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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) • THANK YOU