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Superlattices and Microstructures, Vol. 1, No. 1, 1985 73 Superlattices and Microstructures, Vol. 1, No. 1, 1985 73 ELECTRON ENERGY STATES AND MINIBAND PARAMETERS IN A CLASS OF NON-UNIFORM QUANTUM WELL AND SUPERLATTICE STRUCTURES Vijai K. Tripathi Department of Electrical and Computer Engineering Oregon State University, Corvallis, Oregon 97331, USA Pallab K. Bhattacharya Solid State Electronics Laboratory Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109, USA (Received 13 August 1984 by J.D.Dow) A simple method to compute the carrier energy states, miniband parameters and dispersion characteristics for single and multiple quantum well and superlattice structures is presented. The method utilizes the continuity of the envelope function across the heterojunctions according to the boundary conditions that both the wavefunction # and the particle current density ~'/m* be continuous at each interface. The nonuniform potential distribution encountered in doped or compositionally graded materials is approximated by piecewise constant potential functions. In addition to being conceptually simple, the method is readily adopted to fairly complex structures where other more sophisticated methods such as LCAO, reduced Hamiltonian and tight binding theories may become unfeasible or unmanageable. It is shown that for an arbitrary stepped potential variation, the eigenvalues (or the energy states) of quantum wells or a finite number of coupled quantum wells can be found by utilizing a transverse resonance method which is readily implemented on a digital computer for the computation of these eigenvalues. For the case of periodic superlattices, the miniband parameters and the dispersion characteristics are computed from a suitably defined transmission matrix associated with a unit cell of the superlattice which may itself consist of multiole layers. Typical results for the computed parameters for several wells and simple, biperiodic, binary and polytype superlattices consisting of various AlxGa1_xAS and InxGa1_xAS alloys are presented. I. INTRODUCTION particularily for thin structures, the A considerable amount of work has wave equation approach with appropriate been done in recent years on the evalua- boundary conditions 11'12 is conceptually tion of the electronic properties of simple and can be readily applied to the various compositional and doped quantum multilayered wells and simple and 1 -I0 13 well and superlattice structures. polytype superlattice problems. In One of the most successful methods used this paper a unified systematic approach in deriving the eigenvalue equations for that utilizes a transverse resonance quantum wells and dispersion equations condition for quantum wells and a for periodic structures has been the use transmission matrix method for periodic of plane wave type solutions with structures is presented. assumed I-3 or derived 6'7'11'12 boundary conditions or connection rules. Even II. THEORY though more accurate methods such as e---~genvaluesT of the quantum LCAO 5, reduced Hamiltonian 9 and other wells and multiple coupled quantum wells and the miniband parameters of periodic microscopic theories 8 are available quantum well structures are found by 0749 6036/85/01 0073+07 S02.00/0 © 1 985 Academic Press Inc. (London) Limited 74 Superlattices and Microstructures, Vol. I, No. 1, 1985 solving the one-dimensional Schrodinger results for analogous systems in layered wave equation with the potential media, cascaded guided wave structures variation specified by the conduction and other engineering problems. It is and valence band discontinuities 14. It to be noted that we do not necessarily is assumed that each layer contains a have to utilize this analogy in that all sufficient number of atomic sublayers the results can be obtained by such that the effect of atomic constructing the solution of the wave potentials in each region can be assumed equation subject to the given boundary to be at least partly included in the conditions. effective mass. Each layer acts like a In the notation of linear system crystal modified at most by a slowly theory the state vector x] varying potential and the effective mass characterizing the one dimensional wave function is a modulated Bloch wave system is a solution of: with an envelope governed by the d Schrodinger equation. In addition, the x] = [A]x] (2) layers are assumed to be sufficiently thin with low doping levels such that A x I they are modelled by wells and barriers where x] = x2 ] = I d~ ] and then the of uniform thickness even in the • *dz presence of an applied bias voltage. 3m The interface connection rules for the characteristic matrix is given by effective mass wave function at the abrupt heterojunctions correspond to the 0 -jm continuity of the slowly varying [A] = [_j 2(E-V) 0]. me state ~2 envelope function introduced by Bastard 6 and subsquently reinforced by White et. variables x I and x^z are continuous at 11 al. Alternately, the connection rules the heterojunctions between individual layers. derived by Kroemer and Zhu 12 may be utilized which will obviously lead to The solution of the wave equation slightly different results depending in each layer can now be expressed in upon the discontinuity in effective mass terms of a propagation constant and a across the interface. That is, the wave characteristic impedance which are given function in the jth region is a solution by of W y = / ~*(V-E) (3) d2~j 2mj (E-Vj) ~j=0 (I) ~2 dz ~ + and *2 d,j Z = / m ~ (4) with * dz and ~ being continuous • v / (~7-E~v) . m. The cnaracterlstlc impeaance parameter 3 represents the ratio [~/1 d~] across the heterojunction if the jm*dZ continuity of envelope function as associated with an electron wave derived by Bastard 6'7 for type I super- traveling in the positive z direction. lattices (e.g., GaAs-A£.Ga I .As)is to be It should be noted that in the regions utilized. It should be mentloned that both boundary conditions conform to the where E<V, Y is real and Z o imaginary continuity of the probability current and the solution represents evanescent density. In addition, for the case of modes. Each layer of width £ is type II (InAs-GaSb type) superlattices completely characterized in terms of ¥ where the periodic part of the host and Z given above and the solution of material Bloch function cannot be compl~x boundary value (multiple neglected, the corresponding connection interface) problems can be facilitated rules for the envelope function" can be by recalling some of the transmission utilized. and translational properties of x]. me The solution of the above equations transmission matrix for the two for simple cases of potential wells and variables is given by superlattices are well known. For more complex stepped or graded potential variation due to compositional changes in the structure, it is convenient to solve for the eigenvalues of quantdm wells and dispersion characteristics of jm d at z=£ periodic structures by utilizing known Superlattices and Microstructures, Vol. 1, No. 1, 1985 75 E:J s I ....I d~ (5) * = ~-j=l I z~sinh~j ~j coshxj gj jm at z=0 (8) The translational property is Then the dispersion equations for the characterized in terms of the periodic structure are readily found in impedance .(~/ I, ~_~)d~ seen at a distance terms of the elements of the ABCD matrix and is given by "m A+D £ from a discontinuity. For example, if - cos k d (9) I d~ 2 Z£~ [~/jm* ~] is specified at a given where d is the total length of each unit N plane (z=£ I) then the impedance at a cell (= j~1 £j) and k is the wave number. distance £ from that plane is given by The derivation of the dispersion Z£1+Zotanh¥£ equation then simply ammounts to multiplying the transmission matrices to I d~ = Zo Z + Z£ tanh¥£ find overall transmission parameters A jm* ~z- at z=Z1-£ o I (6) and D and is given in Table II for the cases of a simple and an ABC polytype III. QUANTUM WELL STRUCTURES superlattice. For the case of isolated or coupled quantum well structures, represented by a finite number of stepped potential V. RESULTS AND DISCUSSION regions, the eigenvalues are found by The eigenvalue equations for utilizing the transverse resonance quantum well structures are of the form FI (E,Pl,O~,..o~z n )=0 and the dispersion condition. That is, the net (~/ I d~) equations for periodic structures are of m* dz or total impedance is either zer6 or the form F2(E,Pl,P2,..pn)-cos kd=0. In infinity at every plane along the both cases pl..pn are known variables structure. In order to illustrate the such as effective mass and potential procedure, we consider the case of the associated with the individual layers. symmetrical stepped potential strucutre The effect of nonparabolicity of the shown in Table I, case (d). The total conduction and valence band of each impedance at the axis of symmetry is layer can be included in the effective readily found by utilizing the translational property of impedance as mass 14. A convenient method to solve these equations is the use of standard given by eqn. (6) and is found to be multiple variable optimization programs 15 Here we define a utility functionIF112 for quantum well problems I Z 3+Z 2tanhx2£2 1 ZI Z2 Z2+Z3tanhY2£2 + Z1tanh71£1___ zl and IF2-cos kd~ 2 for superlattice = ~-- | Z3+Z2tanh~2£ 2 " dispersion problems and seek the minima at the t ZI+Z2 z2+z3tanh72£ 2 tanh¥1£1 (zeros) of the utility function with axis --2-- of symmetry energy as a variable. The eigenvalues for some quantum well structures and the miniband parameters such as the Setting this impedance equal to 0 or ~ dispersion diagrams are shown in Figs.
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