ivchenko_bookreg

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

131

In the following the centers of QDs in the 3D array are indicated by the translation vectors a. The nonlocal material equation relating Pexc and E is obtained if we present the excitonic contribution to the dielectric polarization as a sum of individual contributions (3.125) over a,

4πPexc(r) = G(ω)	a	pa Φa(r) , pa =	Φa(r ) E(r ) dr , (3.138)

where Φa(r) = Φ(r − a). In this subsection we neglect the overlap of exciton envelope functions Ψa and Ψa with a = a so that excitons excited in di erent dots are assumed to be coupled only via the electromagnetic field. We seek for Bloch-like solutions of (3.126) satisfying the translational symmetry

EK (r + a) = exp (iKa) EK (r) ,

(3.139)

Pexc,K (r + a) = exp (iKa) Pexc,K(r) ,

where the wave vector K is defined within the first Brillouin zone. The exciton-polariton dispersion ω(K) can be shown to satisfy the equation

Det||δαβ − Rαβ (ω, K)|| =

0 ,

(3.140)

where α, β = x, y, z, δαβ is the Kronecker symbol and, for QD lattices,

Rαβ (ω, K) = G(ω)

q02

|IK+g |2 Sαβ (K + g)

,

(3.141)

v0

(K + g)2

−

q2

IQ =

g

Φ(r)eiQr dr , Sαβ (Q) = δαβ −

q2

,

(3.142)

QαQβ

g are the reciprocal lattice vectors and v0 is the volume of the lattice primitive cell.

Equations (3.140, 3.141) can be derived by using the two equivalent approaches: (i) to express the exciton dielectric polarization Pexc(r) in terms of the electric field, E(r), and find solutions of the wave equation for E(r); (ii) by using Green’s function of the wave equation, to express the electric field in terms of the exciton polarization and write a system of self-consistent equations describing electric-field-mediated coupling between the excitons excited in di erent QDs. In the first approach, we substitute (3.138) into (3.126) and

expand the vector function EK (r) in the Fourier series as follows
EK (r) =	ei(K+g)r EK+g .	(3.143)
	g
The integral (3.138) can be transformed into
pa = eiKa	IK+g EK+g ≡ eiKa Λ .	(3.144)

g

132 3 Resonant Light Reflection and Transmission

The sum a Φa(r)eiKa satisfies the translational symmetry similar to (3.139) and can be presented as

Φa(r) eiKa =	ei(K+g)r	I		(3.145)
		K+g .
a	g	v0

The system of linear equations for the space harmonics EK+g can be written in the form

2

2

2

I

K+g ˆ

[(K + g)

− q

] EK+g = G(ω) q0

S(K + g) Λ ,

(3.146)

v0

ˆ

where the vector Λ is introduced in (3.144) and S(Q)Λ is a vector with

the components S

2

αβ (Q)Λβ . Dividing both parts of (3.146) by (K + g) −

2

q

, multiplying them by IK+g and summing over g we arrive at the vector

ˆ

ˆ

equation Λ = R(ω, K)Λ, where the matrix R is defined by (3.141), and hence

at the dispersion equation (3.140).

In the second approach, we use the 3D Green function (3.127) rewriting

it as

1

exp [iQ(r − r )]

G

(r

r ) =

,

(3.147)

3D

−

V

Q2

−

q2

Q

where V is the 3D QD-lattice volume. Green’s function allows to express the electric field E(r) of the normal light wave via the polarization in the form of (3.130) with the initial field amplitude set to zero. For the Bloch solutions (3.139) one has pa = eiKap0. Taking a = 0 in (3.144) and using (3.130,3.143) we obtain

			G3D(r − r )
p0 = G(ω)	dr Φ0(r)	dr		(3.148)
× 1 + q−2 grad div		p0	Φa(r ) eiKa .

a

If we now use equations (3.142, 3.145) and the presentation (3.147) of Green’s

ˆ

function we finally come to the equation p0 = R(ω, q)p0 and rederive (3.140). In the particular case of spherical QDs with the radius R exceeding the

Bohr radius aB one has

2R 3/2				sin QR
IQ = π						.	(3.149)
	aB			QR[π2 − (QR)2]
Then (3.141) can be transformed into
				Ω2
Rαβ (Ω, K) = ξ					σαβ (Ω, K) ,		(3.150)
			Ω2 − 1

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

133

f (|K + b|R) Sαβ (K + b)

σαβ (Ω, K) = Ω2 − Ω2(K + b) ,

b

	ω		64 ωLT			R 3			, f (x) =	π2 sin x	2
Ω =		, ξ =									,
	ω0		π ω0				a			x(π2 − x2)

(3.151)

(3.152)

Ω(Q) = cQ/(nbω0). Equation (3.140) is equivalent to the three separate equations Rj (Ω, K) = 1, where Rj (j = 1, 2, 3) are eigenvalues of the matrix Rαβ . The further simplification follows taking into account a small value of the parameter ξ since, in semiconductors, the ratio ωLT/ω0 typically lies between 10−4 and 10−3. Then one can change the factor Ω/(Ω + 1) in (3.150) by 1/2.

For high-symmetry points of the Brillouin zone, the symmetry imposes certain relations between the Rαβ components, and the eigenvalues Rj can be readily expressed via them. Table 3.2 illustrates these relations for the points Γ, X, L, W, K and U of a face-centered-cubic QD lattice. The coordinates of the points in the reciprocal space are given in the first column (in terms of π/a). Note that in this case the lattice constant a and the unit-cell volume v0 are related by v0 = a3/4.

Table 3.2. Nonzero components of the matrix Rαβ and dispersion equations written in terms of Rαβ for di erent K points in the Brillouin zone of a face- centered-cubic QD lattice.

K (π/a)	Nonzero components of Rαβ	Dispersion equations
Γ (0, 0, 0)	Rxx = Ryy = Rzz	Rxx = 1
X (0, 0, 2)	Rxx = Ryy , Rzz	Rxx = 1, Rzz = 1
L (1, 1, 1)	Rαα = Rxx, Rαβ = Rxy (α = β)	Rxx − Rxy = 1,
		Rxx + 2Rxy = 1
W (1, 0, 2)	Rxx, Ryy = Rzz	Rxx = 1, Ryy = 1
K (3/2, 0, 3/2)	Rxx = Rzz , Ryy , Rxz = Rzx	Rxx ± Rxz = 1, Ryy = 1
U (1/2, 1/2, 2)	Rxx = Ryy , Rzz , Rxy = Ryx	Rxx ± Rxy = 1, Rzz = 1

According to (3.141, 3.150) the dispersion near the point K satisfying the condition cK/nb ≈ ω0 should be characterized by a giant anticrossing between the branches of bare transverse photon and exciton modes. Note that the anticrossing can be described with a high accuracy by retaining in the sum over b in (3.151) the two terms due to b = 0, −(4π/a)(0, 0, 1) if the resonant

Bragg condition is satisfied for the (001) planes, i.e., if (2π/a)(c/n ) ≈ ω , or

√ b 0 two terms due to b = 0, −(2π/a)(1, 1, 1) if (π 3/a)(c/nb) ≈ ω0.

3.2.3 Di raction by a Planar Array of Quantum Dots

We assume the array of QDs to be regularly packed in one plane normal to z. For simplicity, we consider here the normal incidence of the light and a quadratic lattice of spherical or cubic QDs with the translational vectors a

134 3 Resonant Light Reflection and Transmission

being integer combinations of the basic vectors a1 = a(1, 0, 0), a2 = a(0, 1, 0). In this case the electric field can be expanded in the 2D Fourier series as

E(r) = Eb(z) exp (ibρ) ,	(3.153)

b

where b = lb1+mb2 are the 2D reciprocal-lattice vectors, b1 = (2π/a)(1, 0, 0), b2 = (2π/a)(0, 1, 0), l and m are integers, a is the constant of the 2D QD quadratic lattice.

The integral in (3.125) can be transformed into

Φa(r) E(r) dr = eiba Eb(z)Φ(ρ, z) exp (ibρ)dρdz (3.154)

b

= ϕb(z)Eb(z) dz ≡ Λ1 ,

b

where ρ = (x, y),

ϕb(z) =

Φ(ρ, z) exp (ibρ)dρ

(3.155)

and we used the identity exp (iba) = 1. We will also use the expansion

Φa(r) =

1

ϕb(z) exp (ibρ) ,

(3.156)

a2

a

b

where a2 is the 2D unit cell area.

The function Eb(z) satisfies the equation

d2

2

q2

1 +

1

0

+ qb

Eb(z) = −G(ω)

grad div

b ϕb(z)Λ1 ,

(3.157)

dz2

a2

q2

where qb =

,

q2 − b2

2

∂

b = −KαKβ , Kx

= bx, Ky = by , Kz = −i

∂

(3.158)

.

∂rα∂rβ

∂z

The solution can be presented as

Eb(z) = E(0)eiqz δb,0

(3.159)

+

iq2

dz eiqb|z−z |

1 + q−2grad div b ϕb(z )Λ1 ,

0

2qba2 G(ω)

where E(0) is the amplitude of the initial wave. Multiplying the both parts of (3.159) by ϕb(z), integrating over z and summing over b we obtain

Λ1 = Λ10						(3.160)
+		iq2			dzdz eiqb|z−z |ϕb(z) 1 + q−2grad div	b ϕb(z )Λ1 ,
	b	0
		2qba2 G(ω)

3.2 Reflection and Di raction of Light from Arrays of Quantum Wires and Dots						135
where
Λ10 = E(0)		ϕ0(z)eiqz dz = E(0)		ϕ0(z) cos qz dz .	(3.161)

Let us denote by β the star of the vector b. If b = lb1 + mb2, the star β contains the vectors ±lb1 ± mb2, ±mb1 ± lb2 of equal absolute values. For l = m = 0 the star consists of eight vectors, otherwise it has four vectors (l = m = 0 or l = 0, m = 0 or l = 0, m = 0) and one vector in the particular case l = m = 0. Then the second term in the right-hand side of (3.160) can

	be rewritten				iq02nβ
	G(ω)		β				dzdz eiqβ |z−z |ϕβ (z)
					2qβ a2
								1 ∂2
	× 1 −	β2			Λ1, + 1 −					ϕβ (z ) ,
									∂z 2 Λ1,
		2q2						q2

where Λ1, , Λ1, are vectors with the components (Λ1,x, Λ1,y , 0) and (0, 0, Λ1,z ),

respectively, nβ

is the number of vectors in the star β and β2 = b 2

, q

β ≡

q

.

0

0

0

, 0) we obtain

| |

b

Taking into account that Λ1 = (Λ1,x

, Λ1,y

iq02nβ

1 −

β2

dzdz e

|ϕβ (z)ϕβ (z )

−1

Λ1

0

− G(ω)

iq

z

−

z

= Λ1 1

β 2qβ a2

2q2

β |

= Λ0

ω0 − ω − iΓ

.

(3.162)

1 ω˜0 − ω − i(Γ + Γ0)

Here ω˜0 is the normalized exciton resonant frequency, the di erence between ω˜0 and ω0 consists of two terms

	q2πa3		nβ	1	−	β2	(3.163)
δω1 = ωLT 2a2B		β B1 qβ				2q2

×dzdz sin(qβ |z − z |) ϕβ (z)ϕβ (z ) ,

	q2πa3		nβ	1	−	β2
δω2 = −ωLT 2a2B		β B2 æβ				2q2

×dzdz e−æβ |z−z |ϕβ (z)ϕβ (z ) ,

B1 and B2 represent the sets of stars β with real and imaginary qβ , respectively, æβ = Im{qβ }. The exciton radiative damping rate is given by

	q2πa3		nβ		1	−	β2		(3.164)
Γ0 = ωLT 2a2B		β B1 qβ					2q2 λβ2 ,

136 3 Resonant Light Reflection and Transmission

λβ =	ϕβ (z) cos qβ z dz = Φ(r) cos (bρ + qβ z) dr .	(3.165)

The light waves di racted in the backward and forward directions are written as

Eb(r) exp [i(bρ − qbz)] and

Eb(t) exp [i(bρ + qbz)] .

(3.166)

b

b

The amplitudes E(r)

, E(t) are given by [3.53]

Eb

b

b

1 −

E

(3.167)

= i 2qba2

ω˜0 − ω − i(Γ + Γ0)

q2

,

(r)

2

3

ωLTλbλ0

(0)

q

πaB

KrKr

Eb

= δb,0 + i 2qba2

ω˜0 − ω − i(Γ + Γ0) 1 −

q2

E ,

(t)

2

3

ωLTλb

λ0

(0)

q

πaB

KtKt

= Ki

j Kj Ej , Kr = (bx, by , −qb), Kt = (bx, by , qb). While

where (KKE)i

deriving (3.167) we

took into account that Λ(0)

= 0 and

1,z

eiqbz (−id/dz)ϕb(z) dz = −qb

eiqbz ϕb(z) dz = −qbλb .

One can check that (3.167) satisfies the energy-flux conservation law. Really, for zero dissipation, i.e. for Γ = 0, we have

qb |Eb(r)|2 + |Eb(t)|2 = q|E(0)|2 .

(3.168)

b B1

For short-period 2D QD lattices, where qa 1 and B1 contains only one element β = 0, we obtain for the reflection and transmission coe cients

	E0(r)		iΓQD		, t =	E0(t)
r =		=					= 1 + r ,	(3.169)
	E(0)		ω˜0 − ω − i(Γ + ΓQD)			E(0)

where the exciton radiative damping, Γ0 ≡ ΓQD, is given by [3.51, 3.54]

ΓQD =	3π	ΓSQD	(3.170)
	(aq)2

and ΓSQD is the exciton damping in a single QD, see (3.135). For qa 1, the latter reduces to

ΓSQD = 6		ωLT (qaB)3		2
				drΦ(r) .
	1

The superradiance factor 3π/(aq)2 substantially enhances the radiative damping rate in dense QD arrays. This factor evidences that identical QDs located at distance a from each other emit coherently the secondary light.

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

137

The dependence of the reflection spectra on the number of equidistant QD planes, N , was studied theoretically in [3.53]. If the period d satisfies the Bragg condition at the exciton resonance frequency ω0, then the halfwidth of the reflection spectrum is almost linearly increasing as a function of N . This is similar to the enhancement by a factor of N of the radiative damping of the superradiant mode in resonant Bragg MQW structures, see (3.92).

The theory presented in the above two subsections can also be used to generalize the theory of resonant di raction of γ-radiation by nuclei from bulk crystals [3.55] to synthesized multilayers like the nuclear multi-

57 ˚ ˚ ˚ ˚ ×

layer [ Fe(22 A)/Sc(11 A)/Fe(22 A)/Sc(11 A)] 25 studied by Chumakov et al. [3.56]. The developed approach takes into account a contribution of only one confined-exciton resonance. This is valid if the separation between the exciton size-quantization levels is much larger than the bulk value of the exciton longitudinal-transverse splitting, ωLT. In the opposite limit of extremely large bulk-exciton translational e ective mass one can use the local material relation D(r) = æ(r, ω)E(r). This was done by Sigalas et al. [3.57] for phonon-polaritons in a 2D lattice consisting of semiconductor cylinders.

Similar formula can be derived for the amplitude reflection coe cient from a grating of QWRs. In particular, for a short-period grating of thin QWRs and the polarization parallel to the wire principal axis, y, the radiative damping ΓQWR is given by [3.51]

1			ΦQWR(ρ) d2ρ	2
ΓQWR = ΓQW				,	(3.171)
	dx		ΦQW(z) dz

where dx is the spacing between QWRs, ΓQW is given by (3.18), ΦQW(z) =

√

SΨexcQW(r, r), ΦQWR(ρ) = Ly ΨexcQWR(r, r), ΨexcQW and ΨexcQWR are the single-

QW and single-QWR exciton envelope functions, Ly is the wire (macroscopic) length. Note that the exciton radiative dampings in a single QWR and in an array of QWRs are related by: ΓQWR = (2/qdx)ΓSQW R.

3.2.4 Two-Dimensional Quantum-Dot Superlattices

In the previous subsection the resonant optical reflection from a lateral array of QDs was calculated neglecting the overlap of the exciton envelope functions excited at di erent dots. Now we extend the theory allowing an exciton to tunnel coherently from one potential minimum to another [3.58].

We consider a QW with a periodic 2D potential V (x, y) = V (x + a, y) = V (x, y + a) acting at an exciton like at a single particle. It has no e ect on the exciton internal state, i.e., x, y are the in-plane coordinates of the exciton center-of-mass. Note that here a is the lateral period, not the QW thickness as in other parts of this book. For simplicity, we assume the potential V (x, y) to have the point symmetry of a quadrate: V (x, y) = V (±x, ±y) = V (y, x). Due to the potential V (x, y) the exciton energy spectrum is transformed from the parabolic dispersion Eexc(kx, ky ) = 2(kx2 + ky2)/(2M ) in an ideal

138 3 Resonant Light Reflection and Transmission

QW with V ≡ 0 into a series of 2D minibranches (M is the exciton in-plane translational e ective mass). The two-particle envelope function is written as

Ψexc(re, rh) = ψ(ρ)F (ρe − ρh)ϕe(ze)ϕh(zh) .

(3.172)

Here the functions ϕe, ϕh describe one-particle quantization of an electron (e) and a hole (h) along the growth axis z, ρe,h is the position of the electron or hole in the interface plane, F describes the relative motion of the electron-hole pair within the exciton in an ideal QW in the absence of an additional lateral potential (in the following the 1s-exciton level is considered), the envelope ψ is a function of the exciton center-of-mass position ρ = (x, y). It should be noted that one can neglect the e ect of the lateral potential V (ρ) on the exciton internal state if the e ective 2D Bohr radius is small in the scale of the potential variation. We consider the lowest electron and hole subbands in a symmetric QW in which case ϕe(ze) and ϕh(zh) are even functions of z if the origin z = 0 is taken at the QW central plane.

Under normal incidence of the light the excitonic states are excited at the Γ point of the 2D Brillouin zone, kx = ky = 0. The envelopes, ψ(ρ), of these states are enumerated by the discrete index ν. They are periodic with the SL period and can be expanded in the Fourier series

ψν (ρ) = a	b	cb	exp (ibρ) , cb	= a	Ω0	ψν (ρ) exp (−ibρ) dρ (3.173)
1		(ν)	(ν)	1

over the space harmonics with the wave vectors equal to the 2D reciprocal vectors b = (2π/a)(l, m) introduced in (3.153). We choose the normalization condition

|ψ|2 dxdy = 1 ,

Ω0

where Ω0 is the unit cell, say the area −a/2 < x, y < a/2. Thus, the expansion coe cients cb satisfy the condition

|c(bν)|2 = 1 . b

Moreover, taking into account that the states with ν = ν are orthogonal we
obtain
	) = δνν ,
cb(ν) cb(ν
b
from which it also follows that
cb(ν) cb(ν) = δbb .		(3.174)
ν

Note that by an appropriate choice of the phase factors the Bloch functions at the Γ -point can be assumed real. Because of the high symmetry of the

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

139

potential V (ρ), the coe cients c(bν) are also real and the asterisk in (3.174) can be omitted.

In bulk semiconductors with the allowed interband optical transitions, only s-excitons, 1s, 2s etc., are allowed in the dipole approximation. Due to a similar reason, under normal incidence of the light upon the 2D lateral SL, the dipole-allowed excitonic states are invariant with respect to any operation from the point group of a quadrate (the representation Γ1). In the following the index ν enumerates only such states. For them the coe cients cb in (3.173) with the reciprocal vectors belonging to the same star β, see the definition of a star in the previous subsection, coincide and the index b in cb can be replaced by β: cb ≡ cβ .

To complete the Maxwell equations we use the material relation (3.124) where the exciton contribution, Pexc, to the dielectric polarization is con-

nected with the total electric field by
4πP	(r) = æ				Ψ ν (r, r)					(3.175)
		ω			−	−	Λ		,
exc	b		LT	B	ω0ν	ω iΓν		ν
				ν

Λν = dz dρ Ψexcν (r , r ) E(r ) .

Ω0

Here ω0ν is the exciton resonance frequency in the state ν, the integration in the plane (x, y) is carried out over the unit cell Ω0. In (3.175) we take into account a set of excitonic states, as distinct from (2.202) where the linear response contains only one pole.

By using the Green function approach we can express the amplitude, Er, of the normally reflected (backscattered) wave via the vectors Λν

Er = iΓ0QW			c0(ν)η0				Λν ,	(3.176)
			−		−
				ω		iΓν
	ν ω0ν

where c(0ν) is the expansion coe cient in (3.173) for b = 0, the set of values Λν satisfies the system of linear equations

ΓQW

[Aνν + i(Bνν

+ Cνν )] .

(3.177)

Λν ω0ν

−

ω

−

iΓν

Λν = Λν0 + i

ν

Here

Λν0 = E0c0(ν)η0 , η0 =

ϕe(z)ϕh(z) cos qz dz ,

(3.178)

ΓQW is the exciton radiative damping in an ideal QW given by (3.18),

Aνν + iBνν = β

B1 nβ qβ 1 − 2q2

cβ cβ

η0

+ iIνβ

,

(3.179)

q

β2

(ν) (ν )

ηβ

2

140 3 Resonant Light Reflection and Transmission

Cνν = − β B2 nβ æβ			1 − 2q2		cβ cβ	)	Jνβ ,	(3.180)
		q		β2	(ν) (ν
ηβ =	ϕe(z)ϕh(z) cos qβ z dz ,

B1, B2 are subsets of the set of stars β introduced in (3.163) and satisfying the condition |b| < q or |b| > q, respectively, nβ is the number of vectors in

the star β, qβ =

, æβ =

,

q2 − β2

β2 − q2

Iνβ = η0−2

dzdz Seh(z, z ) sin qβ |z − z | ,

Jνβ = η0−2

dzdz Seh(z, z )e−æβ |z−z | ,

Seh(z, z ) = ϕe(z)ϕh(z)ϕe(z )ϕh(z ) .

While

deriving

(3.179), (3.180)

we took into

account that the values of

b2

, b2 , b b

y

averaged over the vectors b = (b

x

, b

y

) belonging to the star β

x

y

x

are equal to β2/2, β2/2, 0, respectively.

For a function ψ(ρ) belonging to a basis of any representation di erent from Γ1 the coe cient c(0ν) is zero. Therefore, the reflection of the light wave is indeed mediated only by the invariant excitonic states Γ1. Then the vectors Λν , Λ0ν , Er are parallel to E0 and we can replace the vectors in (3.176)– (3.178) by scalar amplitudes. Then, if we divide both parts of (3.176) by E0, we find the reflection coe cient r = Er/E0.

If the spacing between resonant frequencies of optically active excitons exceeds the exciton damping then the reflection coe cient is a sum of individual terms

rν (ω) =

iΓQWc(ν)2

(3.181)

0

,

ω˜0ν − ω − i(Γν + Γ0ν )

where

2

β2

(ν)2

q

ηβ

1 −

(3.182)

Γ0ν = ΓQW β B1 cβ

nβ qβ

η0

2q2 ,

ω˜0ν − ω0ν = Γ0QW(Bνν + Cνν ) .

(3.183)

In this case Γ0ν is the total radiative damping rate of the exciton ν and ω˜0ν is its renormalized resonant frequency. It is worth to mention that for 2π/a > q the subset B1 consists of one element b = 0 while other reciprocal vectors belong to the subset B2. In this particular case Γ0ν equals to ΓQWc(0ν)2. For the sake of convenience, we define the ν-exciton oscillator strength as

fν	= c(ν)2	(3.184)
	0

because the squared coe cient c(0ν) enters into the numerator in (3.181). The sum of oscillator strengths is conserved because according to (3.174) one has