22 April 1996
PHYSICS
LETTERS
A
Physics Letters A 213 (1996) 191-196
‘Band nonparabolicity in quasi-periodic Fibonacci heterostructures M. Palomino-Ovando a Institute
de Fisica,
h Centro de Investigaciones
a*1,Gregorio H. Cocoletzi a, C. Perez-L6pez b*2
Universidad
Autdnoma
en Dispositivos Apartado
de Puebla, Apartado
Semiconductores,
Institute
Postal 1651, Puebia
Postal J-48, Pueblo
de Ciencias,
72000, Mexico
Universidad
Autdnoma
de Puebla,
72000, Mexico
Received 18 December 1995; accepted for publication 5 February 1996 Communicated by L.J. Sham
Abstract Nonparabolic dispersion relations of electrons in the reformulated Schriidinger equation and a 2 x 2 transfer matrix approach are applied to the study of quasi-periodic Fibonacci semiconductor heterostructures. Energy minibands and resonant tunneling amplitudes are investigated and compared with the parabolic case to show the importance of nonparabolic deviations in Al,Gal-,As/GaAs superlattices.
1. Introduction Novel artificial quasi-periodic Fibonacci heterostructures have attracted the attention of recent publications since the systems exhibit interesting unique electronic [ 1,2] and optical [ 31 features which differ
from those of the periodic [4] and the disordered systems [ 51, as well as from those of their individual components. Studies of the wave functions and the electronic energy levels in quasi-periodic superlattices have demonstrated that the electronic states [ 21 are neither extended nor localized and that the transmission coefficient of electrons exhibits a very rich self-similar structure. The first experimental realization of these quasi-periodic materials was achieved by Merlin et al. [6] recently. The unique properties ’ Permanent address: Facultad de Ciencias Ffsico MatemPicas, UAP, Apdo. Post. 1152, Puebla 72000, Mexico. * Permanent address: Escuela Preparatoria “B. Jukez”, UAP, Puebla, Mexico.
of these novel materials and their possible technological applications are reasons to further explore such systems. On the other hand, recent experiments of luminescence [7] in semiconductor quantum wells have shown the peaks to exhibit a dependence on the orientation of the substrate surface. The shifts of the peaks were explained [8] in terms of the nonparabolic band dispersion relation of electrons in each layer. Band nonparabolicity has been introduced in the effective mass Hamiltonian of electrons in semiconductor heterostructures very recently [ 81. The modified S&r&linger equation and the corrected boundary conditions have been used to study energy levels of particles in quantum wells [ 81 and miniband structures [ 71 of Kronig-Penney-type superlattices, to show the deviations from the parabolic dispersion relations. In this work we investigate quasi-periodic Fibonacci heterostructures and superlattices accounting for nonparabolic dispersion relations of electrons in
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192
M. PalvmLw-Ovundo
GaAs
et d/Physics
AlGaAs
Fig. I. Schematic representation of a Fibonacci superlattice of the fourth generation F, = 4, with the conduction band discontinuities indicated, yielding potential wells (GaAs) and barriers (AlGaAs) for the electrons. The coordinate system is also depicted.
each layer. To be precise, we explore the resonant tunneling phenomena and the energy minibands in artificial lattices of type-1 semiconductors. The Hamilton operator of the Schrtidinger equation is modified by the first nonparabolic term which is proportional to k4, as recently reported [ 81. A 2 x 2 transfer matrix approach is exploited to calculate the transmitt~ amplitudes of electrons in the Fibonacci heterostructures and the dispersion relation of the quasi-periodic semiconductor superlattices and the results are compared with the parabolic case. This report contains in Section 2 the outline of the theory. There we define the system and introduce the nonp~abolic effects in the effective mass H~ltonian. In Section 3, we discuss the results and reach conclusions.
Letters A 213 (1996) 191-196
(AlGaAs) and wells (GaAs) for electrons and holes. Here we solve the rn~ifi~ Schr~ing~r wave equation in each layer to construct the transfer matrix and calculate the properties. The recently reformulated [ 81 Schriidinger equation is based on the Kane [9] twoband model, which considers that the kinetic energy of the electrons in the conduction band may be expanded up to an arbitrary order in k2. However, the simplest nonp~a~lic correction to the dispersion relation is to include a k4 term and neglect higher order terms. In this work, the dispersion relation of electrons includes the first correction term, so the new time-independent Sc~~dinger equation may be written as (fli2a2V
- h4a4V4 + V)!P(r)
= W(r).
(1)
The -I- sign is chosen for the barrier and the - for the well [ 81, a2 = 1/2m, V is the potential energy and u4 is the positive nonparabolic coefficient. For this new differential equation the wave function is still continuous but its derivative is not, instead the following boundary condition is applied [ 81, u2&’ ( o- ) - fi%z4&” = U2~~/(0~)
(o- )
+ ~zu4~~~/~(0+),
(2)
where C.& &$band ffzW,~4~ are constant parameters for the barriers and wells, the symbols ’ and “’ stand for the first and third derivative and O- and O+ indicate that the functions are calculated on the left and right of the interface, respectively. Here we outline the transfer matrix approach for the one-dimensiona solution of Eq. ( 1) . The wave function has the form AeiRz, where qr has four solutions with only two of them having physical meaning [ 81. The solutions are
2. Fo~alism d = q&,,ax{1 - [’ + (V - E)/(&nax - W]“2}. The quasi-periodic Fibonacci heterostructures considered in this work are generated along the z-direction with their interfaces being parallel to the xy-plane of a right-handed 3D-coordinate system, Fig. 1. The generation is provided by two elementary seed blocks, L. and S, where each block is constructed by two semiconductor layers of GaAs and AlGaAs, having thicknesses d*L (dAs) and ds (dB) in block L (S), respectively. The conduction and valence band structures display discontinuiti~ as the particles move from one layer to another, giving rise to potential barriers
(3) In the potential well V = 0, the maximum of the wave vector is qz,max= ~2,,,/fi*Z~4,,, and the maximum of the energy is E= u~~/~u~~. For the barrier V = Vo with q&m = u~b/~*4u4b and Em = &,%av#, + 6. Solutions for values of qz larger than qz,mm are nonphysical and shall be ignored [ 81. To obtain the transfer matrix, let us consider either a barrier or a well and write the wave function as a su~~sition of traveling electronic waves,
M. Palomino-Ovando et al./Physics $j(z)
=
A+ei4:~” + A-e-iY:jz,
(4)
and define the function 4j (2) = a*jlG;!(z) + Uhj@j”( z ) . After Ref. [ 111, one may write the matrix equation
Letters A 213 (1996) 191-196
193
Now we calculate the dispersion relation of superlattices with periods d of different Fibonacci unit cells. If periodicity is present, we may invoke Bloch’s theorem [ IO] to write [ 111
(8)
Eq. (5) expresses a relationship between $ and Cpat the right z r and left z’ boundaries of the well (or barrier) with Mj = AjT(dj)A,” being the transfer matrix and T( z ) = diag( eiqzz,eeiez ) a diagonal matrix constructor. The matrix elements of M are
M,j =
cos (qzjdj)
1
Sin(
(6) where aj = a2j - a4jri2qzj, with j = b, w. Using the continuity of Ic/ and Cp,one can construct the transfer matrix of an arbitrary number of alternating potential barriers and weils as a simple matrix multiplication. The result may be expressed as M = M,M,_t ’ ’ -Mt. The indices n and 1 label the nth and first layers in the structure. We now turn to define the Fibonacci sequence as &=[Ls],
....
SN=SN-*SN-2.
Notice that each block contains the same semiconductor layers A and B correspondingly, the potential well and barrier (see Fig. 1), with the only difference being the thickness of layer A. The generation of an arbitrary qu~i-peri~ic ~angement is provided by the generating Fibonacci recurrence formula, Fo = 1, Fi = 1, Fz = 2, . . . , FN = FN_~ + FN-2. If we define the matrices MAL and MEL representing layers A and B of block L, and similarly for block S, we may write for St, Mt = ML = MSLMAL and for $2, M2 = M~ML. Here it is convenient to adopt the ratio of the layer thicknesses to be the inverse of the golden mean A = (ds + dAL)/(dB + d,ts) = (6 - 1)/2. By invoking the recurrence formula it is easy to see that the transfer matrix of a Fibonacci heterostructure may be written as MN = M,v_zM~-t.
[M-leipd]
qzjdj) 42.F.j 17 1
-
Cos(qzjdj)
SI = ILI,
where p is the one-dimensional (possibly complex) Bloch’s wave vector. If we denote the transfer matrix of a period by M and use Bloch’s theorem, we may write
(7)
[~]
=O.
(9)
2
As usual, from this equation one obtains cos(pd) = tr( M) /2. Here tr( M) is the trace of M. Next we calculate the tunneling amplitude. We consider that the left boundary of the heterost~cture is at z = 0. Electrons encountering this boundary have finite probability of reflection and transmission, The wave function in terms of the incident and reflected amplitudes is given by Eq. (4). Using the wave function+(z) in&z) ~ddefiningZ(0) =~(O)/#(O), we can obtain $
= [Z(O)ik,cr,
- l]/[Z(O)ik,a,
+ I].
(10)
Then the probability of reflection is R = ]A-/A’l*. Finally, by the conservation of probability current, the tunneling amplitude can be written as ITI = 1 - 1RI*. For the layered system under consideration we easily find Z(0) = fM{,Z(n)
+ &l/&Z(n)
+ M;*, (11)
with M’ = M-t = AT(-d and Z(n) = l/ ikza,. In these equations, k, = qz and qz is given in Eq. (3) with V = 0.
3. Results and discussion The formalism presented in the previous section is applied to study the miniband structure and tunneling amplitudes in the quasi-periodic Fibonacci heteros~ctures. For the numerical calculations we consider layers of AlGaAs and GaAs. The masses are
M. Palomino-Ouundo er al./Physics
ok 0
x‘Re(pd)”
’
n
Letters A 213 (1996) 191-196
1
Fig. 2. Dispersion relation for the first nonparabolic minibands (solid lines), for three Fibonacci superlattices and comparisons with the parabolic case (dotted lines). ds = 10 A, d&V= 16.18 A, and L~AL = 32.36 A.
1
2
3
4
5
6
ORDER OF THE FIBONACCI SEQUENCE Fig. 3. The allowed energies as function of the Fibonacci sequence. da = 30 A, dAS = 48.5 A, and dAL = 97 A.
nz; = 0.092 m and rnc = 0.067 m in the barrier and
in the well, respectively. The potential barrier height is V = 150 meV. The nonparabolic dispersion relation for the first minibands is presented in Fig. 2 for three Fibonacci superlattices and compared with the parabolic cases. As FM is increased, the superlattice period is effectively enlarged, and consequently, the energy miniband width and forbidden minigap are drastically affected. The former is reduced while the latter is augmented, both by several tens of meV. It can be noticed in the figure that the differences between nonand parabolic results are only prominent for high-order minibands, that is, band nonparabolicity becomes nonnegligible as the energy increases. To complement the discussion on the miniband structures, we should mention that the miniband width decreases by several hundreds of meV and eventually becomes an energy level of an isolated well as the period is increased as a function of the barrier width. An interesting feature of the nonparabolic energy miniband structure for the Fibonacci superlattices is depicted in Fig. 3 for several FM numbers. The layer thicknesses are ds = 10, dAS = 16.18 and dAL = 32.34 A. The notorious effect appearing in this figure is how the minibands split into smaller minibands as FN increases, with the number of minibands equaling the Fibonacci numbers. We should mention that this feature shown by the first nonp~a~lic miniband is also exhibited by the parabolic results and the deviations of nonparabolicity are present for high order
0
250
500
ENERGY(meV) Fig. 4. Transmission coefficient, in logarithmic scale, versus the energy of the incident electron for the second, third and fourth Fibonacci generations. dB = 50 A,dAs = 80.90 A, and dAr. = 191.80 A.
Fibonacci numbers. We now turn to the discussion of the nonparabolicity effects on the electronic tunneling phenomena. To show how the resonant phenomenon is affected by the variations of the effective heterostructure thickness, in Fig. 4 the electron ~ans~ssion ~plitude is depicted for three Fibonacci arrangements as functions of the energy of the incident electron. when FN = 2, a symmetric structure is obtained, with two barriers separated by a single well, the chosen parameters allow only two resonant peaks. The increase of the number of barriers and wells is a function of FM and consequently, the number of resonant peaks is also in-
M. Palomirw-Ovando et al./ Physics Letters A 213 (1996) 191-196
195
/,,/53.0 N=
2=0.45
is
I -20 0
I
250
500
ENERGY(meV)
I
Fig. 6. Transmission coefficient, in logarithmic scale, versus the for different Al concentrations X. FN = 3, dg = 20 A, dAs = 32.36 A, and dAL = 64.72 A.
100
energy of the incident electron
84 ENERGY(meV) Fig. 5. Tunneling electron
for
(/AL = 64.72
amplitude
different
mention that, as the number of periods is increased in a finite superlattice of a Fibonacci sequence, the resonant tunneling amplitude tends to reflect the energy miniband structure, with the corresponding forbidden
‘lo
versus the energy of the incident
FAJ. do
= 20
A,&
A to exhibit the self-similarity
= 32.36
A,
and
feature.
creased. A remarkable feature appearing in these systems is how the peaks do not split with the increase of the number of wells as in the periodic case. This fact is clearly depicted, for example, by the curve with FN = 3 for the low energy peaks. The effect is understood if one realizes that the potential well width is different for each seed block, giving origin for different energy levels which may or not interact with levels of the adjacent wells depending upon their proximity. Additionally, an interesting result of the Fibonacci arrangements is the appearance of the self-similarity spectra in the tunneling phenomena, as shown in Fig. 5 for FN = 12, 15, 18. The same behavior is exhibited by the parabolic theory calculations. As the Fibonacci sequence is increased, the resonant peaks become sharper, consequently the tunneling time increases, as corroborated by a simple uncertainty principle calculation. One should remark that the difference on the peaks’ height is a consequence of the loss of periodicity and that the same features are present in the parabolic theory. Moreover, it is important to
gaps. Another interesting effect to explore is how the change of the Al concentration x affects the tunneling phenomena. As x increases, the semiconductor band gap is effectively enlarged, leading to larger valence and conduction band discontinuities, and consequently, the potential barrier height for both electron and hole is augmented. For this situation, the isolated potential wells favor the confinement of electrons, and the resonant tunneling amplitudes become sharper and shift in energies. The effect is illustrated by the results for the transmission coefficients in Fig. 6, for a structure where FN = 3 with the mass m = (0.067 + 0.083~)~ and the band discontinuity [ 141 E(Al,Gal_,As) - E(GaAs) = 0.70x eV for 0 6 x < 0.45. To end with the results, a comparison between the non- and parabolic cases is presented in Fig. 7. For the figure, FN = 5 with only one period considered, the low-energy peaks have different size while the high-order nonparabolic resonant tunneling amplitudes suffer low energy shifts. This latter effect agrees with the reported results of the electron energy in quantum wells [ 81. Notice that the deviations from the parabolic results are important and consequently they should not be ignored. We have presented studies of quasi-periodic Fi-
M. Palomino-Ouando
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et al./Physics
Leirers A 213 (1996) 191-196
Acknowledgement MPO and CPL acknowledge the financial support of CONACyT-Mexico. The work of GHC was partially supported by CONACyT-Mexico under contract # 48 1 lOO-5-5264E. ‘j
P;IONPARABOL*C References
‘PARABOLIC FN= 5
1
III M. Kohmoto, L P Kadanoff and C. Tang, Phys. Rev. Lett. 500
ENERGY(meV) Fig. 7. Transmission coefficient for parabolic (dotted line) and nonparabolic (solid line) cases. F,v = 5, dn = 20 A, d4s = 32.36 A, and t/AL= 64.72 A.
bonacci heterostructures taking into account nonparabolic dispersion relations of electrons in each semiconducting layer. The reformulated Schriidinger equation, the modified boundary conditions and the transfer matrix approach have been applied to calculate the energy minibands and the resonant tunneling amplitudes. The minibands and the transmission coefficients show quantitative differences as compared with the parabolic results. As discussed in the text, the variation of parameters like the aluminium concentration allows for the choice of the potential barrier height. This fact favors, for instance, the electron confinement in isolated potential wells and the sharpness of the resonant amplitudes of the transmission coefficients in finite superlattices. Additionally, the layer thickness may be selected to provide a well-defined and desired energy miniband structure that allow technological applications. Concluding, quasiperiodic Fibonacci heterostructures are interesting artificial materials with a great possibility of applications in the field of microelectronics. Moreover, nonparabolicity effects should be included in the quantum theory of transport properties in superlattices, especially for structures with thin layers.
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