Artificially Modulated Structures © H.L. Tuller, 2001 Quantum Wells E C hν n=3 n=2 n=1 E V L If we approximate well as having infinite potential boundaries: k = n π for standing waves in the potential well L h 2 k 2 h 2 n 2 We can modify electronic E = 2 m * = 8 m * L 2 transitions through quantum wells © E. Fitzgerald-1999 7 Photodetectors/Solar Cells E-h pairs generated by photons with energy h γ ≥ E g are separated by the built-in potential gradient at the p-n junction. The current voltage characteristics are given by I = I o [ exp ( qV kT ) − 1 ] − I p where I p is the photo-induced “reverse current.” Junctions/Functions © H.L. Tuller, 2001 Junction Function Application P/n Metal/semiconductor Injection/diffusion/collection Blocking (reverse bias) p-n rectifier, switch p-n-p transistor Acceleration/breakdown Tunneling Avalanche and tunnel diodes Injection/confinement/recombination LED, injection laser Generation/separation Solar cell, photodiode Separation/confinement High electron mobiity devices Quantum devices M/I/S Inversion/depletion/accumulation MOSFET 8 - - 3.225 1 The Capacitor d A C A Qd V C Q A Qd EdxV A Qt dxE E o o o d d oo d t d o o ε ε ε εε ρ ε ρ ε ρ = == == === =⋅∇ ∫ ∫ − − − − 2 2 2 2 +V + + + + + + - - - - I=0 always in capacitor ρ E V t d/2 d/2 © E. Fitzgerald-1999 3.225 2 The Capacitor • The air-gap can store energy! • If we can move charge temporarily without current flow, can store even more • Bound charge around ion cores in a material can lead to dielectric properties •Two kinds of charge can create plate charge: •surface charge •dipole polarization in the volume •Gauss’ law can not tell the difference (only depends on charge per unit area) © E. Fitzgerald-1999 1 3.225 3 Material Polarization χ ε ε εεε εε +=+= = =+= 11 E P EPED o r or o P is the Polarization D is the Electric flux density or the Dielectric displacement χ is the dielectric or electric susceptibility + + + + + + + - - - - - - + + + + + - - - - - - - - E P d A C or εε = All detail of material response is in ε r and therefore P © E. Fitzgerald-1999 3.225 4 Origin of Polarization • We are interested in the true dipoles creating polarization in materials (not surface effect) • As with the free electrons, what is the response of these various dipole mechanisms to various E-field frequencies? • When do we have to worry about controlling – molecular polarization (molecule may have non-uniform electron density) – ionic polarization (E-field may distort ion positions and temporarily create dipoles) – electronic polarization (bound electrons around ion cores could distort and lead to polarization) • Except for the electronic polarization, we might expect the other mechanisms to operate at lower frequencies, since the units are much more massive • What are the applications that use waves in materials for frequencies below the visible? © E. Fitzgerald-1999 2 3.225 5 Application for Different E-M Frequencies Methods of detecting these frequencies Cell phones λ=14-33cm DBS (TV) λ=2.5cm Other satellite, 10-50GHz λ=3cm-6mm (‘mm wave’) Fiber optics λ=1.3-1.55µm ‘MMIC’, pronounced ‘mimic’ mm wave ICs In communications, many E-M waves travel in insulating materials: What is the response of the material (ε r ) to these waves? © E. Fitzgerald-1999 3.225 6 Wave Eqn. with Insulating Material and Polarization ( )( 0 EPED t E Bx t PE JBx t D JHx t B Ex insulating o nonmag rrrr r r rr rr r rr r r εε ε ε =+= ∂ ∂ =∇ ∂ +∂ +=∇→ ∂ ∂ +=∇ ∂ ∂ −=∇ 2 2 22 2 00 2 t E ct E E r r ∂ ∂ = ∂ ∂ =∇ ε εεµ k n c k c k c c rE rE erEeeEeEE optical r r r titiriktrki →= = −=∇ === −−•−• ε ω ε ω εω ϖϖϖ 2 2 2 2 2 2 0 )( 0 )( )( )( So polarization slows down the velocity of the wave in the material © E. Fitzgerald-1999 ) → 3 3.225 7 Compare Optical (index of refraction) and Electrical Measurements of ε Material Optical, n 2 Electrical, ε diamond 5.66 5.68 NaCl 2.25 5.9 H 2 O 1.77 80.4 Only electronic polarization Electronic and ionic polarisation Electronic, ionic, and molecular polarisation Polarization that is active depends on material and frequency © E. Fitzgerald-1999 3.225 8 Microscopic Frequency Response of Materials • Bound charge can create dipole through charge displacement. • Hydrodynamic equation (Newtonian representation) will now have a restoring force. • Review of dipole physics: - + d r dqp r r = Dipole moment: +q -q p r Applied E-field rotates dipole to align with field: Exp r rr =τ Torque θ cos EpEpU r r r r =⋅−= Potential Energy © E. Fitzgerald-1999 4 3.225 9 • For a material with many dipoles: Microscopic Frequency Response of Materials )( EpENpNP r r r r r αα == (polarization=(#/vol)*dipole polarization) α=polarizability 0 so, ε α χ ε χ N E P o == r r Ep r r α= Actually works well only for low density of dipoles, i.e. gases: little screening For solids where there can be a high density: local field E ext E loc For a spherical volume inside (theory of local field), o extloc P EE ε3 r rr += © E. Fitzgerald-1999 = 3.225 10 • We now need to derive a new relationship between the dielectric constant and the polarizability Microscopic Frequency Response of Materials + = −= +== 3 2 r extloc extoextor extoextor EE EEP PEED ε εεε εεε Plugging into P=NαE loc : ( ( ( 2 3 1 3 2 +=− + =− ror ext r extoextor N ENEE ε α εε ε αεεε Clausius-Mosotti Relation: oor r N νε α ε α ε ε 332 1 == + − Where v is the volume per dipole (1/N) Macro Micro © E. Fitzgerald-1999 ) ) ) 5 3.225 11 Different Types of Polarizability • Atomic or electronic,α e • Displacement or ionic, α i • Orientational or dipolar, α o Highest natural frequency Lowest natural frequency Lightest mass Heaviest mass oie αααα += ti o eEE ω − = As with free e-, we want to look at the time dependence of the E-field: KxeE t xm t x m −− ∂ ∂ = ∂ ∂ τ 2 2 Response Drag Driving Force Restoring Force ( m K m eE m K m eE x KxeExm exx KxeExm o o oo o ooo ti o = − = − = −−=− = −−= − ω ωω ω ω ω 22 2 2 )( && So lighter mass will have a higher critical frequency © E. Fitzgerald-1999 + ) 3.225 12 Classical Model for Electronic Polarizability • Electron shell around atom is attached to nucleus via springs + E r + E r p rK K r ti olocii erreEZKrrmZ ω − =−−= assume, && Z i electrons, mass Z i m © E. Fitzgerald-1999 6 3.225 13 Electronic Polarizability − = i o o mZ K m eE r 2 ω 2 2 , oe i eoe m eZ ω αωω =<< 0, eoe αωω =>> ( 22 2 oe i e m eZ ωω α − = ( 22 2 oeo o i o EE m eZ p α ωω = − = ( 22 ; i oe oe o o mZ K m eE r ω ωω = − = ; ti oi epperZqdp ω =−== − If no Clausius-Mosotti, ( 2 22 2 11 n m eNZN oeo i o e r = − +=+= ω ωεε α ε ε r ω ω oe 1 ( 2 2 1 oeo i m eNZ ωε + © E. Fitzgerald-1999 ) ) ) ) ) 3.225 14 QM Electronic Polarizability • At the atomic electron level, QM expected: electron waves • QM gives same answer qualitatively • QM exact answer very difficult: many-bodied problem () h 01 10 22 10 10 2 ; EEf m e e − = − = ω ωω ωα E 1 E 0 f 10 is the oscillator strength of the transition (ψ 1 couples to ψ o by E-field) For an atom with multiple electrons in multiple levels: () h 0 0 0 22 10 0 2 ; EE j f m e j j j j e − = − = ∑ ≠ ω ω ωα © E. Fitzgerald-1999 7 8 3.225 15 Ionic Polarizability • Problem reduces to one similar to the electronic polarizability • Critical frequency will be less than electronic since ions are more massive • The restoring force between ion positions is the interatomic potential E(R) R Nuclei repulsion Electron bonding in between ions Parabolic at bottom near R o )( 2 )( 2 o o RRk R E F RRk E −= ∂ ∂ = − = R o - + klijklij CkxF εσ =⇒= © E. Fitzgerald-1999 3.225 16 Ionic Polarizability - + E loc + - p u+ u- • 2 coupled differential eqn’s • 1 for + ions • 1 for - ions ( ( 22 2 22 2 , , 2 11 1 , ωω α α ω ωω ωω − = == = − = == +−= + = −=−= −− −+ −+−+ oi i oioo oi oi o o ti o ti oloc loc M e Eewp M K M eE w ewweEE eEKwwM MM M uuwuuw && &&&&&& Ionic materials always have ionic and electronic polarization, so: ( 22 2 ωω ααααα − ++=+= −+ oi eitot M e © E. Fitzgerald-1999 ) ) ) . → 3 3.225 7 Compare Optical (index of refraction) and Electrical Measurements of ε Material Optical, n 2 Electrical, ε diamond 5.66 5.68 NaCl 2.25 5.9 H 2 O 1 .77 80.4 Only electronic. ωα © E. Fitzgerald-1999 7 8 3.225 15 Ionic Polarizability • Problem reduces to one similar to the electronic polarizability • Critical frequency will be less than electronic since ions are. potential well L h 2 k 2 h 2 n 2 We can modify electronic E = 2 m * = 8 m * L 2 transitions through quantum wells © E. Fitzgerald-1999 7 Photodetectors/Solar Cells E-h pairs generated