ivchenko_bookreg

3.1 Optical Reflection from Quantum Wells and Superlattices

121

a

a

Fig. 3.4. Optical anisotropy ρh and ρl of a GaAs/AlAs QW, as a function of well width a for the growth direction (a) [110] and (b) [113]. Dotted lines correspond to a calculation based on the 4×4 Luttinger Hamiltonian (finite barriers). In the other two calculations, the 6×6 Hamiltonian was used, for infinitely high barriers (dashed) or finite barriers (solid). Experimental points in (b) were taken from [3.35] (circles) and [3.36, 3.39] (squares). From [3.40].

122 3 Resonant Light Reflection and Transmission

this subsection is to demonstrate that, in zinc-blende-lattice multi-layered heterostructures grown along the [001] principal direction, the interfaces induce a remarkable in-plane anisotropy, namely nonequivalence between the

¯

[110] and [110] axes. The anisotropy can manifest itself both in optical and transport phenomena but we restrict the consideration to the optical manifestations only.

In a zinc-blende-lattice bulk semiconductor, say GaAs as a representative of III-V compounds or ZnSe as a representative of II-VI compounds, any lattice site is a center of the tetrahedral point symmetry. For any anion atom the right-hand-side bonds always lie in the same 110 -like plane, say the (110) plane, whereas the bonds on the left lie in another plane which is perpendicular to the previous one.

Now we shift from a bulk semiconductor to a single heterojunction CA/C A . If the compositional materials CA and C A contain the identical anion atoms, labelled A, the interface is formed by one atomic plane of anions neighboring the C atomic plane on one side and the C plane on the other side. As compared to bulk semiconductor crystals, the symmetry of a single heterojunction is reduced to the point group C2v (Chap. 2). In this

¯

group the directions [110] and [110] are nonequivalent. Thus, we conclude that, in general, a single heterojunction gives in-plane anisotropy. The particular mechanism and order of magnitude of the anisotropy can be obtained by using the microscopical model discussed below, and we shift the symmetry analysis from a single to double heterojunction, or to a QW. The symmetry of an ideal QW is higher than that of a single heterojunction because it contains the mirror rotation S4 about the [001] axis by 90◦. The second interface also induces the in-plane anisotropy. However, for the left and right interfaces,

¯

the role of the axes [110] and [110] is interchanged, their contributions to the anisotropy cancel each other and, as a result, an ideal QW is isotropic in the interface plane, at least in the linear optics.

The next step is to consider an interface between two materials lacking common anions and cations, C = C , A = A . The heteroboundary CA-on- C A consists of two nonstandard planes containing anions of one material and cations of another material. There are two possibilities of the transition from the layer C A to the layer CA, either via the chemical bond C -A in the sequence of atomic planes −C −A −C A−C−A− (interface of kind a) or via the chemical bond A -C in the sequence −A −C −A C−A−C− (interface of kind b) where the standard and nonstandard bonds are indicated by the symbols − and , respectively. The structure is assumed to grow from the leftto the right-hand side. Taking into account the similar alternative for the other interface, C A -on-CA, we conclude that, for a heteropair with no common atom, there exist four kinds of ideal QWs labelled as La-Ra, Lb-Rb, Lb-Ra and La-Rb. Here a, b indicate the interface chemical bond and R,L indicate the rightand left-hand-side interfaces. The first two have symmetrical interfaces related by the S4 operation, such QWs are isotropic in the interface

3.1 Optical Reflection from Quantum Wells and Superlattices

123

plane. The remaining two kinds are characterized by the low symmetry C2v , the left and right interfaces contribute di erently to the in-plane anisotropy and these contributions do not compensate each other. It is accepted that under growth without a special control the interface is predominantly cationterminated and the QWs are nominally La-Rb-like. Of course, one cannot exclude interfacial disorder and some admixture of a-like bonds at the right interface and b bonds at the left interface. At present technology allows the interfacial control in the process of layer-by-layer growth and fabrication of QW structures with both anion and cation-terminated interfaces [3.41].

Optical anisotropy in heterostructures without common cations and anions was predicted by Krebs and Voisin [3.42] and observed in GaInAs/InP QWs [3.43,3.44]. They used the experimental setup typical for measurements of birefringence and dichroism in bulk uniaxial crystals (Fig. 3.5a). In this setup the initial light linearly polarized by the polarizer P is focused onto the sample S, the transmitted light is filtered through an analyzer A and a spectrometer and finally detected by a photomultiplier. The sample can be rotated around the growth axis. In the particular experiment presented in Fig. 3.5 the sample contained 80 InGaAs QWs separated by InP barriers [3.44]. The curve (b) with the left-hand side scale shows the angular dependence of the transmitted intensity in the setup with parallel polarizer and analyzer with θ being the angle between the analyzer and the sample [100] axis. The curve (c) shows the ratio between the transmission signals in the crossed and parallel polarizer-analyzer configurations. From the curve (b) we conclude that the absorption exhibits a maximum and a minimum at the angle ±45◦, respectively, or, in other words, when the light is polarized along

¯

the [110] and [110] axes. The curve (c) confirms this conclusion because the transmitted signal measured in the crossed configuration P -A reaches maxima at the angles equal to 0 and 90◦ when the polarization plane lies in the

¯

middle between the axes [110] and [110]. From the maximum and minimum values of the transmission one can find the absorption coe cients K110 and

K110¯ . Remarkably, nearly no anisotropy e ect is observed in InGaAs/AlInAs MQWs where the compositional materials share a common anion.

Let us calculate the lateral optical anisotropy of a quantum-well structure CA/C A at the absorption edge, i.e., under the hh1-e1 interband transitions with zero 2D electron wave vector (kx = ky = 0, Γ point). In the envelopefunction method, the electron and hole e ective Hamiltonians within the well or in the barrier have the same form as in the corresponding bulk materials and, in particular, possess cubic symmetry Td. The low symmetry of the interface is taken into account by including the additional term in the boundary conditions for the valence-band wave-function envelope, see (2.42).

Neglecting the relativistically small corrections, we shall use for the conduction-band electron commonly accepted boundary conditions (2.10), i.e., the continuity of the envelope and of its normal derivative divided by the e ective mass. In this case the electron wave function at the bottom of the

124 3 Resonant Light Reflection and Transmission

θ

Fig. 3.5. Scheme of the experimental set-up for polarization-resolved transmission measurements (a), transmission signal at ω = 925 meV, as a function of the angle θ between the sample [100] axis and the analyser, with parallel polarizer and analyzer (b) or with crossed polarizers (c). The small deviation from π/2 rotation invariance in (c) is due to a very small optical anisotropy generally observed in InP substrates.

˚	˚
The studied sample S is 80-period InGaAs(45A)/InP(68A) MQWs. From [3.44].
e1 subband can be conveniently written as
ψ±(e11)/2 = ϕe1(z)|Γ6, ±1/2 ,	(3.113)

where |Γ6, ±1/2 are the ↑ S and ↓ S Bloch functions, S is the coordinate function of the Γ1 representation of group Td, and ϕe1(z) is given by (2.8).

For the four envelope functions ϕm (m = ±3/2, ±1/2) of the hole wave function

3.1 Optical Reflection from Quantum Wells and Superlattices

125

1

ψ = √

m ϕm(z)|Γ8, m

(3.114)

S

we shall use the extended boundary conditions

= ϕ ,

=

ϕ

+ 2

ϕ

(3.115)

ϕ

m

ϕ

t

l-h

J

J

y }mn

,

m

m

m

m m

√3

{ x

n

n

taking into account the mixing between the heavyand light-hole states at the (001) interface, which is allowed by the interface symmetry even under normal incidence of the hole, i.e., for kx = ky = 0 (Sect. 2.1.2). The notation used in writing the boundary conditions is m, n = ±3/2, ±1/2,

±3/2 = a0	m0 d			, ±1/2 = a0	m0 d			,
	mhh		dz		mlh		dz

mhh, mlh are the heavyand light-hole e ective masses, which are di erent in the CA and C A materials (here and subsequently the quantities pertaining to the C A material are primed), Jα are the angular-momentum matrices for J = 3/2 in the Γ8 basis (Table 2.1), and tl-h is a dimensionless parameter of heavy-light–hole mixing introduced in (2.42). For tl-h = 0, the pair of the Kramers-conjugate states at the bottom of the hh1 hole subband contains an admixture of m = ±1/2 states:

(hh1)	= F (z) |Γ8, ±3/2 ± iG(z) |Γ8, 1/2 ,	(3.116)
ψ±3/2

where |Γ8, m are the Bloch functions. Note that the second of the boundary conditions (3.115) for function G(z) has the form

1 dG			=	1	dG		+	tl-h	F .	(3.117)
mlh dz				m
						dz		a0m0
				lh

Within the QW the real envelope functions F (z) and G(z) can be written as

F (z) = A cos khz + B sin khz ,	(3.118)
G(z) = C cos klz + D sin klz ,

and in the barrier layers, z > a/2 and z < −a/2, they decay exponentially

F (z) = F (±a/2)	exp [−æh(|z| − a/2)] ,	(3.119)
G(z) = G(±a/2)	exp [−æl(|z| − a/2)] .

Here A, B, C, D are z-independent coe cients, the point z = 0 is chosen at the well center,

kh = (2mhhε/ 2)1/2, kl = (2mlhε/ 2)1/2 = (mlh/mhh)1/2kh ,

æ

h

= [2m

(V

−

ε)/ 2]1/2, æ

l

= (m

/m

)1/2æ

h

,

(3.120)

hh

lh

hh

126 3 Resonant Light Reflection and Transmission

ε is the hole energy, and V is the barrier height, i.e., the valence-band o set at the interface. Note that the barrier height for conduction-band electrons is ∆Eg −V , where ∆Eg is the di erence between the band-gap widths in the compositional materials. In a common-anion structure the mixing coe cients tLl-h and tRl-h at the leftand right-hand interfaces, respectively, coincide, and therefore coe cients B and C in (3.118) vanish identically, and F (a/2) =

F (−a/2), G(a/2) = −G(−a/2). For tLl-h = tRl-h, the F (z), G(z) functions do not possess definite parity under sign inversion of z.

Å

Å

Fig. 3.6. Relative anisotropy of the absorption coe cient in a periodic QW structure Ga0.47In0.53As/InP vs. dimensionless heavy-light-hole mixing coe cient tRl-h at the right-hand interface calculated for fixed values of tLl-h. The solid lines are

˚

calculated for wells with thickness a = 100 A, and the dashed line, for a = 70

˚

A. [3.45]

According to (3.113, 3.116, 3.118) and Tables 2.2, 2.3, the interband optical transitions allowed under normal incidence of the linearly polarized

3.1 Optical Reflection from Quantum Wells and Superlattices

127

light, e z, involve excitation of electron-hole pairs (e1, −1/2; hh1, 3/2) and (e1, 1/2; hh1, −3/2). The transition rates are proportional to [3.45]

|M−1/2,3/2(e)|2 = |M1/2,−3/2(e)|2				(3.121)
= M02	I12 + 3 I22	+ √3 I1I2 cos 2φ		,
	1	2

where M0 is a constant, φ is the angle between the plane of polarization and the [110] axis, and

I1 = ϕe1(z)F (z)dz ,

I2 = ϕe1(z)G(z)dz .

In accordance with symmetry considerations, light absorption passes through

¯

extrema at φ = 0 (e [110]) and φ = π/2 (e [110]). According to (3.121), the absorption anisotropy in a periodic quantum-well structure is described by the relation

K[110] − K[110]¯

2

I1I2

2

I2

ρ ≡

K

[110]

+ K ¯

= √

I2

+ (I2

/3)

≈ √

I

1

,

(3.122)

[110]

3

1

2

3

where the definition of ρ di ers from (3.99) by a factor of 2. In a QW grown of materials with no-common cations and anions, and with no electric field present, the optical anisotropy is dominated by the di erence between heavy-light–hole mixing coe cients tl-h for the CA/C A and C A /CA heteroboundaries. It is assumed that the potential barriers at the leftand right-hand interfaces are the same. Figure 3.6 plots the absorption-coe cient anisotropy vs mixing parameters for the left-hand (tLl-h) and right-hand (tRl-h) interfaces. The calculation was done using (3.122) for Ga0.47In0.53As/InP

˚	˚
QWs with thicknesses of 100 A	(solid lines) and 70 A (dashed line). The

main parameters of the materials are listed in Table 3.1. The band o sets used are ∆Ec = 0.262 eV and ∆Ev = 0.348 eV for the conduction and valence bands, respectively. For a fixed value of tLl-h, the tRl-h dependencies presented in Fig. 3.6 reverse sign at tLl-h = tRl-h, which corresponds to a symmetric QW. A decrease in well thickness enhances the influence of interfaces and, hence, makes a steeper ρ(tRl-h) dependence for a narrower QW.

Table 3.1. The band parameters of Ga0.47In0.53As and InP used in the calculation of curves in Fig. 3.6.

	Eg (eV)	me(m0)	mhh(m0)	mlh(m0)
GaInAs	0.81	0.041	0.377	0.052
InP	1.42	0.077	0.65	0.12

It was shown in the previous subsection that the in-plane anisotropy appeared due to the heavy-light hole√mixing in (hhl)-grown QWs if the Luttinger band parameters γ2, γ3 (or 3B, D) are di erent. We see now that,

128 3 Resonant Light Reflection and Transmission

in no-common-atom (001)-grown QW structures with interfaces of di erent kinds, the in-plane anisotropy originates from heavy-light hole mixing taking place at the interfaces. The e ect of an electric field on the anisotropy is considered in Sect. 3.4.3.

3.2 Reflection and Di raction of Light from Arrays of Quantum Wires and Dots

In the previous sections we could trace the modification of the excitonpolariton concept in long-period MQW structures. Here we outline the framework for similar analysis of structures containing regular arrays of QWRs and QDs. The shift from MQWs to 3D lattices of QDs or QWRs allows to bridge the gap between multilayered structures and photonic crystals. The latter are defined as periodic dielectric structures with the period being comparable to the wavelength of the visible-range electromagnetic waves. In the simplest realization, a photonic crystal is thought of as a periodic lattice of spheres of dielectric constant æa embedded in a uniform dielectric background æb (see reviews [3.46, 3.47]). Other potential realizations are a 3D lattice of resonant two-level atoms [3.48] or semiconductor microcrystals embedded into the pores of periodic porous materials [3.49] (see also [3.50]). We pay more attention to the dots and present short information containing the wires because the both procedures are quite similar.

3.2.1 Rayleigh Scattering of Light by a Single Quantum Wire or Dot

We start with a single QD imbedded in an infinite barrier material of the dielectric constant æb. The excitonic states in the QD are 0D, or quasi-0D, due to the quantum-confinement e ect. We consider a narrow frequency region near a particular exciton size-quantization level and solve the scattering problem of an incident electromagnetic wave on this QD. In the resonant frequency region the dielectric response to an electromagnetic wave is nonlocal and our goal is to show how the theory makes allowance for such kind of nonlocality and give the corrections to the line position and natural linewidth similar to the values ω˜0 and Γ0 in (3.18).

We use the Maxwell equations

∆E − grad divE = −		ω		2
				D ,
		c
divD = 0				(3.123)

for the electric field E and the displacement vector D. In linear response the relation between D and E reads [3.51]

D(r) = æbE(r) + 4πPexc(r) ,

(3.124)

3.2 Reflection and Di raction of Light from Arrays of Quantum Wires and Dots			129
4πPexc(r) = p G(ω) Φ(r) , p =	Φ(r ) E(r ) dr .	(3.125)

Here G(ω) is the same one-pole function as that in (3.11) with ω0 being the bare QD-exciton resonance frequency, æb is the background dielectric constant, Φ(r) denotes the envelope function Ψexc(re, rh) of an exciton excited in the QD at coinciding electron and hole coordinates: Φ(r) = Ψexc(r, r). In addition, we neglect anisotropy of the bulk-exciton longitudinal-transverse splitting and consider, in fact, the case of the Γ6 ×Γ7 exciton. For simplicity, the QD background dielectric constant æa is assumed to coincide with æb. Clearly, the length scale of non-locality is set by the exciton envelope function which is of the same order as the QD size.

It follows from (3.124) that div E = −(4π/æb) div Pexc which allows to rewrite the first equation (3.123) as

∆E(r) + q2E(r) = −4πq02 1 + q−2 grad div Pexc(r) .

(3.126)

We remind that q0 = ω/c, q = q0nb and nb = √æb.

Instead of the 1D Green function used for QW structures we use now the

3D Green function
G	3D	(r	−	r ) =	exp (iq|r − r |)					,	(3.127)
					4π	r	−	r	|
					|
satisfying the di erential equation
(∆ + q2)G3D(r − r ) = −δ(r − r ) .											(3.128)

Green’s function allows to express E(r) via the excitonic polarization as

E(r) = E0eiqr + 4πq02		dr G3D(r − r )	1 + q−2 grad div	Pexc(r ) ,
				(3.129)

where q and E0 are the wave vector and amplitude of the initial light wave. Integration by parts allows to rewrite (3.129) as

Eα(r) = E0,αeiqr + 4πq02

dr

β

Gαβ (r − r )Pexc,β (r ) ,

(3.130)

where we introduced the matrix Green function [3.52]

Gαβ (r) = δαβ +

1

∂2

eiqr

(3.131)

q2

∂rα∂rβ

4πr

1

eiqr

2

rαrβ

1

i

1

= −

δαβ δ(r) +

δαβ + 3

− δαβ

−

−

.

3q2

4πr

3

r2

(qr)2

qr

3

Using (3.125, 3.130) we obtain

130 3 Resonant Light Reflection and Transmission

pα = p0α + Tαβ pβ , p0 = E0 drΦ(r) , (3.132)

Tαβ = q02G(ω) drdr Gαβ (r − r )Φ(r)Φ(r ) .

Equation (3.132), together with (3.130, 3.131), completes the solution of the electromagnetic scattering problem.

ˆ

For spherical or cubic QDs, the matrix T is diagonal and isotropic, Tαβ = T δαβ , with

T (ω) =

−

δω0 − iΓSQD

,

(3.133)

ω0 − ω − iΓ

δω

=

π

a3

ω

LT

Φ2(r) dr

−

q2

drdr cos q r − r |

Φ(r)Φ(r ) ,

0

3 B

2π

|r |− r |

(3.134)

1

sin q|r − r |

Γ

=

q2a3

ω

LT

drdr

Φ(r)Φ(r ) .

(3.135)

SQD

6

B

|r − r |

Here δω0 and ΓSQD are the renormalization of the resonance frequency and radiative damping of the 0D exciton. We return to their interpretation in Chap. 5 while considering the fine structure of exciton levels in nanostructures.

The above procedure can be repeated for the elastic scattering of light by a QWR. In this case one can use the 2D Green function G2D(ρ) = (i/4)H(1)0 (qρ)

satisfying the equation
	∂2		∂2
		+		+ q2 G2D (x, y) = −δ(x)δ(y) ,	(3.136)
	∂x2		∂y2

where H(1)0 (z) is a Hankel function. Under normal incidence of the monochromatic electromagnetic wave upon a single QWR the electric field is written similarly to (3.125, 3.130) as

Eα(ρ) = E0,αeiqρ+

			1			(3.137)
+q02G(ω) dρ Φ(ρ )	β	δαβ +			∂2	G2D(ρ−ρ ) dρ Φ(ρ )Eβ (ρ ) ,
			q2	∂rα∂rβ

where Φ(ρ) is the 1D-exciton envelope function at coinciding coordinates of the electron and hole.

3.2.2 Periodic Arrays of Isolated Quantum Dots

Let us turn to an ensemble of QDs and consider first the photonic (or more precisely, exciton-polaritonic) band structure of a 3D periodic array of QDs (or, simply, QD lattices) and then, in the next subsection, the light reflection from and transmission through a planar quadratic QD lattice.