

dN /dE ~ E

N = 2n/L

DO

SDO

SDO

S

1D 2D 3D

E = ħk2/2m= ħ/2m(kx2+ky2+kz2)

• k is discreet in confinement directions only

Size Effect: Energy Levels and DOS

A.P. Alivisatos, Science 271, 933 (1996)

3d 2d 1d 0d

EnergyD

OS

EF

Bulk Nano atom particle

Size controlled band gap tuningDiscrete Energy levels

CB

VB

Semiconductor

LUMO

HOMO

Band gap

Size Effect:1D-Quantum well states

F.J. Himpsel et al, Adv. Phys. 47, 511 (1998)

Size Effect: Optical Spectra

A.P.Alivisatos, J. Phys. Chem. 100, 13227 (1996)

• Shift to higher energy in smaller size• Discrete structure of spectra• Increased absorption intensity

Size effect: Tunable Band Gap

Optical excitation is significantly enhanced, both, in frequency and intensity, in smaller sizes.

S. Ogut et al, Phys. Rev. Lett. 79, 1770 (1997)

Bulk Si = 1.14 eV GaAs =1.5 eV

Energy Bands

Go to P. 7 – 10 of Doc2

Energy Band Structure: Energy vs. k

Cnn

VCn V n

(h2/2V Ej= cos 2j/N index j = 0, 1, 2 … Define a new index k = 2j/Na: wave vector E(k) = coska, k = eiknan : Bloch wave function (symmetry adapted LCAO)

/a k=0 /a

E

= 2/k = 2a

= ∞ ….

….

….

….

a

01

2

/a k=0 /a

E

Electronic Transition

i

fif

•Direct transition (k=0)•In phase •Added transition dipole•Electronically allowed transition

i

fif

•Indirect transition (k ≠ 0)•Out of phase•Cancelled transition dipole•Electronically forbidden but vibronically allowed

Electric Transition dipole moment if = <f |er i>

•Band width: overap of wave functions•Slope dE/hdk = hk/m = vg: group velocity of electron

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.

Absorption spectra: Direct and Indirect Transition

Size Effect: Enhanced Absorption

k

E

E

N(E)

For quantum dot,•Energy levels: discrete•DOS: delta function

• xp ~ h• x: well defined• p=hk: Not well-defined• k is not an exact quantum number for QD

•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

Selection rulek’ = kk = 0) k’ = k + qk ≠ 0)Energy relationship hv = Eg hv = Eg + hv(q)Interaction electronic: two body vibronic: three bodyTransition rate fast ~ 10 -7 sec slow ~ 10-2 secRadiative efficiency high lowExample GaAs (Eg (dir.) =1.4 eV) Si (Eg (ind.) = 1.1 eV)

(Eg (dir.) = 3.37 eV)

E

kEg

q

hv

phonon

• P. 26, 27 of doc2(Optical properties of semiconductor

nanoparticles)

• P. 18 of doc2• (optical properties of metal

nanoparticles)

Charge Quantization

•Charging energy: Ec = e2/2C >> kT At T =300K kT = 26 meV C<< 3.1x10-19 F C = 4d 4= 1.1x10-10 J -1 C2m-1

•For charge quantization, the diameter of dot (d) must be << 28 nm

ee

d

N=0

321

Tunneling Spectroscopy of InAs QD

Ec=0.11 eV: single electron charging energyEg=1.02 eV: nanocrystal band gap

d = 32A

T=4.2K

U. Banin et al, Nature, 400, 926 (2000)

S-like

P-like

STM

Optical

• P. 21, 22, 23 of doc2 forConduction through metal

nanoparticles.

• P. 30 forComparison table

Property: Melting Temperature of Nanocrystal

A.P.Alivisatos, J. Phys. Chem. 100, 13227 (1996)

Y.J. Lee et al, J. Comp. Chem 21, 380 (2000), Phys. Rev. Lett. 86, 999 (2001)

• 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

Property: Thermodynamic Behaviors of Metal Clusters

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