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• Quantum Wells and Superlattices Quantum Wells • Quantum Wells 2•. QuantumQuantum W wellsell Sta andtes ( QsuperlatticesWS) and Quantum Size Effects Qualitative explanationQuantum… wells k zz •2D conducting system (x-y plane), size - y quantization in z, k y Dd d ! λF x kx Metals: λ 0.5 nm • F ! Electronic structure •Semiconductors: λF 10 100 nm in parabolic subbands ! − ikx x ik y y "(x, y, z) = !n (z)e e 2 2 2 2 ! n k x + k y E(n•,Filmk ,k ) = deposition + technology (e.g. molecular-beam epitaxy) ! x y 2D2 2 quantum wells with c-Si, GaAs 4 26 • Quantum wells and superlattices •2D electron density of states (per unit area): (2m! m!)1/2 DOS (!) = g g x y 2D s v 2π 2 ! effective masses fo x and y motion gs spin degeneracy, 2, Si, GaAS gv valley degeneracy (degeneracy of states at bottom of conduction band); 1 for GaAs (single valley); 2 or 4 for Si (indirect gap; depending on number of valleys involved) GaAS: ! ! ! 3D mx = my = ml = 0.067me direct gap semiconductor Si{100}: indirect gap semiconductor, more complicated m! 0.19 m transverse t ! e m! 0.92 m longitudinal l ! e 27 • Quantum wells and superlattices •Calculate λF Electron density (areal) na ! ! 1/2 (mxmy) !F na = DOS2D(!F )!F = gsgv 2π!2 2π 4πn 1/2 k = = a ⇒ F λ g g F ! s s " GaAS {100}: λF = 40 nm Si {100}: λ = 35 110 nm (depending on n ) F − a 28 • Quantum wells and superlattices •Square-well potential Infinite , z > d, z < 0 • ∞ V (z) = 0 , 0 < z < d 1D particle in a box Ψz(0) = Ψz(d) = 0 nπ Ψ = A sin k z , k = z z z d •Finite Maximum number of bound states: V0 2m! V d2 1/2 n = 1 + Int z| 0| d max 2π2 !" ! # $ 29 • Quantum wells and superlattices •Electron energy: 2 2 ! 2 ! 2 ! = !n(z) + ! kx + ! ky 2mx 2my - quantized kinetic energy for 2D free e motion energy levels ! ! mx, my effective masses for (x,y) 2 2 ! π 2 n = 1, 2, 3... !n(z) = ! 2 n 2mzd effective mass for motion in z direction ! (z) 50 meV (d=10 nm) • GaAs, 1 ! • ! n2 n ∝ 30 2. Quantum Well States (QWS) and Quantum Size • QuantumEffects wells and superlattices Qualitative explanation… kz n = 1, 2, 3... !n subbands k y D kx Electronic structure in parabolic subbands ikx x ik y y "(x, y, z) = !n (z)e e 2 2 2 2 ! n k x + k y E(n,k ,k ) = + finite well x y 2D2 2 zero point4 energy • No allowed state until ! = !1(z) infinite well • DOS constant until ! = ! 2 ( z ) subband starts to be populated infinite well 31 • Quantum wells and superlattices Sandwitch thin layer GaAs (d 5 nm, ! 1.5 eV) • ! g ! between thick layers AlxGa1-xAs (! 1.42 + 1.26x eV 2 eV for x 0.45) electrons g ! ! ! •Similar unit-cell parameters! possible epitaxial growth one over the other. conduction •The band gap of AlGaAs (large gap) bands straddles that of GaAs (smaller gap). valence •Electrons in conduction band of GaAs layer confined in z direction, by forbidden energy gap of AlGaAs in both sides holes 32 • Quantum wells and superlattices •Triangular potential well Si-SiO2 junction in MOS Triangular wells - infinite Triangular potential wells are quite common in real devices. Examples: Si MOSFETs; single-interface GaAs/Al0.3Ga0.7As heterojunctions. Idealization: GaAS-n-AlGaAs , z < 0 ∞ V (z) = Ez , z > 0 Linear variation of confining electrostatic potential Quantized energy levels 2 1/3 3 2/3 E 3 2/3 !n(z) ( π!e) (n + ) ; n = 0, 1, 2, ... ! 2 2m! 4 ! z " 33 • Quantum wells and superlattices •Optical absorption: • Bulk ! absorption involves e- states at conduction band minimum and valence band maximum. bulk • QW! quantized levels n2 ! square well n ∝ d2 2/3 !n E triangular well Lowest allowed∝ level, n=1 in square well and n=0 in absorption triangular well threshold •GaAs, "F=10 nm • The threshold energy for optical absorption increases ("100 meV) !sharp exciton (bound e-h pair) peak • GaAs, d=21 nm ! max. number bound states nmax=4 ! 4 peaks d=14 nm ! max. number bound states nmax=3 ! 3 peaks •Absorption profile: exciton peaks occurring at each discontinuity in DOS (peak when a new subband starts to be populated) 34 • Quantum wells and superlattices • Thinest QW !GaAs d=5nm Fine structure: splitting in exciton peak one bound state n=, in valence band split into two 35 • Quantum wells and superlattices 1 j=3/2 states split into two ! conf ∝ m! heavy holes light holes Bulk QW heavy!flat band The light is the heavy now 36 • Quantum wells and superlattices Metals, example Pb islands Building island heights Number of islands Covered area histograms 37 Courtesy: Rodolfo Miranda R. Otero, A. L. Vázquez de Parga and R. Miranda PRB 66, 115401 (2002) • Quantum wells and superlattices Superlattices • Molecular beam-epitaxy ! ordered arrays of heterostructures or homojunctions • d1 material (1) + d2 material (2) ! new period d= d1 + d2 Periodic array of QW’s 38 • Quantum wells and superlattices • Periodic array of semiconductor QW’s. •d1 : thickness of well •d2 : thickness of barrier • Two possibilities: 1- Isolated QW’s, d2 >> d1 # "F Multiple quantum wells (no tunnelling of electrons through barriers) 2- Interacting QW’s, d1 # d2 Superlattice (very different properties from bulk material) 39 • Quantum wells and superlattices • Electronic structure of semiconductor superlattices gaps at nπ k = d energy-width • Tight-binding model ! Bloch function, array N QW’s 1/2 iknd Enveloppe function of nth well centerd Ψ (z) = N − e ψ (z nd) k n − at z=nd, overlap with the two n neighbours. Equivalent to atomic orbitals ! or Wannier functions • Analogy with normal crystals ! energy of electron motion in superlattice !(k) = ! α 2β cos kd i − i − i i-th level of a self-energy overlap cos-like dependence, miniband 40 well, i=1,2,3... integral, >0 integral, >0 • Quantum wells and superlattices 41 • Quantum wells and superlattices Density of states ! Nm 1 (! !i + α) g(!) = cos− − − 2π2 2β ! ! i " , ! ! + α < 2 β | − i i| | i| Smearing corresponding to band width 42 • Quantum wells and superlattices π Gaps at ∓ d nπ • Forbidden gaps at k = (d >> a) d v (k) ! smaller than BZ !no periodic extended state g along z • Bloch oscillations: E d.c. electric field ! oscillatory electron velocity, a.c current in superlattice. !(k) = !0 !1 cos kd ! group velocity of electron: − 1 ∂" "1 vg(k) = = sin kd ! ∂k ! 43 • Quantum wells and superlattices (previous page figure) • Consider ! ( k ) = ! 0 ! 1 cos k d − 1 ∂" "1d ! group velocity of electron: vg(k) = = sin kd ! ∂k ! π Periodic, v = 0 at k = , Bragg scattering g | | d ! ˙ • Apply E !k = eE k = ko eEt/! ⇒ − ⇒ − ! Period for motion in reciprocal space between k = π/d ± 2π ! T = d eE Bloch oscillatrions observed if T < τ (scattering relaxation time) , not possible to observe in bulk crystals (d=a) ; but superlattice d 10 100a ! − 44 • Quantum wells and superlattices • Stark ladder : applied E field prevents minibands !no Bloch oscillations (no wave function overlap) If eEd 4 β ! | i| miniband field induced width potential displacement 45 • Quantum wells and superlattices • Effects on phonon propagation !acoustic phonons folded back at k = π /d ± (Phonon dispersion Brillouin new boundaries (d: new period superlattice) in a crystal) - Optical phonons ! confined to one or other layer of material: confined modes • Doping superlattice !periodic array of homojunctions or p-n junctions (nipi structure) intrinsic layer - Band-edge modulations determined by doping level: between n and p $ dopant concentration ! $ Eo modulation energy (next page figure) !eff = ! E ! “indirect” minimum gap in real space g g − 0 46 • Quantum wells and superlattices E N , doping concentration Nipi structure o ∝ d spatial variation of n- and p- doping spatial variation (in z) of conduction and valence bands nipi semimetallic 47 • Quantum wells and superlattices Nipi structure • To estimate the modulation consider the homojunction as a capacitor: semiconductor !! dielectric constant ! C/A 0 ! d/2 Charge per unit area, Q/A = Nddn = Nadp widths of p-type and n-type regions (assume the same for - and + regions) 2 2 2 e Nddnd e Ndd C = Q/V E0 = (dn = dp = d/2) ⇒ " 2!!0 4!!0 electrostatic potential difference=E0/e 48 • Quantum wells and superlattices Nipi structure • Very high doping ! modulation very pronounced, nipi semimetal Band modulation and ef f can be controlled varying charge • !g densities in n- and p- layers ! injection of excess e- or holes electrically or optically reduce E0 by neutralization of charge donors or acceptors. e- -hole pairs after absorption of photons eff Light intensity $ ! ! g $ 49.
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