Lateral confinement of carriers in ultrathin semiconductor quantum wells

N. Shtinkov*, P. Desjardins, R.A. Masut

Departement de Genie Physique and Groupe de Recherche en Physique et Technologie des Couches Minces (GCM), Ecole Polytechnique de Montreal,

C.P. 6079, Succursale “Centre-Ville”, Montreal, Que., Canada H3C 3A7

Abstract

We investigate theoretically the effects of one-dimensional lateral confinement on the electronic states in strained ultrathin quantum wells

(QWs). We studied two monolayer (ML) thick InAs/InP (001) QWs containing 3 ML thick one-dimensional (wire-like) InAs islands from 2

to 45 ML wide, oriented along the k010l and k110l directions. Localized (wire) and extended (QW) states are found in the energy spectra of

the structures. The wire localization leads to a band-gap reduction and to an increase of the heavy hole–light hole splitting. Coupling between

extended and localized states is observed at certain wire widths, indicating that these structures cannot be regarded simply as consisting of

independent regions with different QW widths.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Electronic structure; Lateral confinement; Quantum well

1. Introduction

The morphology of heterostructure interfaces has a large

impact on the electronic and optical properties of the

structure and is of crucial importance for its suitability for

device applications. The vast majority of the numerous

studies on the electronic structure of semiconductor

quantum wells (QWs), however, assume perfectly abrupt

interfaces with only a small number being devoted to the

investigation of structures with realistic interfaces. Most of

the attention has been focused on theoretical investigations

of the effects of interface roughness, segregation, and

interdiffusion [1]; a few works have also been devoted to

step-like interfaces [2]. To the best of our knowledge,

localized states due to lateral carrier confinement from

monolayer (ML) steps at the interfaces have not been

theoretically studied until now, although experimental

observations have emphasized the importance of this effect

in heterostructures [3]. The lack of studies on this problem is

partly due to its numerical complexity, since the presence of

interface steps reduces the dimensionality of the structure

and greatly increases the required computational effort.

In the present work, we study theoretically the one-

dimensional (1D) lateral confinement of carriers in ultrathin

InAs/InP (001) QWs due to ML steps at the QW interfaces.

The electronic properties of these structures are very

sensitive to ML width fluctuations, making them a suitable

test structures for studying carrier localization [3]. In

Section 2, we develop a novel efficient technique for

calculating the electronic structure of 1D structures. In

Section 3, we present and analyze the results of our

calculations and discuss their implications on the properties

of ultrathin QWs. The effects of the wire width and

orientation on the energies and the spatial distributions of

electronic states are investigated.

2. Model and method

We consider 2 ML thick InAs/InP (001) QWs in which

one of the interfaces is planar and the other exhibits two

1 ML steps, delimiting an InAs wire with a thickness of

3 ML. We study structures with wire orientations along the

k010l and k110l directions. The two-dimensional (2D)

projections of structures with 6 ML wide wires are shown

schematically in Fig. 1. As a unit cell, we use an atomic

chain along the k001l direction (shown in the figure with

rectangles). Thus, we have a ‘sandwich’ structure, consist-

ing of two physically different domains: the semi-infinite

regions with the 2 ML thick QW at the left and the right

sides and the finite 3 ML thick QW in the middle. We

calculate the Green functions of these two media using an

efficient iterative algorithm [4], and use the surface Green

function matching (SGFM) approach [5] to obtain the Green

0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0026-2692(03)00073-9

Microelectronics Journal 34 (2003) 459–462

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* Corresponding author. Tel.: þ1-514-3404711x2964; fax: þ1-514-

3403218.

E-mail address: nshtinkov@polymtl.ca (N. Shtinkov).

function of the entire system. The energies and the spatial

distributions of the electronic states are calculated from the

obtained Green function.

This approach has several major advantages over

conventional supercell techniques. The calculation is

always carried out in the subspace of layer projections;

therefore, the computational effort depends only on the size

of the unit cell (i.e. the thickness of the considered slab in

the k001l direction), but not on the lateral size of the wire. In

contrast to algorithms specifically designed for diagonaliza-

tion of very large supercell Hamiltonians which compute

only a limited number of eigenvalues, our method allows us

to investigate the entire energy spectrum (discrete and

continuous) of the system. Finally, the proposed approach is

not limited to the described sandwich structure, but can also

be used for other systems where a SGFM description is

available.

For the calculations, we use the semi-empirical sp3sp

tight-binding (TB) model, including the spin – orbit

coupling, with the parameters from Ref. [6]. The atomic

positions are considered the same as in a 2 ML QW as

calculated from the elasticity theory (i.e. the additional

distortions due to the presence of the wire are not taken into

account). This is a reasonable approximation, since in this

work, we aim at investigating the general trends rather than

obtaining precise numerical results. The modifications of the

TB parameters to account for the compressive strain in the

InAs layer include scaling the two-center integrals with the

bond length [6] and changing the on-site energies of the p

orbitals [7]. The valence band offset is 0.3 eV for the

unstrained heterojunction, leading to a strained band offset

of 0.38 eV. The zero energy is at the InP valence band

maximum. The structures are considered infinite in the

(001) plane and periodic in the wire direction. In the k001ldirection, we have imposed periodic boundary conditions,

choosing the width of the entire slab to be 20 ML (2 ML

InAs and 18 ML InP), which is sufficient to ensure weak

coupling of the neighboring QWs.

Fig. 1. A schematic view of structures with 6 ML wires along the (a) k010l and (b) k110l directions. Empty circles denote cation (In) atoms and small/big full

circles denote P/As anion atoms.

Fig. 2. (a) Energies of the conduction and (b) valence band localized states in k010l and k110l wires. The dotted lines denote the 2 ML QW band edges. 1 ML in

the k110l direction equalsffiffi

2p

ML in k010l:

N. Shtinkov et al. / Microelectronics Journal 34 (2003) 459–462460

3. Results and discussion

We investigate the conduction and valence band states

of structures with wire lateral widths ranging from 2 to

45 ML. The energies of the electron and hole states of

k010l and k110l wires, calculated at zero wavevector with a

precision of 1 meV, are shown in Fig. 2. Two types of

states are observed in the conduction band of the structures

(Fig. 2a). The extended 2D QW states form the continuous

part of the spectrum at energies above 1.333 eV. This band

arises from the ground state of the 2 ML InAs/InP (001)

QW. The localized (or bound) 1D states are confined in the

wire region, with energies below the QW band edge of

1.333 eV. In the valence band (Fig. 2b), a similar behavior

is observed for the heavy hole and light hole states, where

the QW band edge energies are 0.118 and 0.033 eV,

respectively. The bound states shift away from the 2 ML

QW band edge energies with increasing wire widths,

causing a reduction of the optical gap of up to 93 meV for

45 ML k010l (32 ML k110l) wires. The heavy hole states

are more sensitive to the wire width, which leads to an

increase of the heavy hole–light hole splitting of up to

33 meV compared to a 2 ML QW. From Fig. 2 it is seen

that only the energies of the localized heavy hole states

exhibit a significant anisotropy. The localized electron and

light hole states energies are the same (within the

calculation precision of 1 meV) in k110l and k010l wires

of equal width.

A question of practical importance is how the 3 ML QW

regions affect the states of the 2 ML QW, i.e. the extended

states of the structure. To study this effect, we have

calculated the spatial distribution of the conduction band

extended state at 1.333 eV in wires of different widths. In

Fig. 3, we have shown three representative cases: 5, 15, and

20 ML k110l wires. In the 15 ML wire, the amplitude of the

QW state in the wire region is strongly suppressed, and the

structure can be viewed as consisting of two non-interacting

components: the wire and the QW. In the other two cases,

however, a coupling between extended and localized states

is observed, leading to the appearance of mixed states,

which are distributed both in the QW and in the wire

regions. Note that the coupling also changes the energy of

the states, which is evident from the shift in the localized

states energies as they approach the band edges (see Fig. 2).

This effect can be observed where the localized states have

energies close to those of the extended states.

In the valence band, the localized light hole states have

energies within the heavy hole band, which lies below

0.118 eV. Therefore, coupling between extended heavy hole

and localized light hole states is observed. At energies

below 0.033 eV, the presence of the wire increases the

coupling between the extended heavy hole and light hole

states. With four different types of states, the valence band

structure is dominated by coupling between states of

different real space localization (well–wire) and character

(heavy hole–light hole).

4. Conclusion

We have studied theoretically the 1D lateral confinement

of carriers in ultrathin QWs, using the semi-empirical sp3sp

Fig. 3. Contours of equal probability amplitude for the QW conduction-band-edge state in structures with (a) 5 ML, (b) 15 ML, and (c) 20 ML k110l wires. The

InAs/InP interfaces are shown with thin solid lines.

N. Shtinkov et al. / Microelectronics Journal 34 (2003) 459–462 461

TB model and the SGFM technique. We consider 2 ML

thick InAs/InP (001) QWs in which one of the interfaces is

planar and the other exhibits two 1 ML steps, delimiting an

InAs wire with a thickness of 3 ML. The electronic states in

structures with wire lateral widths ranging from 2 to 45 ML

and orientations along the k010l and k110l directions are

investigated. Two sets of states are identified: localized 1D

states, whose amplitudes are concentrated mainly inside the

wire, and extended 2D states, whose amplitudes are

localized in the QW, but are strongly suppressed in the

wire region. The presence of ML steps at the interfaces leads

to a significant band-gap reduction (up to 0.1 eV) and to an

increase of the heavy hole– light hole splitting. The

comparison between structures with different wire orien-

tations shows that only the heavy hole states exhibit

significant anisotropy. At certain wire widths, coupling

between extended and localized states occurs, for which the

resulting states’ wavefunctions are distributed in both the

QW and the wire. The presence of interface steps also

increases the coupling between heavy hole and light hole

states in the valence band. Our results show that ultrathin

QWs with step-like interfaces cannot be viewed as simply

consisting of regions with different QW widths, since they

have a complex electronic structure, dominated by coupling

between states of different real space localization (well–

wire) and of different character (heavy holes–light holes).

Acknowledgements

This work was financially supported by the Natural

Sciences and Engineering Research Council of Canada.

P.D. also acknowledges support from the Canada Research

Chair program.

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