The energy spectra and wavefunctions of MSQW

Superlattices and Microstructures 40 (2006) 29–37 www.elsevier.com/locate/superlattices

The energy spectra and wavefunctions of MSQW Zhi-Yong Zhang Department of Physics, Science School, Beijing University of Aeronautics and Astronautics, Beijing 100083, People’s Republic of China Received 7 May 2005; received in revised form 20 January 2006; accepted 26 January 2006 Available online 13 March 2006

Abstract On the basis of the quasistationary state approximation, the asymptotic transfer method (ATM) was extended to calculate the energy spectra, envelope functions of carriers and dispersion relations of Ga1−x Alx As multiple sawtooth quantum wells (MSQW) in the presence of an electric field applied along the growth direction and a magnetic field parallel to the MSQW interfaces. It turned out that the energy spectra revealed the Landau-like behavior of carriers under a magnetic field. The envelope functions of carriers displayed obvious fluctuation and the dispersion relations showed a non-parabolic shape. c 2006 Elsevier Ltd. All rights reserved.  Keywords: Multiple sawtooth quantum wells; Asymptotic transfer method; Crossed electric and magnetic fields; Quasistationary state approximation

1. Introduction During the last few years, heterostructures and superlattices have been studied intensively due to their novel physical properties and potential device applications. For example, with the use of the transient photovoltaic effect that occurs in sawtooth superlattices (multilayer gradedgap structures), a new type of photodetector has been proposed and studied [1–5]. An accurate description of the energy spectra and wavefunction of sawtooth quantum wells and superlattices is thus essential to the understanding of the physics of the new type of structure, particularly in the presence of external fields. Although theoretical calculations of the energy spectrum and wavefunction of semiconductor quantum wells and superlattices under crossed electric and magnetic fields have obtained some

E-mail address: zyz [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter  doi:10.1016/j.spmi.2006.01.003

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Fig. 1. Sketch of the structure of the band edges of Ga1−x Alx As multiple sawtooth quantum wells consisting of ten SQWs.

results [6,7], so far the theoretical investigation of multiple sawtooth quantum wells (MSQW) under crossed electric and magnetic fields has not yet been reported. In the case of MSQW, because of the non-periodicity of the system and a large number of wells, as well as the non-flat shape of the band edges, it is very difficult to determine the energy spectrum of the system by the theoretical method usually used, especially under the effect of external fields. It is pointed out that, in terms of the succession of a great number of transfer matrices, subbands and wavefunctions can be solved consistently for semiconductor multilayer systems in which the band edges of a conduction band and a valence band are not flat (oblique lines or curves) [8–10]. Obviously, by means of this method, it is possible to take two factors into account at the same time. One factor is the declination of the edges of both a conduction band and a valence band in the non-periodicity of systems, and the other factor is the effect of external fields. In this paper, we calculate the energy spectra and envelope functions as well as the dispersion relations of carriers in MSQW under crossed electric and magnetic fields by using the asymptotic transfer method (ATM). In Section 2, we simply review the ATM and generalize it to suit MSQW systems. In Section 3, the results of numerical calculations for the energy spectra and envelope functions of the electrons and heavy holes as well as their dispersion relations are given, followed by a brief discussion. The asymptotic transfer method (ATM) and approaches used in this paper are of universality and can be extended to deal with the microstructural properties of multiple quantum wells and heterostructural systems in which band edges have any complex shapes. 2. Method and theory We divide the width of the whole system into j sublayers. If j is large enough, the width of each sublayer is very small. Within each sublayer, the lth sublayer for instance, the energies Vsl (z) and V pl (z) of the edges of a conduction band and a valence band can be considered as constant: Vsl and V pl . So the band edges of the whole system could be expressed approximately by a series of flat steps. The envelopes of these flights of stairs are simply the real shapes of the band edges. Now it is easy to see that, after this type of division, systems in which the band edges are not flat are approximately equivalent to multi-sublayer systems. Therefore, in terms of the successive product of a series of transfer matrices, the energy spectra and wavefunctions of the whole system can be determined. We call this type of method the asymptotic transfer method (ATM) [8]. As an illustration of using ATM, we consider an MSQW system that consists of ten Ga1−x Alx As sawtooth quantum wells terminated by barriers that have finite height but infinite width at both ends. The structure of the band edges of the system is shown in Fig. 1. Within

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each well, the concentration of Al is proportional to the coordinate z, resulting in a sawtooth shape of both a conduction band and a valence band. At the end of each well, the energies of the conduction band and the valence band have the extreme values. We choose the z-axis to lie along the growth axis of the MSQW. A constant electric field F is applied along the z direction. A constant magnetic field B applied along −y direction is described by the vector potential A = (−Bz, 0, 0). In our calculations we use the l − k effective mass theory to describe the electrons and holes in the MSQW. According to ATM, the potentials are approximated by a piecewise-constant potential. Within each sublayer, the lth sublayer for instance, the Hamiltonian is given by   2 1 ˆx ∓ e Bz l + Py2 + Pz2 + Vs,l p ± eFz l P (1) Hˆ e,h = 2m ∗e,h (z l ) c where m ∗e,h (z l ) is the effective mass of electrons or holes and Vs,l p are the energy values of the band edges of the conduction band or the valence band in the lth sublayer, respectively. Because the Hamiltionian is independent of x and y components, the intrinsic states of the Hamiltonian in the lth-sublayer can be written as Ψle,h (z) = exp[i(K x x + K y y)]Φle,h (z).

(2)

By substitution of the states (2) into the Schr¨odinger equation Hˆ Ψ = EΨ , for electrons we can obtain   2m ∗e (z l ) m ∗e (z l )ωc2 (z l ) e l 2 ¨ Φl (z) + (z l − z 0 ) Φle (z) = 0, ε(z l ) − Vs − eFz l − 2 h¯ 2 (z l−1 ≤ z ≤ z l ). (3) In (3), Φle (z) is the envelope wavefunction of electrons in the lth-sublayer. Here, ε(z l ) = E −

h¯ 2 K y2 , 2m ∗e (z l )

ωc (z l ) =

eB , m ∗e (z l )c

z0 =

h¯ cK x . eB

Let Sl =

V (z l ) + V (z l−1 ) 2

(4)

where V (z l ) = Vsl + eFz l +

h¯ 2 K y2 m ∗e (z l )ωc2 (z l ) (z l − z 0 )2 + . 2 2m ∗e (z l )

Eq. (3) is given by 2m ∗e (z l ) Φ¨ le (z) + [E − Sl ]Φle (z) = 0. h¯ 2 The general solution for Eq. (5) is given by Φle (z) = Al exp(iK l z) + Bl exp(−iK l z),

(5)

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where

 2m ∗ (z )(E − Sl )/h¯ , K l =  e∗ l i 2m e (z l )(Sl − E)/h¯ ,

E > Sl E < Sl .

In terms of the two continuity conditions of wavefunctions as well as their derivations in the interface of two sublayers, the amplitudes of the wavefunctions of the lth sublayer can be connected to those of the (l + 1)th sublayer via transfer matrix Tl :   Al+1 Al = Tl . (6) Bl+1 Bl The transfer matrix Tl is [11]  1 (1 + θl ) exp[−i(K l+1 − K l )z l Tl = 2 (1 − θl ) exp[i(K l+1 + K l )z l

(1 − θl ) exp[−i(K l+1 + K l )z l (1 + θl ) exp[i(K l+1 − K l )z l

 (7)

where θl = K l m ∗e (z l+1 )/K l+1 m ∗e (z l ). By means of the successive product of ( j + 1) transfer matrices, we can obtain 

 A0 A j +1 =M , B j +1 B0

where M =

j

Tl .

(8)

l=0

In (8), A0 , B0 and A j +1, B j +1 are the amplitudes of wavefunctions in barriers that are at both ends with finite height and infinite width. In the presence of an electric field, the barriers also decline. But if the electric field is not very strong, it is impossible that the carriers will pass through the barriers by tunneling and the lifetime of the carriers will be very long. So, the carriers can be approximately regarded as infinite in width and constant in height, and this is named the quasistationary state approximation. Because the wavefunctions must be bounded, the amplitudes of the exponentially increased waves in both infinite barriers must be zero, i.e., A0 = 0, B j +1 = 0. Then we obtain     A j +1 M11 M12 0 0 = and A j +1 = M12 B0, M22 B0 = 0. (9) =M 0 M21 M22 B0 B0 The condition that there are non-trivial solutions is M22 = 0. The energy spectra of electrons in MSQW under crossed electric and magnetic fields can be determined by Eq. (9). For each definite energy level, the corresponding wavefunction can be obtained by means of Eq. (6). For heavy holes, there are equations analogous to those for electrons. According to the formula given in Ref. [12], εg (x) = 1519.2 + 1247x (meV), where εg (x) is the forbidden gap of Ga1−x Alx As and x is the concentration of Al in Ga1−x Alx As. We take the Kane matrix element of GaAs to be Π = (h¯ /m 0 )iS| Pˆz |z = 1.076 × 10−4 (meV cm) [8]. The band offsets used are 60% and 40% of the band gap difference between the conduction band and the valence band, respectively. If we take x = 0.2, the extreme energy values of the conduction band and the valence band in each sawtooth well are Vsmax = 249.4 × 60% = 149.64 (meV) V pmin

= −249.4 × 40% = −99.76 (meV).

(10) (11)

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Fig. 2. The variations in electron energies (a) and heavy hole energies (b) versus the electric field strength for the lowest three energy levels under the condition of K x = 0, K y = 0 and B = 0.

˚ Jd = 50, m = 10, j = Jd m = 500. Where D is the The parameters taken are D = 50 A, width of each sawtooth well and is divided into Jd = 50 sublayers. m and j are the numbers of wells and the whole sublayers, respectively. The band edges of both the conduction band and the valence band in the lth sublayer are Vsl (z) = Vsl = 149.61 × [(l − 1)/Jd ]

(12)

V pl (z)

(13)

=

V pl

= −99.76 × [(l − 1)/Jd ].

It is noticed that we have taken the variations of both the conduction band and the valence band into account. This is approximately equivalent to taking account of the variations in the effective mass of electrons and heavy holes. In the lth sublayer, we take the effective mass approximation to be [8] 1 2Π 2 = (εg + ε − V pl )−1 and 2m ∗e (z l ) 3 m ∗h (z l ) = [m ∗h (GaAs) + 0.31x(l − 1)/Jd ]m 0 = [0.45 + 0.062(l − 1)/Jd ]m 0

(14)

where we take m ∗h (GaAs) = 0.45m 0. m ∗e (z l ) and m ∗h (z l ) are the effective masses of electrons and heavy holes in the lth-sublayer, respectively. m 0 is the mass of free electrons. 3. Results and discussions We calculate the energy spectra of the electrons and heavy holes in ten sawtooth quantum wells with the absence and presence of an electric field applied along the z or −z directions, respectively. Let K x = 0, K y = 0, B = 0; the energy variations of the electrons and heavy holes versus electric field strength are shown in Fig. 2 for the lowest three energy levels. Fig. 2(a) is the energy of the electrons and (b) is the energy of the heavy holes. The zero point of energy is situated at the edges of the conduction band and the valence band of the infinite barrier on the left-hand side, respectively. In our calculations, the energy of the heavy holes is taken to be a positive value.

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Fig. 3. The variations in electron energies (a) and heavy hole energies (b) versus magnetic field intensity for the lowest three energy levels under the condition of K x = 0, K y = 0 and F = 10 kV/cm. The band profiles and squared carrier wave functions of MSQW under an electric field of 10 kV/cm and a magnetic field of 10 T. The valence band profile and squared ground state heavy hole wavefunction of MSQW under the above conditions (c). The conduction band profile and squared ground state electron wavefunction of MSQW under the above condition (d).

From Fig. 2, it can be seen that the presence of the electric field introduces a Stark shift, i.e., the energy levels have a shift. The energies of the electrons and heavy holes for the ground state and the excited state increase and decrease, respectively, with an increase in the electric field. The energies present a stronger dependence on the electric field, which shows that the Stark shift is remarkable in the absence of a magnetic field. Under the condition B = 0, the energy interval for different energy levels is very compact and the energy differences between adjacent energy levels tend to be equally separate. In particular, the first three bound states’ energies achieve constant values under negative bias (at −30 kV/cm), as seen in Fig. 2(a) and (b). This result shows that in well regions, potential energy (conduction and valence bands) that decreases (or increases) with this electric field at each zone edge of the periodic structure is about 300 meV. This means that this amount of energy is very near the band discontinuity between two adjacent wells. Let K x = 0, K y = 0, F = 10 kV/cm; the energy variations of the electrons and heavy holes versus a magnetic field under the condition of a fixed electric field are given in Fig. 3(a) and (b) for the lowest three energy levels. It turns out that the energy spectra show the Landaulike behavior of carriers under a magnetic field. Our calculations show that, as magnetic field increases, the energies of magnetic energy levels increase and decrease for the electrons and

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Fig. 4. A plot of the squares of moduli of the heavy hole wavefunctions (a) and the electron wavefunctions (b). The zero point of z/D is chosen to be situated at the left interface of the first well on the left-hand side of a ten-well system under the condition B = 0. (a) F = −30 kV/cm (full curve) for the lth excitation state and F = 30 kV/cm (broken curve) for the sixth excitation state; (b) F = −20 kV/cm (full curve) and F = −30 kV/cm (broken curve) for the third excitation state.

heavy holes, respectively. This is in agreement with the results calculated from a single quantum well by Brum et al. [7]. But the zero point of energy is different from that in Ref. [7]. For electrons, the energy difference between two adjacent energy levels tends to be widened by an increase in magnetic field. For heavy holes, the situation is just the opposite of that for electrons. The band profiles and squared carrier wave functions of MSQW under an electric field of 10 kV/cm and a magnetic field of 10 T are shown in Fig. 3(c) and (d). The maximum of squared wave functions increases and decreases with an increase of z/D for electrons and heavy holes under the above-mentioned conditions. Then, we calculate the wavefunctions that correspond to the energy levels determined by the above calculations, so as to obtain the probability of occupation at the z-position for electrons and heavy holes in MSQW. A plot of the squares of moduli of the heavy hole wavefunctions that correspond to the first and the sixth excited energy levels is shown in Fig. 4(a). The full curve is used to plot the first excitation state for F = −30 kV/cm and the broken curve is used to plot the sixth excitation state for F = 30 kV/cm under the condition B = 0, respectively. A plot of the squares of moduli of the electron wavefunctions that correspond to the third excitation state is shown in Fig. 4(b) under the condition B = 0 for F = −20 kV/cm (full curve) and F = −30 kV/cm (broken curve). From our previous work [8–10], it was known that an electric field leads to a localization effect of wave functions for carriers. Our results show that the wavefunctions of the electrons and heavy holes display obvious fluctuation. We can see that the fluctuation of the probability of occupation at the z-position of the electrons and heavy holes

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Fig. 5. The variations in electron energies (full curve) and heavy hole energies (broken curve) versus the wave vector K x for the lowest three energy levels under the condition K y = 0 and B = 10 T. (a) F = 0 for the electrons and heavy holes; (b) F = 30 kV/cm for the electrons and F = −30 kV/cm for the heavy holes.

strengthens with increasing energy. For the electrons the fluctuation increases under the effect of the electric field applied along the −z direction and then reduces the conduction band edge and the maximum of the squares of moduli increases with an increase of z/D, whereas for the heavy holes the fluctuation increases under the effect of the electric field applied along the z direction. This reduces the valence band edge and the maximum of the squares of moduli decreases with an increase of z/D. Finally, let F = 0, F = 30 kV/cm and F = −30 kV/cm, the E–K x curves, i.e., the dispersion relations are given in Fig. 5(a) and (b). The full curves are the dispersion relations of the electrons and the broken curves are those of the heavy holes. Our results show that the motion of the carriers in the z direction is closely related to that in the x–y plane, so that the dispersion relations for the energy of a ground state and a lower excitation state have non-parabolic shape for both the electrons and the heavy holes. In the absence of an electric field, because of the effect of a magnetic field, the motion of carriers is confined to cyclotron orbits in a plane (z–x plane) perpendicular to the magnetic field. The variation in energy is remarkable, as indicated in Fig. 5(a). In the presence of a strong bias (at ±30 kV/cm), the effect of the electric potential and the sawtooth potential is predominant. The carriers cannot complete the entire cyclotron orbits, and the energy variation in the carriers tends to slow down with the variation in K x , as shown in Fig. 5(a). In Fig. 5(a), we take K y = 0, B = 10 T, F = 0 for both the electrons and the heavy holes. In Fig. 5(b), we take K y = 0, B = 10 T, F = 30 kV/cm for the electrons and F = −30 kV/cm for the heavy holes. The energy value labeled at the longitudinal coordinate on the left-hand side of the figure is the value of the electron levels, and that labeled on the righthand side of the figure is the value of the heavy hole levels. Particularly, if B = 0, according 2 to Eq. (4), V (Z l ) = Vsl + eF Z l + 2mh∗¯ (Z ) (K x2 + K y2 ), the motion of the carriers in each x–y l l plane is free and independent of that in the z direction. Also, if B = 0, K x = K y = 0, then V (Z l ) = Vsl + eF Z l , and the motion of carriers is only confined to the z direction under the effect of the electric field potential and the sawtooth potential. 4. Conclusions We have presented the calculation results of the energy spectra, the wavefunctions and the dispersion relations for multiple sawtooth quantum wells in the presence of crossed electric

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and magnetic fields. It turns out that the energy spectra showed the Landau-like behavior of carriers under a magnetic field. The wavefunctions of carriers display obvious fluctuation and the dispersion relations show a non-parabolic shape. Some of the features were described under the presence of the external fields, including the coupling between the electric and magnetic effects and the interaction of the motion in both the z direction and the x–y plane. Acknowledgements The author would like to thank Professor Guoyi Qin for his help. This research is partly supported by the National Science Foundation of China. References [1] J.A. Brum, P. Voisin, G. Bastard, Transient photovoltaic effect in semiconductor superlattices, Phys. Rev. B 33 (1986) 1063–1066. [2] C. Sitori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, U. Ostarle, GaAs/Alx Ga1−x As quantum cascade lasers, Appl. Phys. Lett. 73 (1998) 3486–3488. [3] K. Ohtani, H. Ohno, Mid-infrared intersubband electroluminescence in InAs/GaSb/AsSb type-II cascade structures, Physica E 7 (2001) 80–83. [4] E. Ozturk, Y. Erg¨un, H. Sari et al., The self-consistent calculation of Si delta-doped GaAs structures, Appl. Phys. A. 73 (2001) 749–754. [5] Y. Erg¨un, M. Hostut, S.U. Eker, I. S¨okmen, Intersubband electron transition across a staircase potential containing quantum wells:light emission, Superlatt. Microstruct. 37 (2005) 163–170. [6] W.J. Fan, J.B. Xia, Subband structure of GaAs/AlGaAs superlattices under crossed electric and magnetic fields, Acta Phys. Sin. 9 (1990) 1465–1472. [7] J.A. Brum, L.L. Chang, L. Esaki, Band structure of quantum wells under crossed electric and magnetic fields, Phys. Rev. B 38 (1988) 12977–12982. [8] G.Y. Qin, The asymptotic transfer method in the envelope function approximation, J. Phys.: Condens. Matter. 1 (1989) 7335–7345. [9] G.Y. Qin, Electric-field-induced localisation of type-II coupled quantum wells, J. Phys.: Condens. Matter. 2 (1990) 5723–5728. [10] G.Y. Qin, Z.Y. Zhang, Calculations on subbands and optical transitions of semiconductor multiple quantum wells in electric field, Chin J. Semicond. 11 (1991) 649–656. [11] L.A. Cury, A. Celeste, J.C. Portal, Calculation of the diamagnetic shift in resonant-tunneling double-barrier GaAs–Alx Ga1−x As heterostructures, Phys. Rev. B 38 (1988) 13482–13485. [12] M.H. Meynadier, C. Delalande, G. Bastard, M. Voos, Size quantization and band-offset determination in GaAsGaAlAs separate confinement heterostructures, Phys. Rev. B 31 (1985) 5539–5542.