Optical properties of near-surface exciton quantum wells - Cinvestav

Sociedad Mexicana de Ciencia de Superficies y de Vacío

Superficies y Vacío 13, 134-139, Diciembre 2001

Optical properties of near-surface exciton quantum wells N. Atenco-Analco, B. Flores-Desirena, A. Silva-Castillo, F. Pérez-Rodríguez, Instituto de Física de la Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, México

An overview of theoretical investigations on near-surface semiconductor quantum wells, whose optical properties are considerably affected by the interaction of the exciton with the sample surface is given. Near-surface quantum wells with both weak and strong quantum confinement of excitons are considered. When the exciton quantum well is very close to the sample surface, exciton dynamics is determined not only by characteristics of the quantum well, but also by the interaction of the exciton with the sample surface. Optical spectra for various systems of near-surface quantum wells are discussed. In addition, the optical manifestation of magnetoexcitons in near-surface quantum wells is also commented. Keywords: Exciton quantum wells; Quantum well surface optical properties; Magnetoexcitons

Semiconductor thin films and quantum wells are commonly grown on substrates and may have a cap layer, overlying them. Therefore, their optical properties depend on the characteristics of both the substrate and the cap layer. If the thickness of the cap layer is sufficiently small (of the order of the exciton radius), the optical properties of the semiconductor structure may be affected considerably by the interaction of the exciton with the sample surface. In this paper we present an overview of the theoretical approaches (Secs. 2-4) applied for interpreting optical properties of near-surface exciton quantum wells in both weak (Sec. 3) and strong (Sec. 4) confinement regimes. Also, we discuss about optical techniques (reflectivity, 45 degrees reflectometry, diffuse reflection, for example) which are quite appropriate to characterize that kind of confined systems.

1. Introduction Excitons in confined systems, such as quantum wells (QWs), have attracted the attention of researches because of their potential applications to optoelectronic devices. There exist two qualitatively different regimes of exciton confinement in quantum wells: weak and strong confinement regimes. In the regime of weak confinement of exciton (also known as thin film regime), the effective width d of the quantum well is much larger than the exciton radius a0 ( d >> a0 ) and, therefore, the center-of-mass motion of the exciton is quantized. In this regime the relative motion of the electron-hole pair is practically the same as in the bulk except for a small distortion near film boundaries [1,2]. Because of the interaction of the exciton with the interfaces, transition (exciton-free) layers appear near film boundaries. Therefore, the confinement of the exciton center-of-mass motion occurs in an effective length deff smaller than the actual film thickness ( deff ≈ d − 2 l, l is the effective size of one exciton-free layer inside the film ). The quantization of the exciton center-of-mass in thin films manifests itself as a resonance structure in optical spectra (reflectivity, transmission, and absorption). It is interesting that this resonance structure can also be explained as an effect of multimode interference of exciton polaritons [2-4]. In the regime of strong exciton confinement, the thickness of the quantum well d is smaller than the exciton radius a0 ( d ≤ a0 ) and the motion of the electron and hole is quantized separately in the direction perpendicular to the quantum-well plane. Due to the in-plane Coulomb interaction, the electron-hole pair in a quantum well forms a quasi-2D exciton. The optical response of excitons in quantum wells has been investigated in numerous works (see, for example, [5-7] ). Quasi-2D excitons in quantum wells couple to light and produce resonances in reflectivity spectra, which correspond to exciton-polariton modes with polarizations perpendicular to the plane of incidence (T mode), parallel to the in-plane component k|| of the wave vector for the incident light (L mode), and perpendicular to the quantum-well plane (Z mode).

2. General formulation of the problem The optical properties of excitons in near-surface quantum wells (quantum wells on substrate) can be described by employing the density matrix approach of Stahl and Balslev [8], which leads to the set of differential r equations for the coherent-amplitude vector Y (rre , rre ) of r electron and hole with coordinates rve and rh , respectively: r

v r

(H vc − h(ω + iν ) )Y = Mˆ vc (rr) E ( R) ,

(1)

r

where E is the electric field of the electromagnetic wave of frequency ω, Mˆ vc is the interband-transition dipole density, ν is a phenomelogical damping coefficient, Hvc is a two-band Hamiltonian with gap Eg , which is given by H vc = E g −

h2 2 h2 2 r ∇e − ∇ h + Ve (re ) + Vh (rvh ) + 2 me 2m h

r r +Vc (r ) + Vs (re , rh ) .

134

(2)



Here, Vc (r ) = −e 2 / ε s r is the Coulomb potential, r is r r r r r the magnitude of the vector r = re − rh = ( ρ , z ) , R = r r (me re + mh rh ) /(me + m h ) is the radius vector of the exciton center-of-mass, εs is the low-frequency dielectric constant of the medium, me and mh are respectively the electron and hole effective masses. The quantities Ve (rve )

pair. One of the mostly applied models to interpret optical spectra in terms of the concept of the exciton center-ofmass quantization is the so-called adiabatic approach [9]. r r r Within this model, the coherent amplitude Y ( re , rh ) is expanded as r r r r r r Y ( re , rh ) = ∑ Yλ ( R )ϕ λ ( Z , r ),

v and Vh ( rh ) are the confining potentials for the electron

λ

where λ = {n, l , m} is the complete set of quantum r numbers, ϕ λ ( Z , r ) are solutions of the Schrödinger equation given by

and the hole. In the case of QW geometry these confining potentials have steplike form in the z direction. Finally, the quantity Vs (rve , rvh ) denotes the potential due to the surface. An intrinsic contribution to Vs (rve , rvh ) is the image potential, Vim (rve , rvh ) , which appears because of the dielectric mismatch at the sample surface (Z=0): e2 ε s − 1 1  1 1 Vim (rve , rvh ) =   + ε s ε s + 1  4  z e z h  1  − ρ 2 + ( z e + z h ) 2 

 h2 2 e2 r r ∇r − + Vs (Z , r )ϕ λ ( Z , r ) = − εsr  2µ  = E λr (Z )ϕ λ ( Z , rr )

  − 

(7)

r Here, the eigenfunctions ϕ λ ( Z , r ) and the eigenenergies E λr (Z ) associated with the relative motion depend parametrically on the center-of-mass coordinate Z. According to the adiabatic approximation, the expansion v r coefficients Yλ (R ) in Eq. (6) satisfy the equation:

(3)

The polarization of the medium is related to the coherent amplitudes as r r r r r r r P ( R) = 2∫ Mˆ vc (r )Y ( R, r )dr .

(6)

 h 2 2  r r r ∇ Yλ ( R ) = E g + E λ ( Z ) − h (ω + iν ) − 2M  

(4)

In order to calculate optical functions of near-surface quantum wells, the system of equations (1)-(4) together with Maxwell equations,

r r = 〈 Mˆ vc 〉 ∗λ E ( R) , where

(8)

r ω2 v ω2 r ∇ × ∇ × E − ε ∞ 2 E = 4π 2 P , c c

〈 Mˆ vc 〉 ∗λ = ∫ drrϕ λ∗ ( Z , rr) Mˆ vc (rr ) .

(9)

(5)

A simple model for the transition dipole density is the shell model [8] which is defined as

should be solved. The quantity ε∝ in Eq. (5) stands for the high-frequency dielectric constant. In solving Eqs. (1)-(5), r the boundary condition Y = 0 , when the electron or the hole is at the vacuum-QW interface (ze=0 or zh=0 ), as well as the condition of continuity for tangential components of the electric and magnetic fields of the electromagnetic wave are applied.

M0 r Mˆ vc (r ) = δ (r − r0 ) , 4πr 2

(10)

with r0 → 0 . Hence, only s-excitons couple to light. Sufficiently far from the surface (V s → 0) the energy E λr ≈ − E b3,λD , where E b3,λD is the binding energy for the λsate 3D exciton. It is convenient to introduce the quantity U λ ( Z ) = E λr ( Z ) + E b3,Dλ which plays the role of an effective surface potential in Eq. (8) for the exciton translational motion. Finally, the excitonic polarization can be written as

3. Weak confinement of exciton 3.1. Thin films on substrate Several theories (see, for example, Refs. [1-4, 9-13]) have been developed to describe the optical manifestation of the quantization of the exciton center-of-mass in thin semiconductor films. The main problem encountered here is associated with the complicated behavior of the exciton near film boundaries because of its interaction with them. The exciton-interface interaction leads to the coupling of the relative and translational motions of the electron-hole

r v r r P ( R ) = 2∑ 〈 Mˆ 〉 λ Yλ ( R ) . λ

(11)

Below, we shall consider only the contribution of the r ground-state exciton (λ={1,0,0}) to the polarization P [Eq.

135



(11)]. For simplicity, the sub-index λ={1,0,0}, indicating the exciton ground state, will be omitted. So, in this case r v r r the polarization P ( R) = 2〈 Mˆ 〉Y ( R) . In high-quality semiconductors, the no-escape boundary condition for the electron and hole at the surface and the image potential make the exciton surface potential U (Z ) be repulsive [9,14]. This intrinsic potential decays towards the interior of the film over distances of the order of the exciton radius. The simplest model for an intrinsic potential is an infinite barrier at certain distance l from the surface of the order of a0. Therefore, in films of III-V and II-VI compounds, which are characterized by a relatively large exciton radius a0 ∼ 3-13 nm, the size l of exciton-free layers reduces notably the effective thickness of the nonlocal film [1-5]. Besides, the resonance structure of the optical spectra of this kind of materials turns out to be very complicated due to the quantization of the center-of-mass of both heavy-hole (hh) and light-hole excitons [5]. The study of exciton center-of-mass quantization is simpler in CuCl thin films [11-13] which have a large binding energy

The reflectivity Rs, calculated in Ref. [15], exhibits a rich resonance structure associated with size-quantized transverse exciton polaritons. The z component of the wave vector for these polaritonic modes satisfies the Fabry-Perot relation Re kzd= nπ ( n is an integer ) at frequencies very close to the eigenvalues

 h 2π 2 n 2 hω Tn = E g − Eb3D +  2  2Md

ω Ln = ω Tn + ω LT ,

film ( 0 < Z < d ) one can assume that E ( Z ) = − E b

3D

(i.e. U ( Z ) = 0 ) in Eq. (8). Then, the exciton-surface interaction is described by using only the boundary r v condition P = 0 (or Y = 0 ) at Z=0 and Z=d.

3.2. Extrinsic transition layers The center-of-mass quantization of excitons can also occur in extrinsic transition layers of bulk semiconductors [9,16,18-23] and thin films [4,24,25]. Indeed, the form of the surface potential U (Z ) can be altered either by surface treatments such as electron and ion bombardment, intense illumination, heating, doping, and a bias electric field, or unintentionally during the process of crystal growth. The extrinsic transition layers are characterized by space charge that produces an inhomogeneous macroscopic electric field r Ein (Z ) with which excitons interact. Hence, the potential

In the recent work [15], spectra of s- and p-polarized reflectivity (Rp , Rs) at 45-degrees angle of incidence, and spectra of 45-degrees reflectometry [16] ( ∆45 = Rp -Rs2 ) for a CuCl film (d= 16.5 nm) on a MgO substrate were calculated by using a nonlocal dielectric function,

r 4ω T χ (k ,ω ) = h where

ωT = (Eg − E

(12)

M 02 | ϕ 100 (r = 0) | 2 , hω T 2 2 2 ωT + k − ω − 2iνω M

3D b

)/h.

(16)

where ω LT is the longitudinal-transverse splitting. The longitudinal resonances are not easily detected in the spectrum of p-polarization reflectivity, Rp , because of the relatively large damping at frequencies above the longitudinal one (ω > ωL )[11-13] and their interference with transverse resonances. In spectra ∆45 = Rp -Rs2 of 45degrees reflectometry, transverse resonances are not so pronounced as in the simple reflectivities Rp and Rs [15]. This fact permits to observe the longitudinal resonances (dips) clearly in ∆45 spectra.

angstroms. Hence, CuCl thin films allow to maintain the bulk-like behavior up to a very small film thickness ( d ∼ 7 nm >> a0 ). In the case of CuCl, the presence of excitonfree layers near film boundaries can be neglected in interpreting its optical spectra. So, inside the whole CuCl

r r ε (k , ω ) = ε ∞ + 4πχ (k , ω ) ,

(15)

of the quantized transverse exciton in the thin film. In the p-polarized geometry longitudinal polariton modes, besides transverse modes, are also excited. The frequencies corresponding to the size quantization of the longitudinal polaritonic modes are

Eb3 D ∼ 190 meV and a very small exciton raidus a0 ∼ 7

r

  

(13)

r r Vs (re , rh ) in Eq. (2), describing the interaction of the

electron-hole pair with the surface, is given by

r r r r r Vs (re , rh ) = Vim ( Z , r ) + eEin (Z ) ⋅ r .

(14)

(17)

Far enough from the surface the mechanism of this interaction is related with the quadratic Stark effect. It introduces a coordinate-dependent shift of the exciton binding energy, and consequently, an extrinsic contribution to the excitonic surface potential U (Z ) . This contribution can be attractive, unlike intrinsic potential, and a potential well, where the exciton center-of-mass motion is quantized,

The spatially dispersive dielectric function [Eq. (12)] is straightforwardly obtained from the system of equations (5), (8)-(11) with U ( Z ) = 0 and polaritonic fields,

r r r r r r P(R) and E (R) , proportional to exp[ik ⋅ R] , and assuming | ω − ω T |