1 Universidade de Aveiro Departamento de Física Nikolai A. Sobolev, Svetlana P. Kobeleva Physics and technology of nanosize structures 014/015 Национальный исследовательский технологический университет «МИСиС» National University of Science and Technology "MISIS". QUANTUM WELLS AND SUPERLATTICES.1. QUANTUM WELL ENERGY LEVELS.1.1. Conduction electron energy levels Let us consider the simplest situation when a single layer of material A is embedded between two thick layers of material B. Assumptions: E g (A) < E g (B); Type-I band offset (Fig. -0). Fig. -0. (a) Dispersion relation for a typical, direct gap, bulk semiconductor. ω is the energy and k the wavevector close to the centre of the Brillouin zone. Only the bands closest to the Fermi level are shown. (b) When a confining potential is applied, the bands split into discrete energy (doublet) levels, but the spin quantum numbers are preserved. hh denotes heavy hole, lh light hole, and e electron.
The situation is exemplified by the pairs of materials GaAs/AlGaAs, InGaAsP/InP, InGaAs/InAlAs, GaSb/AlSb etc. In the calculations, we have to introduce the continuity condition at the interfaces, namely, that n (z ) and 1 n( z) m ( z) z should be continuous. NB!! Rather than the continuity condition of the derivative of the wave function as derived in quantum mechanics textbooks it is necessary for conservation of particle current to use the continuity of 1 n( z). m ( z) z In the infinitely deep well approximation (V c (B), see Fig. -1), ψ = 0 in the confining layers B, hence, ψ = 0 at the interface. Fig. -1. Therefore, the Schrödinger equation takes the form ( z) n E ( z) n n (1) m A z and the solution is very simple: E n n m L A z, n = 1,, 3, ()
3 ( ) sin n z A z (3) n L z In the finite-well case (Fig. -), the Schrödinger equation can be solved either. Note, that the problem has an inversion symmetry around the center of the well which we now takes as the center of the coordinates. Fig. -. The Schrödinger equation can only have even or odd solutions: or n () z Acoskz for z L/(inthewell) Bexp zl/ for zl/ (in the barrier) Bexp zl/ for zl/ " n ( z) Asin kz for z L/ (in the well) and the same solutions in the barrier, (4) (5) k where 0, E n V E, V0 E 0 n m m A B (6) From the continuity conditions at z L/ we obtain k m A kl tan, (7) m B
4 k kl cotan m A (8) m B The equations can be solved numerically or graphically. A very simple graphical solution can be developed if ma m B. Then, using Eq. (6), Eqs. (7) and (8) can be transformed into implicit equations in k alone: kl k kl cos for tan 0, (9a) k 0 kl k kl sin for tan 0, (9b) k 0 where mv k (9c) 0 0 This equations can be visualized graphically (Fig. -3). Fig. -3. There is always one bound state. The number of bound states is 1 Int m V L A 0, (10) where Int[x] indicates the integer part of x. Solutions are located at the intersections of the straight line with slope k 0 1 with curves
5 cos kl kl y with tan 0 sin kl kl y with tan 0 ; solid curves; even wave functions, or ; dashed curves; odd wave functions. The important limiting case of infinitely high barriers can be found again by n putting k 0 = in Fig. -3. There is then an infinity of bound states with k. L Even solutions are ψ n ~ cos kz, with kl =(n + 1)π; odd solutions are ψ n ~ sin kz, with kl = nπ. NB!! The situation is much more complicated in real semiconductor materials, especially with respect to the hole states in the valence band..1.. Two-dimensional density of states Besides the energy quantization along the z-axis, the main property of the quantizing films is the bidimensionality in the density of states (DOS). As is well known from the solid-state physics, 1 D( E) de D( k) dk kdk (11) ( ) In the parabolic approximation E k m, and Eq. (11) yields m D (1) NB!! The density of states of a given quantum state E n is therefore independent of E and of the layer thickness (see Fig. -4).
6 The total density of states at a given energy is then equal to that given by Eq. (1) times the number n of different k z states at that energy: D D = n D Fig. -4. The D DOS shows discontinuities for each E n. The 3D and D DOSs are equal for these energies. NB!! Further differences between the 3D and D DOSs: 3D systems: DOS 0 at the bottom of the energy band; D systems: DOS remains finite. Consequence: All dynamic phenomena (such as scattering, optical absorption, and gain) remain finite at low temperatures and low kinetic energies. However, when numerous levels are populated or when one looks at transitions involving large values of n, such as in thick layers, there is no difference between D and 3D behaviours. (This is a clear analogy to the correspondence principle between quantum and classical mechanics.)
7.. EXCITONS AND IMPURITIES IN QUANTUM WELLS..1. Excitons and shallow impurities The mathematical formalism of weakly bound free excitons (or Wannier- Mott excitons) in bulk semiconductors is essentially the same as that of the hydrogen atom. The effective Rydberg and the Bohr radius are given by 4 me m 1 Ry r r (8 ) m 0 r 4 m a 0 a, B me r m B r r Ry, (13) where m r is the reduced mass of electron and hole ( 1/ m 1/ m 1/ m ), r e h r is the relative permittivity of the semiconductor, Ry = 13.6 ev, and a B = 0.59 Å. Remember that one Rydberg is the energy difference between the lowest bound level and the continuum edge, and the Bohr radius is the distance between the electron and the hole (or electron and proton in the hydrogen atom). From a B 100 Å, one infers that the wave function and energy levels of excitons and impurities are quite modified in a QW where the thickness is usually of the order or smaller than the Bohr diameter a B. In the limiting exact D case where L << a B, one should obtain the usual D Rydberg value Ry D 4 Ry (14) 3D for the infinitely deep well.
8 The energy levels are then given by R, Ry3 / n 1/ (15) n D D Absorption spectra of 3D and D excitons are compared in Fig. -5. Fig. -5. The increase in exciton binding energy has a profound influence on QW properties. It allows GaAs based QWs to have their optical properties dominated by exciton effects even at RT!
9 Example: bulk GaAs Binding energy of the free exciton E(FE) = 3.7 mev; Ionization energy of a shallow donor E(SD) = 6 mev; Empirical Haynes rule E(BE) 0.1 E(SD) Binding energy of a donor bound exciton E(BE) 0.6 mev. This value is to be compared to kt(4. K) = 0.36 mev. The RT excitons in QWs allow very promising features such as optical bistability, four-wave mixing, and large electro-optic coefficients.
10.3. TUNNELING STRUCTURES, COUPLED QUANTUM WELLS, AND SUPERLATTICES The particle can stick it's foot through the door and say "It's pretty murky in there" and back off. C. Clark.3.1. The double-well structure Treating the wave function overlap in a sufficiently thin barrier as a perturbation, one finds the perturbation matrix element to be where V ˆ 1 1 H 1 Vˆ, (16) Ĥ is the electronic Hamiltonian, ψ 1 and ψ the unperturbed wave functions of single wells 1 and, V the confining potential of QW (see Fig. -6). Fig. -6. The Schrödinger equation is then E 1 V V 1 1 E 1 V 1 V 1 a a 1 0, (17) 0
11 where V ˆ ˆ 1 1 V( z) 1 V1 ( z), ˆ ( ) ˆ 1 1 1 (, (18) V V z V z) 1, so that ε = E 1 + V 1 V 1, (19) and the levels are split by the amount V 1..3.. The communicating multiple-quantum well structure or superlattice For N coupled wells, the N-degenerate levels give rise to bands with N states. Let us consider the tight-binding model of the N-well chain (Fig. -7). Fig. -7.
1 With a Bloch-like envelope wave function one obtains the energy as ( q) E s t cosqd, (0) i i i i The bandwidth is 4t i and rapidly increases as the barriers get narrower. Assuming usual Born-von Karmann periodic conditions, one finds that q can only take discrete values q = i / (N d), i = 1,,, N (1) Therefore, the superlattice band can accommodate N electrons with different q s (which are the quantum numbers of this problem). NB!! The SL effect introduces a profound change in the D DOS: The dispersion of the N states in a band destroys the steepness of the square density of states (Fig. -8). Fig. -8. The D limit for N independent wells is retained when the bandwidth 0, i.e. when the overlap matrix elements vanish due to wide barriers.
13.3.3. Tunneling Consider the potential V(z) describing a barrier between two regions of constant potential (taken as zero), see Fig. -9. Figs. -9 (top), -10 (bottom) For a square barrier of height V 0 between z a = a / and z b = + a / one finds by standard quantum mechanics that the probability for transition of a particle through the barrier is given by (for E < V 0 ) 1 1 k T t 1 sinh ( a), 4 k () where k E and m V 0 E. m Let us now consider a double barrier structure with two identical barriers displaced by the distance L (Fig. -10). One can calculate T tot t tot T 1 r expiklt, (3)
14 where ka, t 1 k tan tanh( a). k The interesting result is that for most energies the total transmission is of the order of the square of the transmission of a single barrier, but that for some very especial values T tot = 1, i.e. the double barrier gets totally transparent! This occurs when ( kl ) n 1 (4) t For the important case of thick square barriers this condition is 1 k cotan k L a (5) k This is just the condition for having a bound state in the well of width L a between the two barriers! The double barrier therefore serves as a filter, which only transmits electrons of energy close to the resonance values given by Eq. (5), see Figs. -11, -1. Fig. -11.
15 Fig. -1. Resonant tunneling diode simulator : <http://www.nanohub.org/resource_files/tools/rtd/rtd.swf> In a perfect superlattice the same barrier is repeated periodically with a period L, see Fig. -13. Fig. -13
16 There will be bands of allowed k values around energies for which t kl n. (6) In general, The bandwidth will increase when t 1; Above the top of the barriers, when t is very close to 1, the bands are large and separated by small gaps in energy around Φ t + kl = nπ. Fig. -14 illustrates the situation. Fig. -14. Fig. -15 shows the evolution of the minibands as the barrier and the well thickness are varied.
17 Fig. -15. Fig. -16. Superlattice and SL band structure with minibands in the valence and conduction bands.
18.3.4. n i p i Structures (facultative) A spatial modulation of the doping in an otherwise homogeneous crystal can produce a superlattice effect, see Fig..17, and, as a consequence, a reduction of the Brillouin zone and new energy bands in the SL direction. Fig..17. Charged particles are subject to a self-consistent potential: V() z V () z V () z V () z (7) imp H xc where V imp (z) is the electrostatic potential of the ionized impurities, V H (z) the Hartree potential of electrons and holes, and V xc (z) the exchange and correlation potentials. The first term, V imp (z), can be calculated from the Poisson equation: d V imp ( z) e N D ( z) N A ( z) dz 0. (8) R
19 Similarly, the Hartree potential is H( ) dz 0 R dv z e e( z) h( z). (9) The exchange and correlation term can be calculated in a density-functional formalism: V xc e 4 ( z) 0.611 4 0R 3 N ( z ) 1/3, (30) where N(z) is the electron density. Energy levels for the z-quantized motion have to be calculated selfconsistently through the one-dimensional Schrödinger equation V( z) e,h( z) e,h( z), (31) m z where χ e and χ h are the envelope functions of the electron or hole WFs. Features: 1. In the case of exact compensation (equal number of donors and acceptors, d/ d/ ND( z) dz NA( z) dz, d/ d/ where d is the superlattice period, no free carrier exists in the unexcited sample at low temperatures.. For equal uniform doping levels N A = N D and zero-thickness undoped layers, the periodic potential consists of parabolic arcs and has an amplitude V e 0 NDdn 8 0 R (3)
0 (for GaAs with N A = N D = 10 18 cm 3 and d n = 500 Å, one has V 0 = 450 mev). The quantized energy levels in the potential wells are approximately the harmonic oscillator levels: E 1/ en D 1 e,h n 0 Rm e,h. (33) For electrons, e.g., the subband separation is 40. mev for the above parameters. Since the effective band gap eff E g shown in Fig..17 is given by E = E V + E + E, it is reduced below the bulk material value. eff g g 0 e h 3. When there is unequal doping, free carriers will accumulate in the corresponding potential well (Fig..18). Fig..18.
1 Eqs. (7) and (31) must then be solved self-consistently. The Fermi level can be located at will (Fig..18(b)). 4. For large enough spacings and dopings, the effective band gap can become negative (i.e., d 700 Å for N D = N A = 10 18 cm 3 ). There then exists charge transfer from hole wells to electron wells until a zero gap is attained due to three factors: (i) band filling, (ii) diminishing of the periodic SL potential thanks to the charge neutralization by the transferred charges, and (iii) quantized energy-level modification (Fig..17(c)). 5. Under non-equilibrium conditions such as photoexcitation or carrier injection, electron and hole populations can build up in the wells, leading to charge neutralization and an effective band gap increase. 6. Under such non-equilibrium conditions, electrons and holes are spatially separated and the radiative recombination rate is strongly diminished as compared to the bulk case as for indirect-band-gap semiconductors. At the same time, nonradiative recombination rates are also strongly decreased, leading to reasonable quantum efficiencies. Many of the features that are expected from the n-i-p-i-structures have indeed been observed: variation of the band gap with increasing excitation, change of absorption features with light intensity, tunable luminescence, etc.
.4. SUMMARY OF THE PROPERTIES OF QUANTUM WELLS AND SUPERLATTICES.4.1. Basic physical phenomena D electron gas Step-like density-of-state function Quantum Hall effect Fractional quantum Hall effect Excitons at room temperature Resonant tunneling in double-barrier structure and superlattices Energy spectrum in superlattices determined by choice of potential and strain Stimulated emission at resonant tunneling in superlattices Pseudomorphic growth of strained structures..4.. Important consequences for applications Shorter emission wavelength, reduced threshold current, larger differential gain, and reduced temperature dependence of the threshold current for semiconductor lasers Infrared quantum cascade laser Short-period superlattice quantum well laser Optimization of electron and light confinement and waveguiding for semiconductor lasers D electron-gas transistors (high-electron-mobility transistors) Resonant-tunneling diodes Precise resistance standards
3 SEED s and electro-optical modulators Infrared photodetectors based on quantum size level absorption..4.3. Important technological peculiarities Lattice match unnecessary Low-growth-rate technology (MBE, MOCVD) needed in principle Submonolayer growth techniques used Blockading of mismatch dislocations during epitaxial growth Sharp increase in the variety of heterostructure components.