Nanomaterials – Electronic Properties Keya Dharamvir

Quantum confinement Quantum size effect Energy bands and electronic transition Charge quantization Modifications due to :

Nanostructures STRUCTURESPATIAL DIMENSION CONFINEMENT DIMENSION Bulk30 Surface/ Film (Quantum Well) 21 Nanotubes, -wires (Quantum wire) 12 Nano-particles, clusters (Quantum dots) 03

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:volumeSmall 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 (E c ) >> Thermal energy (kT) Ballistic d<mean free path ( ) Free electron case (3D box):  exp(ikr) where k =2  n/L; E= ħ 2 k 2 /2m N = 2x (4  k F 3 /3)/(2  /L) 3 = Vk F 3 /3  2 electron concentration N = N/V E F = (ħ 2 /2m) k F 2 = (ħ 2 /2m) (3  2 N ) 2/3 ; k F = (3  2 N ) 1/3 F = 2  k F = 2  (3  2 N ) -1/3

Exciton : e-h pair bounded by attractive electrostatic interaction (H atom-like) E EgEg Conduction band Valence band Exciton levels Binding energy: E ex =  e 4 /2  ħ 2 n 2 Bohr (exciton) radius: r = n 2  ħ 2 /  e 2  1/m e +1/m h Si Ge GaAs CdSe KCl E ex (meV) r (nm) E EgEg 0 Exciton binding energy: E ex E g -E ex  n=1 n =2 k

Quantum Confinement Exciton radius R R<< r: Strong Confinement - 1 st term (localization) dominant - Electron and hole are quantized - Energy gap ~1/R 2 eg) Si<4.3 nm, Ge<11.5 nm, GaAs<12.4 R>> r: Weak confinement - 2 nd term (coulomb attraction) dominant - Exciton confinement character L.E. Brus, J. Chem. Phys. 80, 4403(1984) Energy for the lowest excited state relative to E gap E(R) = h 2  2 /2  R 2 – 1.8e 2 /2  R … r dot Particle in a box problem

Density of State: # of states per unit energy range N =2  n 2 /L 2 N =8  n 3 /3L 3 d N /dE = constd N /dE ~ E 1/2 k=2  n/L E = ħk 2 /2m k = (2mE)   ħ N = 2xn/L= k/  = (  ħ)  2mE)  d N /dE = ((2m)  /2  ħ) (E)  d N /dE ~ E  N = 2n/L DOS 1D 2D 3D E = ħk 2 /2m = ħ/2m(kx 2 +ky 2 +kz 2 ) k is discreet in confinement directions only

Size Effect: Energy Levels and DOS A.P. Alivisatos, Science 271, 933 (1996) 3d 2d 1d 0d Energy DOS EFEF Bulk Nano atom particle Size controlled band gap tuning Discrete 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, (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  C n  n V  C n V n (h 2 /  2  V  E j =  cos 2  j/N index j = 0,  1,  2 … Define a new index k = 2  j/Na: wave vector E(k) =  coska,  k =  e ikna  n : Bloch wave function (symmetry adapted LCAO )  /a k=0  /a E = 2   /k = 2a = ∞ ….    a 0 1 2

 /a k=0  /a E Electronic Transition i i ff  if Direct transition (  k=0) In phase Added transition dipole Electronically allowed transition i i ff  if Indirect transition (  k ≠ 0 ) Out of phase Cancelled transition dipole Electronically forbidden but vibronically allowed Electric Transition dipole moment  if = Band width: overap of wave functions Slope dE/hdk = hk/m = v g : 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  x  p ~ 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 rule  k’ = k  k = 0)  k’ = k + q  k ≠ 0) Energy relationship hv = E g hv = E g + hv(q) Interaction electronic: two body vibronic: three body Transition rate fast ~ sec slow ~ sec Radiative efficiency high low Example GaAs ( E g (dir.) =1.4 eV ) Si ( E g (ind.) = 1.1 eV) (E g (dir.) = 3.37 eV ) E k EgEg q hv phonon

P. 26, 27 of doc2 (Optical properties of semiconductor nano particles)

P. 18 of doc2 (optical properties of metal nanoparticle s)

Charge Quantization Charging energy: E c = e 2 /2C >> kT At T =300K kT = 26 meV C<< 3.1x F C = 4  d 4  = 1.1x J -1 C 2 m -1 For charge quantization, the diameter of dot (d) must be << 28 nm e e d N=

Tunneling Spectroscopy of InAs QD Ec=0.11 eV: single electron charging energy Eg=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 for Conduction through metal nanoparticles. P. 30 for Comparison table

Property: Melting Temperature of Nanocrystal A.P.Alivisatos, J. Phys. Chem. 100, (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

THANK YOU