"Advanced Semiconductor Materials Lecture 8, Quantum Wells, Quantum"
Advanced Semiconductor Materials Lecture 8, Quantum Wells, Quantum Wires and Quantum Dots Need for low dimensional structures Carrier confinement Ballistic transport => High performance transistors and lasers Elastic scattering: Energy does not change between collisions Inelastic scattering: Energy changes with collision distance Ballistic transport: At low enough dimensions (< average distance between two elastic scattering), electrons travel in straight lines => Light beams in geometrical optics Outline Quantum wells (Well with finite potential) Quantum wires Quantum dots Sebastian Lourdudoss PARTICLE IN AN INFINITE WELL Consider first the particle trapped in an infinitely deep one- dimensional potential well with a specific dimension Observations • Energy is quantized, Even the lowest energy level has a positive value and not zero • The probability of finding the particle is restricted to the respective energy levels only and not in-between • Classical E-p curve is continuous. In quantum mechanics, p = hk with k = nπ/l where n = ±1, ±2, ±3 etc. π En = h2k2/2m = n2 π2 h2/2ml2 In fact the negative values are not counted since the probability of finding the electrons in n=1 and n=-1 is the same and also E is the same at these values • When l is large, energies at En and En+1 move closer to each other => classical systems, energy is continuous. Sebastian Lourdudoss PARTICLE IN A FINITE WELL Region II: A cos kx (symmetric solutions) (1) ΨII = One dimensional A sin kx (antisymmetric solutions) (2) finite well where k2 = 2mE/h2 h (3) Region III: ΨIII = Be-γx where γ2 = 2m(V0-E)/h2 γ h (4) and (5) Region I: ΨI = Beγx (eq. 6) but by symmetry we use only the single boundary condition at x = l/2 between II and III At x = l/2: Ψ = Ψ and Ψ ´ = Ψ ´ II III II III (7) For the symmetric solutions, i.e., for (1), • (10) and (11) have electron energy A cos (kl/2) = Be-γ l/2 γ (8) E on both sides via k and γ => only Ak sin (kl/2) = Bγe-γ l/2 γ γ (9) discrete E values satisfy boundary (9)/(8): k tan (kl/2) = γ (10) conditions (7) For antisymmetric solutions, i.e., for (2), k tan (kl/2 - π/2) = γ (11) From Coldren and Corzine, (In (11), cot x = - tan (x- π/2) has been used to show the Diode lasers and photonic similarity between symmetric and antisymmeteric integrated circuits, Wiley, characteristic equations Sebastian Lourdudoss 1995 PARTICLE IN A FINITE WELL Observations • The wave functions are not zero at the boundaries as in the infinite potential well • Allowed particle energies depend on the well depth Infinite well •Finite well energy levels < Finite well corresponding infinite well energy levels Energy levels and wave functions • The deeper the finite well, in a one dimensional finite well. the better the infinite well Three bound solutions are approximation for the low- illustrated lying energy values • Quantum mechanical tunnelling possible a) Shallow well with single allowed • Quantum mechanical level kl = π/4 reflection possible at E>V0 N.B: k2 = 2mE/h2h b) Increase of allowed levels as kl exceeds π; here kl = 3π + π/4 π c) Comparison of the finite-well (solid line) and infinite well (dashed line) energies; here kl = 8π + π/4 π Sebastian Lourdudoss ENERGY LEVELS IN A FINITE WELL IN TERMS OF THE FIRST LEVEL OF INFINITE WELL • For infinite well case, En = n2 E1∞ (12) where E1∞ = h2k12/2m (13) = π2 h2/2ml2 (14) • Can we arrive at a similar relation for the finite well case? YES How? Solve (10) and (11) using (12) with (3) & (5): Quantum number in the quantum well: nQW = (En/E1∞)½ (15) Maximum number of bound states: nmax = (V0/ E1∞)½ (16) Example: V0 = 25E1∞ => From (16), nmax = 5 (If necessary, round up to the nearest integer) nQW = 0.886, 1.77, 2.65, 3.51, 4.33 from figure From Coldren and Corzine, Diode lasers and photonic Plot of quantum numbers as a function of the maximum allowed integrated circuits, Wiley, quantum number which is determined by the potential height V0 1995 Sebastian Lourdudoss RELATION BETWEEN ENERGY LEVELS IN A FINITE WELL WITH THE FIRST LEVEL OF INFINITE WELL Example: V0 = 25E1∞ => From (16), nmax = 5 nQW = 0.886, 1.77, 2.65, 3.51, 4.33 from the figure Some figures: • The energy spacing between the energy levels for the quantum wells with thickness ~10 nm is a few 10’ 100’ 10’s to a few 100’s meV meV. • At room temperature kT ~ 26 meV. This means only the first energy levels can be occupied by electrons under typical device operational conditions From Coldren and Corzine, Diode lasers and photonic Plot of quantum numbers as a function of the maximum allowed quantum number which is determined by the potential height V0 integrated circuits, Wiley, 1995 Sebastian Lourdudoss Bound states as a function of well thickness 2m V l * 2 n = 1 + Int e 0 max π h 2 2 Sebastian Lourdudoss Optical absorption/emission in the quantum wells hπ n 2 hπ n 2 2 hπ n 2 2 2 2 2 2 1 1 E − E = E + C i i V C −E − =E + i V i g i m + m 2m l 2m l 2 * * * e 2 2l * h 2 e h 1 1 1 = + m = optical effective mass * m m * * * m eh e h eh Sebastian Lourdudoss Density of states in the low dimensional structures Lower the dimension greater the density of states near the band edge => Greater proportion of the injected carriers contribute to the band edge population inversion and gain (in lasers) Sebastian Lourdudoss Quantum wires Sebastian Lourdudoss Quantum dots • Quantization in all the three directions • With a finite potential, the problem can be treated as a spherical dot like an atom of radius R with a surrounding potential V (r) = 0 for r ≤ R and = Vb for r ≥ R Here r is the co-ordinate • The solutions resemble those for the spectra of atoms • Total number of states * 3/ 2 ( 2m V ) L L L N =t e b x y z 3π h 2 3 Sebastian Lourdudoss Courtesy: W.Seifert Sebastian Lourdudoss Growth modes Epitaxial layer (e) γe/v Complete wetting: γs/e Layer-by-layer or γe/v + γs/e < γs/v Frank - van der Merwe Substrate (s) 2D+3D or Stranski - Krastanow γe/v Non-complete wetting: γs/v γs/e 3D or γe/v + γs/e > γs/v Volmer - Weber γe/v and γs/v: surface energies of epimaterial and substrate, γs/e: interface energy substrate/epimaterial Courtesy: W.Seifert Sebastian Lourdudoss Quantum wire and dot fabrication Coupled QWRs -Evidence for tunneling and electronic coupling shown - Wire is GaAs, barrirer is Formed from reorganisation of a AlGaAs sequence of AlGaAs and strained InGaAs epitaxial films grown on From GaAs (311)B substrates by MOCVD. http://www.ifm.liu.se/Matephys/AAnew/research/iii_v/qwr.htm#S1.7 The size of the quantum dots are as small as 20 nm Sebastian Lourdudoss Nanorods Courtesy: W.Seifert Sebastian Lourdudoss Sebastian Lourdudoss Etched Quantum Dots By E-Beam Lithography GaAs AlGaAs QW AlGaAs GaAs • E-beam lithography used for Au-liftoff etch mask • SiCl4/SiF4 RIE etch • Mask size =15-22 nm • Dot Size= 15-25 nm • Dot Density = 3x1010cm-2 • Etched dots have poor optical quality • Dot density is low • Device applications require regrowth Courtesy: P.Bhattacharya, University of Michigan Sebastian Lourdudoss Researchers Vie to Achieve a Quantum-Dot laser (Physics Today, May 1996) Room temperature quantum dot laser Courtesy: P.Bhattacharya, University of Michigan K. Kamath, P. Bhattacharya, T. Sosnowksi, J. Phillips, and T. Norris, Electron. Lett., 30, 1374, 1996. Sebastian Lourdudoss Tunnel Injection QD Lasers Grown by MBE Active region Single mode ridge waveguide lasers 1.5µm n- Al0.55Ga0.45As 20Å Al0.55Ga0.45As 1.5µm p- Al0.55Ga0.45As W=3µm cladding layer barrier cladding layer L=200-1300µm 18Å GaAs 650Å barriers 750Å GaAs GaAs p-AlGaAs Quantum dots 95Å hωLO ω Courtesy: In0.25Ga0.75As P.Bhattacharya, Injector well University of In0.4Ga0.6As n-AlGaAs Michigan quantum dots 2.5 T=12K Quantum Dot(~980nm) • The laser heterostructures are grown by solid 2 source molecular beam epitaxy PL Intensity (a.u.) C, • The quantum dots are grown at 530° the quantum 1.5 Injector (~950nm) C, well is grown at 490° and the rest of the structure at 630° C 1 • The high strain due to the In0.25Ga0.75As QW limits the number of dot layers to less than 4. 0.5 • The energy separation between the quantum well injector layer ground state and quantum dot ground 0 850 900 950 1000 1050 state is tuned by adjusting the In and Ga charge in Sebastian Lourdudoss QD Wavelength (nm) History of Heterostructure Lasers 1000000 DHS - Diode Heterostructure Threshold Current Density (A/cm2) QW - Quantum Well GaAs pn QD - Quantum Dot 100000 QW Miller et. al. T=300K 10000 DHS QW Dupuis et. al. Alferov et. al. QD Kamath et. al. 1000 DHS QW Mirin et. al. Alferov et. al. Shoji et. al. QW Tsang Hayashi et. al. 100 QD Ledenstov et. al. QW Alferov et. al. QD Liu et. al. Chand et. al. 10 1960 1970 1980 1990 2000 2010 Courtesy: Year P.Bhattacharya, University of Michigan Sebastian Lourdudoss