Excitons in V-Shaped and T-Shaped Semiconductor Quantum Well Wires

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Excitons in V-Shaped and T-Shaped Semiconductor Quantum Well Wires

Dirk Brinkmann, Axel Löffler, Guy Fishman

To cite this version:

Dirk Brinkmann, Axel Löffler, Guy Fishman. Excitons in V-Shaped and T-Shaped Semiconduc- tor Quantum Well Wires. Journal de Physique I, EDP Sciences, 1997, 7 (10), pp.1221-1231.

�10.1051/jp1:1997119�. �jpa-00247394�

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Excitons in V-Shaped and T-Shaped Semiconductor Quantum

Well Wires

Dirk Brinkmann, Axel L6fller and Guy Fishman (*)

Laboratoire de Spectrom4trie Physiqiie (**), Universit6 Joseph Fourier, Grenoble I, BP 87, 38402 Saint Martin d'Hbres Cedex, Flance

(Received 17 February 1997, revised 28 Alay 1997, accepted 23 June 1997)

PACS.73.20.Dx Electron states in low-dimensional _structures (superlattices,

quantum well structures and multilayers)

PACS.78.SS.Cr III-V semiconductors

Abstract. We give _a general framework for describing electronic states in isolated quantum

wires. It provides _a description of both the conduction band and the valence band, taking

full account of the complexity of the r8 valence band. This is applied to T-shaped wires and

V-shaped wires. To make _a useful comparison with experimental results _we calculate the exciton

Rydberg. We show that in the _case of the T-shaped wires hole confinement is due to the interaction with the confined electron and not due _to the confining potential: the red shift of the wire exciton line with respect to that of the quantum well is mainly due to the decrease of the electron confinement _energy and only partially to the increase of the exciton Rydberg. The

experimental results _are reproduced with _no adjustable parameters.

1. Introduction

Among the quantum wells (QWS) liable to describe the potential in quantum physics, the finite _square QW plays _a special role because the Hamiltonian is exactly soluble. This _means that the Schr6dinger equation can be reduced to _a transcendental equation which gives all the coefficients of the analytical shape of the _wave function. Finite square QWS _are not very

common except- in two-dimensional (2D) semiconductors where the electrons _are confined in

one direction, _so that the finite square QW describes _very well the physical situation in the effective _mass approximation, which is the framework of this paper. In 2D semiconductors,

two masses, one inside the well and _one in the barrier, _are _necessary to describe the _con- duction electron but this again leads to a transcendental equation iii. In _a quantum wire,

henceforth called _a wire, the electrons _are confined in _a 2D potential and the problem is

different; the potential is _not separable, except in very special _cases, and the Schr6dinger equation is _a partial differential equation: it is _no longer a trivial problem to obtain the electron energy even for the conduction band. The problem is not academic because of progress in making semiconductors wires (also called one-dimensional semiconductors) which invites comparison between the experimental results, particularly the photoluminescence lines, and the theory [2-16j. In recent years, measurements have been performed _on wires of several

(*) Author for correspondence (e-mail: guy.fishman©ujf-grenoble.fr) (**) ^CNRS ^UMR 5588

@ Les #ditions _de _Physique ₁₉₉₇

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types: V-shaped _quantum wires (VWI) _{[5, 6, 9,}10j, T-shaped quantum ^wires (TWI) _[3,7,16j,

and quantum ^wires ^with Strain-induced lateral confinement (SWI) [2,13j. A theoretical study of SWIS has already been published [13,14j _so that in the present paper we deal only with VWIS and Tlvls. Although the shapes _are _very different these wires have at least three points

in _common: ii the potential is not separable, iii the wires _are isolated and iii) the three types

of wires _are parallel to the >110j direction (or _any equivalent crystallographic direction). From

a theoretical viewpoint the main problem is linked to the fourfold degeneracy of the valence band ii?]. The problem of the valence band dispersion _was first solved by assuming either that the potential is separable [18j, or that the wires form _a superlattice jig], the wires being

oriented along [001j. Recently the valence band dispersion _was calculated in _a VWI [4, 8, lsj,

the wire being oriented along [l10], and in a TWI ill,15]. To perform this calculation _we

use the _same symmetrized Luttinger Hamiltonian _as in references [8,18j and _we take into _ac-

count the _exact (finite) potential. However the luminescence lines _were attributed to excitons

in the wire _so that it is not enough to know the conduction electron _energy and the valence band dispersion: _a meaningful comparison between the experimental and theoretical transition energies requires the knowledge of the exciton Rydberg _as well. The aim of this paper is to present a calculation of the excitonic energies in VWIS and TWIS taking into account the

exact valence band symmetry. This provides an unambiguous understanding of the origin of

(he luminescence lines reported in references [3,5j.

Throughout the present _paper, ^the direction along the length of the wire is taken _as both the z-axis and the spin quantization axis, and the radial vector is _p

= ix,vi. The free electron

mass is taken _as unity.

2. V-Shaped Quantum Wires

We deal with the VWI whose shape is described in reference [5]. ^The ^GaAs ^wire ^is ^surrounded by _a barrier in GaAlAs. The photoluminescence (PL) spectra give the energy of the el-h1

and e2-h2 excitons with which _we _are dealing. We need _to know the exact shape of the VWI to perform the calculation. We start from _a TEM micrograph from reference _>20] and model the shape of the boundary between GaAs and GaAlAs with _an accuracy of the order of _one monoatomic layer: to go beyond this accuracy has _no physical meaning. Figure 1 gives the

potential modelization _we _use.

We then calculate the energies and the _wave functions following the multi-component ^vari-

ational method of reference [21], ^which ^is ^suited to isolated QWS and which _we have _ex- tended for isolated wires along the [l10] direction. We _recover the (envelope) _wave function xe = xe(pe) of the conduction band _[5] and the _wave function xh = xh(ph) of the

valence band [10,15]. In _a wire the jJM) components are strongly mixed _even at k

= 0,

where k is the one-dimensional _wave vector. The hole _wave function _xh has two components xh ~ f$~/~(ph) (3/2 ^£ 3/2) + f~~/~(ph) ^(3/2 ^~ l/2) (the two functions _are Kramers conju- gate). The electron and hole ground state (el-hl) wave functions _are given in Figure 2 and the first excited _state (e2-h2) wave functions _are shown in Figure 3. Both _are given for _a _zero

wave vector. We have obtained the dispersion _curve of the conduction band and the valence

band _as well. The first two conduction bands (el and e2) are described by _an effective _mass of 0.067: We have taken into account the larger _mass in the barrier but this does _not change the

e1 and e2 _masses _up to 0.01 l~~. The first two valence bands (hi and h2) _are quite reasonably

described by _masses mhi " 0.13 and _mh2

" 0.20 respectively _up to a wave vector equal to

o,ol i~~,

as shown in Figure 4. Owing to the structure of _Xh the symbols hi _or h2 merely

index the valence subbands without _any relation to heavy or light holes with the quantization

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AlGaAs (33%)

q j, GaAs S_..

« jj>I_ nl'~~ii~

0~ '""j~~ ^~~'

~~~ ~(~ j~~

i~

~~

Y [A] [I lo]

a)

AlGaAs (35°A)

f

~ GaAs

1

~

AlGaAs

~ ⁰ ²⁰⁰ ⁴⁰⁰ ⁶⁰⁰

~ ~~~°~ ~~~~ ~°°~~

b)

Fig. I. Potential profiles of a) the V-shaped wire (VWI) and b) the T-shaped wire (TWI) studied in this paper. The modelization of the VWI is based _on reference [20]. ^The geometry of the TWI is

that given in reference [3).

axis parallel to the wire. However if the quantization axis is perpendicular to the wire the holes _are mainly heavy _or light with the terms usual in quantum wells [15].

Once the _masses and the envelope functions _are known, _we _are in _a position to calculate the

exciton by _a variational approach of the type often used in QWS _[22]. The exciton Hamiltonian is H = p~/2p e~ler where _e is the background dielectric constant, IIv

= 1/me +1/mh where mh = mhi for the el-hl exciton and _mh

= mh2 for the e2-h2 exciton. With this Hamiltonian,

the potential confinement is taken into account via the trial function il

= xexhe~'~'/~ where _z is the in-wire distance between the electron and the hole and I is _a variational parameter that

plays the role of _a Bohr radius. The form of the hole _wave function leads to a two component exciton _wave function in _a wire. We find that the Bohr radius I is equal to 193 I _for _el-hl _and

203 I _for _e2-h2: _these _values

are consistent with the condition that l~~ (about 0.005 l~~) _is

far smaller than the range of validity of the effective _mass description of the carriers (0.01 l~~)

as shown in Figure 4. Were this condition _not fulfilled, the exciton calculation would have _no meaning. The exciton Rydbergs _are respectively 9.5 and 8.0 mev. This small difference is due to the fact that the Bohr radii _are close because the reduced _masses _/~ _are not very different

(respectively 0.044 and 0.050). Thus the difference between the exciton transition energies results mainly ^from different confinement energies for the excitons el-hl and e2-h2 (called

n~ = I and _n~

= 2 in Ref. [5]).

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'l _I'

CQ

(,

-

©

"

4x10'~

~

,

§ 0

~ ~j ^~i^fl01 ~)

cq

" -'-l

~

~/

S

£

= (,

l

~) ₁₅₀

,

0

j ⁴⁰⁰ ~)

$ ^~°°

~ j( it'°I

~

--i

~ ^8x10'~

i£S'

x10'~~

m~)

150 j) ₃₀₀

~~~ ,~, C)

~ihl11-

Fig. 2. Probability of finding the electron and the hole in their ground state (el-hl) in the V-

shaped wire (VWI) described in reference [12). The _wave vector parallel to the wire is equal to _zero.

a) The squared modulus of the conduction band ground state _wave function. b) The squared modulus of the (3/2 ~ 3/2) _component of the valence band ground state _wave function hi. cl The _same _as in

b) but for the (3/2 _~ l/2) _component. The spin quantization axis is parallel to the wire throughout

the present _paper.

The _exact shape of the VWI used in the experiments has a large influence of the energies of

excitons el-hl and e2-h2 but a smaller influence _on the splitting between these two lines. Thus, although _we did not perform _a calculation _on exactly the _same sample _as in reference [5], ^it makes _sense to compare the calculated splitting to the experimental _one. The energy splitting

calculated from the data of reference [20] ^between ^the êxcitons êl-hl ând ê2-h2 îs ²² ^mev while the experimental splitting in reference [5] is measured to be 20 mev with _a FWHM of the order of10 mev. This fully confirms the interpretation made by Îtinaldi et al. of their two

lines [5].

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S 8x10'~

I _I

"

(

) ₄₀₀ _a)

/ 11101

cu

& _8xl 0'~

~

+ ~

cu

~

"

4xl 0'~

n~)

~ ₄₀₀ b)

~ ~IV l''°~

cu~q 8xl0'~

R~_~ 'l'~

~

4xl 0'~

n~)

( c)

$ 800

~ j( ^jl101

Fig. 3. Probability of finding the electron and the hole in their first excited state (e2-h2) in the

same VWI _as in Figure 2. The _wave vector parallel to the wire is equal to _zero. a) The squared modulus of the first excited state wave function of the conduction band. b) The squared modulus of the (3/2 ~ 3/2) _component of the valence band first excited _state _wave function h2. cl The _same _as in b) but for the (3/2 _~ l/2) component.

3. T-Shaped Quantum Wires

The TWI fabricated with GaAs-GaAlAs is _a different _case. We _are interested in the low excitation PL spectra in Figure 4 of reference [3], ^where ^the ^various ^lines are identified _as originating from the TWI and from the two QWS that form the vertical part ^{of the} ^T (growth

axis in the [001] direction) and the horizontal part of the T (growth axis in the II10] direction)

so that the wire is along the >I -I oj direction. The geometry is given in Figure I. In the

following _we write 001 and l10 _as subscripts to distinguish the two QWS. The width of each

QW is 70 1; _each _[001j _QW is separated by 380 I.

First _we calculate the energies of the QWS whose valence band dispersion _curves _are _given

in Figure 5. The calculation is performed in the axial approximation which gives _averages

over all the directions of the in-plane _wave vector k. The confinement energies at k

= o _are

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m~~ ⁼ ^0.13

%(

~ tIJ

~

UJ m~~ = 0.20

I

.,

0.oo 0.ol 0.02 0.03 o.04

WAVEVECTOR [A~~]

Fig. 4. Dispersion curve of valence band energy of the VWI. The VWI has _no inversion center, _so the spin degeneracy is lifted. The _wave vector is parallel to the wire. The indices hi and h2 correspond

to the hole ground state and to the first excited state but not to the heavy _or light hole which has _no meaning in such _a wire (see Figs. ^I and 2). The _average _masses _mini

= 0.13 and _mh2 ₌ 0.20 are valid

up to _wave vectors about equal to 0.01 i~~, _that _is _larger than the inverse of the Bohr radius of the

exciton el-hl and e2-h2 (see text).

Ee

= 55. I mev for the electron in both the QWS (the conduction electron _mass is isotropic)

and Ehooi

" 12.6 mev and Ehiio

" 7.6 mev, while the _masses _mh of the first valence band in

each QW _are _mhooi

= 0.19 and _mhiio

# 0. ii: these _masses _are used _to calculate the exciton

Rydberg in the _two QWS (see below). For the >001j QW the symbols hhn and lhn _mean heavy

and light hole _as usual. For the ii10j QW the symbols 'hhn' and 'lhn' _mean heavy and light hole only in the axial approximation used here. In the general _case the _wave functions hhn and lhn do not correspond to the _wave functions j3/2 £ 3/2) and j3/2 £ 1/2) respectively, _even at k ₌ o (see Ref. >21j). ^The range where the effective _mass of the hole has _a meaning is of the order of 0.02 i~~ _The _exciton Rydberg Rx calculated by _a method similar to that used in Section 2 is 7.2 mev for the vertical QW and 7.0 mev for the horizontal QW. The Bohr radii

are respectively 137 and 147 I, _whose _inverse _is _smaller _than _0.02 i~~: _thus _the calculation is consistent. The exciton transition energies _are then Ex

# EG + Ee + Eh Rx

# EG + REX

so that Ex is 1579.9 mev for the vertical QW and 1575.1 mev for the horizontal QW (EG is the GaAs bandgap). In reference [3j ^the ^PL energies of the two QWS _are 1579 and 1583 mev with _a FWHM of the order of 2 mev.

Now _we _come to the wire. First _we suppose that the wires _are isolated. The conduction band dispersion _curve looks like that of the bulk: The ID conduction _mass is practically unchanged (0.068 instead ofo.067) _[23j. On the contrary the valence band dispersion is quite different and is given in Figure 6. The hole effective _mass, valid _up to _a _wave vector larger than 0.01 i~~,

is equal to 0.12. Once the energy is known _we _can get ^the wave functions, which _are given for

a zero wave vector in Figure 7. More precisely, Figure 7 gives the probability of finding the

electron in _a wire: for the electron it is valid to suppose the wire isolated [3j. The electron confinement _energy is Ee

= 46.4 mev. The _same figure also gives the probability that _a hole is present, which is the _sum of the _squares of the moduli of the two components. Clearly, _as far

as the hole is concerned the wire cannot be considered _as isolated. The _reason that the hole is poorly localized in the wire is the following: the confinement _energy Ehiio is 5 mev lower than the confinement energy Ehooi (see Fig. 5) and therefore the hole is mainly confined in the l10 QW. Were the TWI built with _axes rotated by go degrees (inversion of the [oolj and [l10j axes) the hole confinement would be different.

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~~

m =0.19

I ...

I

._

~

$o _"'.,

z

0.00 0.02 0.04 0.06 0.08 o-lo

~)

m ₌ 0.17

K

._

~ ".

ijt9 _"I

z ",

uJ

,1 -

0.00 0.02 0.04 0.06 0.08 o-lo

WAVEVECTOR [A-1]

Fig. 5. al Valence band dispersion _curve for the quantum well whose growth axis is in the [001) direction and which forms the vertical part of the T-shaped quantum wire (TWI) described in refer-

ence [3). b) The same for the quantum ^well ^whose growth axis is in the [l10] direction and which forms the horizontal part of the TWI. The _masses _mhooi

= 0.19 and mhiio = 0.17 _are used in the variational

calculation of the two~dimensional exciton Rydberg. The indices hhn and lhn for the growth axis in the [001) direction correspond to the usual heavy and light holes in _a two~dimensional semiconductor. The indices 'hhn' and 'lhn' for the growth axis in the [110] ^direction correspond mainly, but not exactly,

to heavy and light holes in _a two~dimensional semiconductor (see text).

I

i~ / ""_

UJ _,.

I m~ = 0.12 ",,

0.00 0.01 0.02 0.03 0.04 0.05

WAVEVECTOR[A'~]

Fig. 6. Valence band dispersion _curve for the wire of the TWI structure. The calculation is made

as if the wires _were isolated. See text regarding the limit of validity of this description.

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t

'

~~ ~j

~~ ')~~" _a

$ j

~

~~~~

400

° 200 ,,°'

~~~~ ~j

600 0

~~

3X10

~~~~

t

~

~ ~

&

~

£ ~~

'~

0

400,

~~ ^~~~

200 (°

~°01j ⁶⁰⁰ ⁰ ~jh

3x10 'J, _~ j~-<

~''

~- '

q

~ c)

~ j _

xiii

~(jj

J ₆₀₀ ⁰

y'

Fig. 7. Probability of finding the electron and the hole in their ground state (el-hl) in the T-shaped

wire (TWI). The _wave vector parallel to the wire is equal to _zero. al The squared modulus of the conduction band ground state _wave function. b) The squared modulus of the (3/2 ~ 3/2) component

of the valence band ground state _wave function hi. c) The _same _as in b) but for the (3/2 _~ l/2)

component. The calculation _was performed assuming ^the wires, separated by 380 I,

are isolated. The

figure shows that this is true for the electron but not for the hole, which is not localized in _a wire: the hole will be localized by the attraction of the confined electron _as shown in Figure 8.

The hole being weakly localized, the validity of the assumption that the wires _are isolated is

questionable and _one could think of calculating the dispersion curve for coupled wires. However

we must keep in mind that the aim is to calculate the exciton Rydberg ^which _means here _an exciton where only the electron is well localized. The method of the preceding section is not usable and _we _are led to _use another method to calculate the excitonic transition energy

Ex, namely localization of the hole by the confined electron, used by Peter _et al. in 2D semiconductors [24j. This is the _reason why _we have _not calculated the hole dispersion _curve for _a coupled wire: in _any _case neither the latter _curve _nor the periodic _wave function would be useful in the method of Peter et al. Now _we have to extend it here to ID semiconductors.