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

Table I: Schematics and eigenvalue equations for various stepped-potential structures.

m2 2m 1 E £ m2 E

--] ~--v tan " - vT'~'~-~/ (Odd)

ml" or (--t--~ /2m1*E2ml*E /in *(V E) (a) tan Y-~r---'-~ " - I Z ' (even/ Syltmetric Well 11 m2

V~ ,m~l m2 E m3 E

/~ (v3 -E ) • tan £ [ m2 V2 h / m2 ""m3 ;,2 m J 1 + . 1 m 1 (V2-E)(VI-E) (b) Nou-S~et r ical Well.

m VI KA ÷ EE m1* tan £ - -

where K~ - m/F~1*~E m1 (V3-E) m~3 (V2"E) 2m2"

I + / ", tanh ~ 4 2 m2 (',,'3- E)

I m * I tan ~ " - ml ~'~ K; (Odd)

m'~ Z 1/2 {d) Sylmlzetrical stepped tan 2-- " - m2 £ (oven) potential well

b (V2-E) ÷ tanh ~ 2° (y3_E) (E-V2) £2 J B2 [ where~ K - (odd) I + F*(V2-Z) /~2" ( z-v 2 ) m2.(V3_E ) tanh , ,~2 ~2(E

D m 1 V 1 m * m2* I V2 v~ ~Vn_l mn-T i m ~i2 ." vj..~m3.l-" v3 zj - ~) ,,__P-7 (e) General multiple stepped potential structure

Z 1 ' = Z 1 , Z 1' + Z 2 tanh Y2L2 Z2' = Z2 Z 2 + z 1 ' tsnh y2Z 2 ; zj+ I zj zj ÷ z~ ta. TI, 1 zj + Zj. 1 tanh Yjlj ;

Zn' + 2 n " 0 ~igenvalue equation) Super~ttices and Microstructures, Vol. I, No. 1, 1985 77

Table II Schematics and Dispersion Relation for Two Superlattice Structures

1 ~2 m1* 81 m2" (a) 2 I Simple AB type SL + cosh Y2L2 cos 61t I - cos k(¢1~2); 0

m) V2

(b) ABC polytype SL

Z I Z 2 + js~n B1tl{sinn T2~ 2 cosh Y]L3[~"~. ~" ~] Z Z 2 1 t ' I+ 3: slnh "~3~[3 Cosh Y2 2L~ ~1;} - cos k (t I + t 2 + t]]; O

* The dispersion equations for t~ese c~ses and Eor ABCD polytype SL are 9lven in Referenc~ 7.

250 I ! example would be: given a simple GaAs - AI0.~Ga_~ ./-As superlattice, to calculate AIo.8Gao.2As ..~90meV ~ 0.133m° the wel~ and barrier lengths such that \ AIo'3Ga0"7As E~O ~ 0"092mo the superlattice has a desired effective GaAs ~IL I 0.067m ° energy band gap and band width. This 200 1121Ill problem is easily solved by simply o looking at the plots of miniband energy 12 = 20A parameters as a function of the two lengths and the results are found to be the same. 150 E IV. CONCLUSIONS A uni£ied approach to analyze and compute the eigenvalues of a class of nonuniform general stepped potential

=,.. loo quantum well strucutes and miniband I.I,J parameters of superlattices consisting of such coupled quantum wells has been formulated. A transverse resonance method for the derivation of the quantum 50 well eigenvalue equations and a transmission matrix approach for the derivation of the dispersion equation for the periodic structures has been used to compute the characteristic i I I O 20 40 60 parameters of some typical structures. The methodology presented is applicable to fairly complex structures with Figure I Variation of electron subband multiple layers where other more (n=1) energy in a symmetrical stepped- accurate computational methods become potential GaAs-AlxGa1_xAS quantum well. unfeasable. In addition to help 78 Superlattices and Microstructures, Vol. 1, No. 1, 1985

i i , i 250 i ! , , ) i/ 12 AI08Gao 2As 490meV .... t • " -1 I-- ...... % / 200 7'' rlL~I r Ino.4Gao.6As°"" / /" Ga As EI"L~ 0.067m o / 200 / / / / / / >e 150 / E / / I >, 100, >, / / c 60 100 UI UJ / //

/ 50 ~7,v"

50 ml=o 049mo m 2 =0 067mo I I I I I 20 , I I I I 60 40 °20 0 0 TI'/3 211/3 11 (11) A k(ll+ I =)[ radiansJ I , t , L { I 0 20 40 60 Figure 4 E-k diagrams for a (12)K In Ga As - GaAs strained layer 0.4 0.6 Fi@ure 2 Variation of electron subband superlattzce. (n=1) energy with step dimension in a symmetrical stepped potential GaAs-AlxGa1_xAS quantL~ well.

GaAs-A!~Ga.TAs POLYTYPE SL

160

200 '-I L~I I-IFL I '''°'°'°,''Ga As . 11 12 11 15¢ ~ -

E 141 ,-, 150 /~ i1 .. 30 A. 11:30A' 12:50/~ 49o~ev330~ev-- ~c~.13]= o Al.8 ca .. oa, oj __.-.------12-20 A' 0.092mo A1 , 3Ga. 7AS E LIJ 0 0.067m..o CaAs i 11 12

100 ¢1 U,.I 12:20A" I : 50~ 80 50 I I I I 50 40 30 20 10 0 11 [/~] I I I I I I 0 10 20 30 40 50 I I I I 12 I/(] 0 !1/3 2n/3 H k(11+12) [radians] Figure 5 Width of the first miniband as a function of barier thickness 12 in a Figure 3 E-k diagram for a GaAs - A1 0 3Ga0 7As - A1 0 8Ga0 2As AI0.8Ga0.2As- GaAs superlattice. polytype s6perlattice. " " Superlattices and Microstructures, Vol. 1, No. 1, 1985 79 predicting the electronic properties of 4. G. A. Sai-Halasz, R. Tsu and L. various quantum well and superlattice Esaki, Appl. Phys. Lett., 130, 651 structures, the analytical and numerical (]977). techniques described are adaptable to a design synthesis procedure for such 5. G. A. Sai-Halasz, L. Esaki and W. A. structures via parameter optimization or Harrison, Phys. Rev., B18, 2812 other analytical techniques that have (1978). been used for analogous systems 16. Even 6. G. Bastard, Phys. Rev. B24, 5693 though the analysis and results (1981). presented in the paper are directly applicable to type I superlattices, the same procedure can be applied to the 7. G. Bastard, Phys. Rev. B25, 7584 (1982). case of type II superlattices. For the latter case the state variables are 8. I. Ivanov and J. Pollmann, Solid redefined and the expressions for Z i are modified such that the corresponding State Commun., 32, 869 (1979). boundary ~onditions as derived by 9. Y. C. Chang and J. N. Schulman, Bastard v'" are satisfied. That is, all J. Vac. Sci . Technol., 21, 540 of the eigenvalue and dispersion (1982). equations derived in the paper also characterize type II quantum wells and 10. P. Ruden and G. H. Dohler, Phys. superlattices provided the expressions Rev., B27, 3538 (1983). for Zj are modified accordingly.

Acknowledgement - We are grateful fop 11. S. R. White, G. E. Marques and L. J. the help in computinq provided by R. J. Sham, J. Vac. Sci. Technol., ~_~I, 544 Bucolo. One of us (P.K.B.) acknowledges (1982). the partial support of a Rackham grant. 12. H. Kroemer and Q. G. Zhu, Vac. Sci. Technol., 21, 551 (1982).

References 13. L. Esaki, L. L. Chang and E. E. Mendez, Japdn. J. Appl. Phys., 20, I. L. Esaki and R. Tsu, IBM Journal of L529 (1981). Research and Development, 14, 61 (1970). 14. D. F. Welch, G. W. Wicks and L. F. Eastman, J. Appl. Phys., 56, 3176 2. D. Mukherji and B. R. Nag, Phys. (1984). Rev., B12, 4338 (1975). 15. M. J. D. Powell, Computer Journal, 3. R. Dingle, in Festkorperprobleme XV ~, 303 (1965). (Advances in Solid State Physics), Edited by H. J. Queisser (Pergamon 16. G. F. Craven and C. K. Mok, IEEE Vieweg, 1975) p. 21. Trans. MTT, 19, 295 (1971).