Interband resonant tunnelling in quantizing magnetic field

SSC 5071

PERGAMON

Solid State Communications 113 (2000) 599–602 www.elsevier.com/locate/ssc

Interband resonant tunnelling in quantizing magnetic field A. Zakharova Institute of Physics and Technology of the Russian Academy of Sciences, Nakhimovskii Avenue 34, Moscow 117218, Russian Federation Received 4 October 1999; accepted 10 November 1999 by E.L. Ivchenko

Abstract We present an investigation of the interband resonant tunnelling in semiconductor heterostructures in a quantizing magnetic field normal to interfaces taking into account the mixing of electron, light and heavy hole states. The probabilities of the tunnelling transitions between the states of different Landau levels are obtained for a InAs/AlSb/GaSb resonant tunnelling structure (RTS). It is shown that coupling to the heavy hole states results in the additional peaks of the tunnelling probability and the magnitudes of these peaks for the processes with changing the Landau-level index can be of the order of those for the processes with conservation of the Landau-level index. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; D. Tunnelling

Since first experimental realizations of the polytype double barrier structures made from InAs, AlSb, GaSb materials, the effect of interband resonant tunnelling has been intensively studied (see e.g. Refs. [1–15]). Both the InAs/AlSb/GaSb and GaSb/AlSb/InAs resonant tunnelling structures (RTS) were investigated. In Refs. [3,4,6], the results of experimental investigation of the interband resonant tunnelling in the presence of a strong magnetic field are presented. In the magnetic field normal to interfaces, the oscillations of the current versus an applied voltage caused by the interband resonant tunnelling through different Landau levels in the well were observed [3,4]. These oscillations were associated with tunnelling processes with the Landau-level index conservation. Recently, it was shown theoretically that due to the spin–orbit interaction the inter-Landau-level processes can occur without scattering, which results in the additional peaks of the tunnelling probability T and current density [14,15]. In Refs. [14,15], the coupling of electron (light hole) states to the heavy hole states was neglected. The aim of this paper is to investigate the interband resonant tunnelling in a quantizing magnetic field normal to interfaces taking into account the mixing of electron, light and heavy hole states with different Landaulevel indices. The transmission coefficients corresponding to the resonant tunnelling transitions between the states of different Landau levels in the InAs/AlSb/GaSb RTS, whose band diagram is shown in Fig. 1, are calculated using the six-band model and the transfer matrix method.

It is shown that the coupling to the heavy hole states results in one or two additional peaks on the dependencies of T on an incident electron energy E for the tunnelling through the states of each subband of a heavy hole size quantization in the well. The number of additional peaks depends on the Landau-level index and spin of the incident state. Note that the Landau-level mixing effects were also investigated in Ref. [16] under consideration of hole tunnelling processes in a way similar to that described in Ref. [17]. The calculations of the transmission coefficients through the interband RTS are carried out using the envelope function approach and the six-band model described in Ref. [18]. We utilize the same basis functions as in Ref. [15], but disregard the states of the split-off band. Then a 6 × 6 Hamiltonian can be written as ! H^ 12 H^ 22 ^ Hˆ ; …1† H^ 11 H^ 21 where 0

H^ ^7

EC …z† B p p B ˆ B 2 2iPk^z = 3 @ Pk^7

p p 2iPk^z = 3 EV …z† 1 G^ p 22 6ik^7 g2 k^z

1 Pk^^ C p C ; 2 6ik^z g2 k^^ C A EV …z† 1 F^ …2†

G^ ˆ 22k^7 …g1 2 g2 †k^^ 2 k^z …2g2 1 g1 †k^z ;

0038-1098/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00536-0

…3†

600

A. Zakharova / Solid State Communications 113 (2000) 599–602

at the interfaces

c2 ; c3 ; c5 ; c6 ;

…2g 02 1 g 01 †

p 2 c 1 2 6g 02 k^ 1 c3 ; 2z 2

…2g 02 1 g 01 †

p 2 c5 1 2 6g 02 k^ 2 c6 ; 2z

…2g 02 2 g 01 †

2 c; 2z 6

(7)

g 01 ˆ g1 1

P2 ; 6…EC …z† 2 E†

…8†

g 02 ˆ g2 1

P2 ; 3…EC …z† 2 E†

…9†

…2g 02 2 g 01 †

2 c ; 2z 3

where Fig. 1. Conduction and valence band diagram of the InAs/AlSb/ GaSb/ AlSb/InAs RTS.

F^ ˆ 22k^7 …g1 1 g2 †k^^ 1 k^z …2g2 2 g1 †k^z ;

…4†

and 0

H^ ^^

p 0 pk^^ = 3 B p B ˆ B pk^^ = 3 0 @ p 0 22 3k^^ g2 k^^

0

1

C p C : …5† 22 3k^^ g2 k^^ C A 0

jz ˆ Re…c1 j^z c†:

In Eqs. (2)–(5) EC …z†; EV …z† are the conduction and valence band edges, the value P is proportional to the interp band momentum matrix element, k^^ ˆ 7i…k^x ^ ik^y †= 2; k^x ˆ 2i2=2x; k^y ˆ 2i2=2y 1 ueuBx=…"c†; k^z ˆ 2i2=2z; e is an electron charge, and c is the light velocity. The magnetic field B is parallel to the z-axis normal to interfaces, and the vector potential is chosen in the following form: Ay ˆ Bx; Ax ˆ Az ˆ 0: The modified Luttinger parameters gi …z† for the convenience are replaced by 2m0 gi …z†="2 as in Ref. [13], where m0 is the free electron mass. Also we have neglected g-factor of the free electron as in Refs. [14,15], and have supposed that the Luttinger parameter g3 ˆ g2 : This Hamiltonian was derived similar to Ref. [19] supposing that the remote bands of G 1 and G 2 symmetry make the main contribution to the Luttinger parameters. We have assumed that the basis functions are the same for the whole structure and the interband momentum matrix element is a constant value. The envelope functions c j in RTS satisfy the equations X

H^ ij cj ˆ Eci ;

i ˆ 1; …; 6;

The transmission coefficient for each permitted transition from the state to the left-hand side of RTS into the state to the right-hand side of it can be obtained as the ratio of the probability current densities along z-axis averaged over coordinate x for the transmitted and incident waves kjtz l and kjiz l: The equation for jz derived as in Ref. [15] can be written in the following form:

…6†

where E is an eigenvalue. In equation system (Eq. (6)), the envelope functions c 1 and c 4 correspond to us1=2;^1=2 l basis states, respectively, functions c 2 and c 5 correspond to up3=2;^1=2 l basis states and functions c 3 and c 6 correspond to up3=2;^3=2 l basis states. To obtain the solution of Eq. (6) in a whole structure, we derive the boundary conditions by integrating Eq. (6) across an interface as in Ref. [18]. As a result, the following functions must be continuous

…10†

In Eq. (10) c is a column of envelope functions for a given incident or transmitted wave, and j^z is the probability current density operator which coincides with the velocity operator ! v^1 0 i ^ ^ ˆ 1 z v^z ˆ ‰Hz ; …11† 2 zHŠ " " 0 v^2 z where 0

0 B p p B ^ v^z ˆ B 2 2iP= 3 @ 0

p p 2iP= 3 22…2g2 1 g1 †k^z p 22 6ig2 k^7

0

p 2 6ig2 k^^ 2…2g2 2 g1 †k^z

1 C C C: A …12†

The solution for the multicomponent envelope function c in a heterostructure can be written in terms of harmonicoscillator functions fn …x 0 † [14,15,20] 1 0 f1 …z†fn …x 0 † C B C B B f2 …z†fn …x 0 † C C B C B B f3 …z†fn21 …x 0 † C C B cˆB …13† C × exp…iky y†; B f …z†f …x 0 † C C B 4 n11 C B C B B f5 …z†fn11 …x 0 † C A @ f6 …z†fn12 …x 0 † where x 0 ˆ x 2 x0 with x0 ˆ 2ky =s; s ˆ ueuB=…"c†: In each layer of the structure, the solution (13) for n $ 1 is a superposition of four incident and four reflected waves

A. Zakharova / Solid State Communications 113 (2000) 599–602

Fig. 2. Transmission coefficients for the InAs/AlSb/GaSb RTS with ˚ barriers and 65 A ˚ quantum well at zero voltage and B ˆ 15 T: 15 A Curves 1–3 correspond to the transitions between different states as described in the text.

Fig. 3. Transmission coefficients for the InAs/AlSb/GaSb RTS with ˚ barriers and 65 A ˚ quantum well at voltage V ˆ 0:06 V and 15 A B ˆ 15 T: Curves 1, 2, 3 and 4 correspond to the transitions between different states as described in the text.

601

corresponding to the mixed bulk electron (light hole) and heavy states with different Landau-level indices, normal to interfaces components of wave vector kz and spin orientations as described in Refs. [14,15]. Two of these four states have real values of kz in the conduction band for sufficiently large energies and are approximately the electron states with Landau-level indices n and n 1 1and values of spin sz < ^1=2; respectively. The other two states have imaginary values of kz in the conduction band and are the heavy hole states. If n ˆ 0; there are only three mixed bulk states. Two states are approximately the electron states in the conduction band with Landau-level indices 0 and 1 and sz < ^1=2: The residual state is approximately the heavy hole state with Landau-level index 2. For n ˆ 21; there are two mixed bulk states with Landau-level indices 0 and 1. There are approximately the electron and heavy hole states with sz < 21=2: If n ˆ 22; there is only one heavy hole state with the Landaulevel index 0 and sz ˆ 21=2; which is not mixed with the other states. In a heterostructure due to the boundary conditions the mixing of all states for a given number n occurs. (Note that for n , 1 the number of boundary conditions reduces, because some functions fm …x 0 † become equal to zero.) This mixing results in additional peaks of the interband tunnelling probability in RTS. The number of additional peaks, which may be observed, is equal to the number of the heavy hole quasibound states in the quantum well for a given number n. Hence it depends on Landaulevel index and spin orientation of the incident electron state. We obtain the solution for the multicomponent envelope function c in RTS using the transfer matrix technique (see, e.g. Refs. [14,15,20]). Then the probability current density components normal to interfaces for the transmitted and incident waves are calculated utilizing 10–12 and the transmission coefficients for the transitions from each incident state l into each transmitted state k Tkl for the InAs/AlSb/ GaSb RTS are obtained. All dependencies Tkl …E† are calculated at B ˆ 15 T: The transmission coefficients for the ˚ barriers and a 65 A ˚ GaSb InAs/AlSb/GaSb RTS with 15 A quantum well at zero voltage V versus an incident electron energy are shown in Fig. 2a and b. All curves correspond to the intra-Landau-level transitions between the identical states in the conduction band of InAs contact layers through the quasibound states of ground subbands of light and heavy hole size quantization in the well. The other interband resonant transitions are forbidden at V ˆ 0: Curve 1 corresponds to n ˆ 21 and describes the tunnelling of an electron from the state with Landau-level index 0 and sz < 21=2: Curves 2 and 3 correspond to n ˆ 0 and 1 and describe tunnelling from the states with Landau-level indices 0 and 1, respectively, and sz < 1=2: The peaks of T…E† on curve 1 at energies E ˆ 0:092 and E ˆ 0:11 eV are due to resonant transitions through the light and heavy hole states in the well with indices 0 and 1, respectively, and the spin opposite to the magnetic field direction. Two peaks of the tunnelling probability on curve 2 at E ˆ 0:027 eV and E ˆ 0:063 eV

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A. Zakharova / Solid State Communications 113 (2000) 599–602

are caused by the resonant tunnelling through the light hole states with Landau-level index 1 and spin opposite to the magnetic field direction and Landau-level index 0 and spin along the magnetic field, respectively. The third peak at E ˆ 0:01 eV is due to the tunnelling through the heavy hole state with Landau-level index 2 and spin opposite to the magnetic field direction. Finally, peaks of the tunnelling probability on curve 3 at E ˆ 0:096 eV and E ˆ 0:015 eV correspond to the resonant interband transitions of electrons through the heavy hole states in the well with Landau-level index 3 and spin opposite to the magnetic field direction, and Landaulevel index 0 and spin along the magnetic field, respectively. The investigation showed that the coupling of electron (light hole) states to the heavy hole states results in one additional peak on each curve for n ˆ 1 or n ˆ 0and two additional peaks for n . 0: This may lead to additional peaks of the current density. We also calculated the dependencies of the transmission coefficients on the incident electron energy at V ˆ 0:06 V (see Fig. 3a and b) to investigate both the processes with conservation and change of the Landau-level index. Curves 1–3 on Fig. 3 correspond to the same transitions with Landau-level index conservation as those in Fig. 2a and b. Curve 4 corresponds to spin-flip tunnelling processes from the states in the conduction band of the left InAs contact with spin sz < 1=2 and Landau-level index 0 into the states with Landau-level index 1. These processes occur from the same initial states as those described by curve 2 and become permitted due to lowering of the Landau level with index 1 and spin sz < 21=2 in the right contact with voltage increasing. Two peaks of tunnelling probability on curve 4 correspond to the transitions through the same quasibound states in the GaSb quantum well as those on curve 2. The probability of resonant transitions through the heavy hole quasibound states is considerable both for processes with conservation and changing the Landau-level index. Comparison of curves 2 and 4 showed that the probability of the inter-Landau-level transitions can be greater than that with the Landau-level index conservation. Hence, these processes can contribute considerably to the current density. In summary, we have investigated the interband resonant magnetotunnelling taking into account the mixing of electron, light and heavy hole states with different Landau-level indices. The transmission coefficients for the InAs/AlSb/

GaSb RTS have been calculated employing the transfer matrix method. It was shown that the probabilities of electron interband tunnelling transitions through the heavy hole states in the well both for processes with conservation and change of the Landau-level index are considerable that may influence essentially the current–voltage characteristics of RTS made from InAs, AlSb, GaSb in a strong magnetic field.

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