ivchenko_bookreg

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

141

c(ν)2	= 1 .	(3.185)
0

ν

In the method of plane waves defined by (3.173) the Schr¨odinger equation reads

		2β2			+	β	cβ	b β Vbb = 0 ,	(3.186)
		2M − E cβ
Vbb = a2	Ω0		V (x, y) cos		a [(l − l)x + (m − m)y] dxdy .
1						2π

Here E is the exciton energy referred to the bottom of the exciton band in an ideal QW, b = (2π/a)(l , m ) and we remind that β2 = (2π/a)2(l2 + m2).

As an example, we define the lateral potential as a periodic array of disks, namely,

			n1 2				− n1a, y − n2a) ,				(3.187)
	V (x, y) =				v(x
		≡		n
							0	, if ρ > R ,
v(x, y)			v(ρ) =				−v0 , if ρ ≤ R
where n1, n2 are integers and ρ =									. Then one has
						x2 + y2
	v0 R J1		2π							R/a ,
Vlm,l m =					(l − l)2			+ (m − m)2			(3.188)
−	a						2	2

(l − l) + (m − m)

where J1(t) is the Bessel function. The result of calculation is convenient to present in the dimensionless variables

E

v0

R

2

2π

2

ε =

, u0 =

, µ =

,

where E0 =

.

(3.189)

E0

E0

a

2M

R

Figures 3.7 and 3.8 represent calculations of Eν and fν as functions of u0 and R/a for the four lowest excitonic states. The sum of fν over ν = 1-4 is shown by curves 5 in Figs. 3.7b and 3.8b. Since this sum is close to unity in the whole range of u0 up to 1.2 and of R/a from 0 up to 0.5, we conclude that the oscillator strengths for excitons with ν > 4 is negligible. The dotted curves in Fig. 3.7 and 3.8 are calculated in the tight-binding approximation while the dashed curves illustrate calculation in the approximation of almost free excitons.

In an ideal QW, i.e. for u0 = 0, the dimensionless energy of the exciton ν

˜

equals µ2β2 and possesses the values 0, µ2, 2µ2, 4µ2... , the optically active is only the exciton ν = 1. This agrees with the behavior of curves in Fig. 3.7 as u0 tends to zero. For u0 = 0, the lateral potential mixes the space harmonics belonging to di erent stars. As a consequence the states ν = 2-4 become optically active. Because of the sum rule (3.185) the oscillator strength is redistributed from the exciton ν = 1 to other excitons. One can see from

142 3 Resonant Light Reflection and Transmission

ν

Fig. 3.7. The energy (a) and the oscillator strength (b) vs. the relative depth, u0, of the potential disc for four lower exciton states in a 2D SL. The index ν = 1 − 4 enumerates the optically-active exciton Bloch states with kx = ky = 0. The calculation is performed for the lateral array of quantum discs of the radius R = a/4. Solid curves, the exact calculation in the method of plane waves; dotted curves, calculation of E1 (a) and f1 (b) in the tight-binding approximation. Dashed curves 1 and 2 in (b) are obtained in the approximation of almost free excitons. Curve 5 in (b) represents the sum of fν over the four lowest energy states. [3.58]

ν

Fig. 3.8. Dependence of energy Eν (a) and the oscillator strength fν (b) on the ratio R/a for the exciton states ν = 1 − 4 in a 2D SL with an array of quantum discs. The calculation is performed for the relative depth u0 = 0.5. The index ν enumerates the exciton Bloch states with kx = ky = 0; curve 5 in (b) represents the sum of fν over the four lowest energy states. Dotted curves 1 and 2 in (b) are obtained by using the tight-binding approximation. [3.58]

Fig. 3.7b that up to u0 = 0.25 this redistribution occurs only in favor of the state ν = 2. This phenomenon can be interpreted as anticrossing of the states 1 and 2 with increasing u0. For large enough periods, the exciton states ν = 1, 2 with negative values of E are bound-like, they can be approximated by the tight-binding functions

ψν (x, y) = ϕν (x − la, y − ma) ,

n1n2

3.3 Electro-Optics of Nanostructures

143

where ϕν (x, y) are the normalized excitonic functions localized at a single potential V (ρ) and characterized by the uniaxial symmetry. Another important point to be mentioned is that for large a, i.e. for small ratios R/a, the oscillator strength for the exciton ν = 3 is prevailing or, in other words, the state ν = 3 is free-like, it is close to the free exciton state in an ideal QW which is described by (3.173) with c0 = 1, clm = 0 if l = 0 or/and m = 0 and, therefore, f = 1. With increasing R/a the oscillator strength is redistributed in favor of the bound-like states ν = 1 and ν = 2. This redistribution can be used for a qualitative analysis of “stealing oscillator strength” from neutral excitons X to charged excitons X− mentioned by Kheng et al. [2.129] (as far as we ignore that the electrons filling the conduction band are not distributed periodically in the interface plane). Anyway the lateral SL allows to analyze a co-existence of boundand free-like exciton states for di erent values of the disk potential depth and ratio R/a.

3.3 Electro-Optics of Nanostructures

Here we briefly discuss e ects of an applied electric field F on optical properties of nanostructures. The e ect of an in-plane electric field, F (x, y) z, on free carriers and excitons in QW structures is similar to that in a bulk semiconductor, namely, free carriers contribute to the dc electric current and the bound excitonic state becomes nonstationary in moderate electric fields. In order to estimate the corresponding value of F we take into account that, in an electric field, the potential energy of an electron-hole pair changes by

−e F · (ρe − ρh) ,

(3.190)

where ρe,h is the electron or hole in-plane position. Now either of the particles (primarily that with the lower mass) can tunnel through the Coulomb barrier. For 1s-exciton the height of the barrier is the exciton Rydberg EB and its width is estimated as ∆x (EB/|e|F ). One can therefore write for the halfwidth Γ of the exciton bound state

− ln Γ	∆x		EB
				.
	aB		|e|F aB

The broadening of the exciton absorption peak determined by Γ occurs in fields of F (EB/|e|aB) (103 −104) V/cm where no noticeable shift of the absorption peak is observed to occur.

In a field applied perpendicularly to interfaces, F (x, y) z, the electron transport in a SQW structure is suppressed at all. In a periodic MQW structure with thick enough barriers transport along the normal to the layers can occurs due to incoherent hopping from one well to another and, thus, is rather hindered. In this case the height of the barrier for the tunnelling dissociation of excitons equals the potential barrier V . This keeps

144 3 Resonant Light Reflection and Transmission

the excitonic states well defined up to F (V /|e|a) 105 V/cm although the exciton level is meanwhile red-shifted by values comparable and even exceeding the 2D-exciton Rydberg. This is the so-called quantum-confined Stark e ect, see the next subsection. As for SLs with the 3D character of charge-carrier motion, the increasing electric field leads to the free-carrier localization along the growth direction of the SL (Sect. 3.4.2). Sect. 3.4.3 is devoted to modification of the Pockels e ect due to the quantum-confinement and interface-induced in-plane anisotropy of heterostructures.

3.3.1 Quantum-Confined Stark E ect

The electric-field induced change of the 2D-exciton excitation energy consists of three contributions

∆Eexc = δEe1 + δEh1 − δε ,

(3.191)

where δEe1,h1 is the change of single-particle confinement energy and δε is the change of the exciton binding energy. The Stark shift (3.191) mainly comes from the first two contributions.

At low fields, |eF |a ( 2/2m )(π/a)2, the electron (or hole) level shift δEν is quadratic in the field F ≡ Fz . It can be found from the second-order perturbation theory. For the lowest level ν = 1 one obtains

	(eF zν1)2
δE1 = − ν=1			,	(3.192)
		Eν − E1

where zν1 is the matrix element of the coordinate z and Eν are the energies of unperturbed states. In symmetric QWs they have a definite parity and therefore only even ν contribute to the sum (3.192). Moreover, estimations show that the term with ν = 2 is determinative and

δE1 ≈ −

(eF z21)2

.

E2 − E1

For infinitely high barriers z21 =

(4/3π)2a,

E2 − E1 = 3E1 and E1 =

( 2/2m )(π/a)2 and we obtain the estimation

eF a)2

1

4

4

δE1 ≈ −C

(

,

C =

.

E1

3

3π

The factor C = 0.0108295... slightly di ers from the exact result

15

1

C =

− 1

= 0.0108152...

π2

48

obtained in the limit of infinitely high barriers, Ve,h → ∞.

3.3 Electro-Optics of Nanostructures

145

For arbitrary values of the electric field one must solve the following

Schr¨odinger equation
−2me,h	dz2 + V (z) ± |e|F z − Ee,h		ϕ(z) = 0 ,	(3.193)
2		d2

where the sign ± and indices e, h correspond to an electron and a hole, respectively, and the Stark shifts δEe1, δEh1 are given by the di erences between Ee,h and Ee1, Eh1. Airy’s functions Ai(X) and Bi(X) are two linearly independent solutions of this equations. Thus, inside the QW its general solution has the form

2m e

F

1/3

±z − |e|F

ϕ(z) = c1Ai(X) + c2Bi(X) , X =

. (3.194)

2| |

e,h

Ee,h

Note that Airy’s function Ai(X) is defined as

1

0

∞

u3

Ai(X) =

cos (uX +

) du .

π

3

The asymptotic representations of the two functions are as follows

Ai(X → +∞) → 2√πX1/4 exp

−3 X3/2

,

1

2

Bi(X → +∞) → 2√πX1/4 exp

3 X3/2

1

2

and

3 |X|3/2

+

4 ,

Ai(X → −∞) → √π|X|1/4 sin

1

2

π

Bi(X → −∞) → √π|X|1/4 cos

3 |X|3/2

+

4 .

1

2

π

For a high electric field the lower levels in a QW with infinitely high barriers are determined by the transcendental equation

Ai −

2m

eF

1/3

a

E

= 0

|

|

+

(3.195)

2

2

|eF |

or

1/3

Ee,h = −2

|eF |a + µ

2me,h

,

1

(eF )2

where µ ≈ 2.338; 4.088; 5.520... are the roots of equation Ai(−µ) = 0. The variational function, ϕ = 2α3/2ze−αz , used frequently in the case of a triangle

146 3 Resonant Light Reflection and Transmission

potential yields for the lowest-level energy the coe cient µ = 2.47, which di ers from the exact value by less than 6%.

The equations presented for the limit Ve,h → ∞ overestimate the Stark e ect shifts. In a real system, e.g., in a GaAs/Al0.35Ga0.65As QW, the quantum-confined Stark e ect shifts the lowest electron and hole states in

5 ˚

the field F = 10 V/cm by 6 and 15 meV, for a 100 A-thick QW, and by 55

˚

and 81 meV for a = 250 A [3.59].

Fig. 3.9. Schematic representation of electron (above) and hole (below) wave functions for the lowest confined states in the absence of (F = 0) and with (F = 0) electric field parallel to z.

At the same time, the envelope wave functions ϕe,h(z) undergo deformation, as is shown schematically in Fig. 3.9. In low fields, |eF |a E1, the level shift is quadratic in F while the shift of the center of mass z¯e,h F . The existence of barriers prevents the breakup of the exciton. As the field increases, the electron and the hole are pressed towards to the opposite interfaces and both δE1 and z¯ saturate. While the electron and hole binding energy in the exciton is decreased, its change δε is small compared to the sum of single-particle shifts, δEe1 + δEh1. In contrast to the binding energy of the exciton, its oscillator strength does not saturate with increasing the electric field because the overlap integral ϕe(z)ϕh(z)dz vanishes in the limit of very

3.3 Electro-Optics of Nanostructures

147

strong electric fields. The oscillator strength of the e1-h1 exciton decreases monotonously. On the other hand, the vν → cν interband transitions, forbidden in the absence of an electric field by parity conservation, become allowed. In weak fields, the oscillator strength is proportional to F 2 for the transitions v2 → e1 and v1 → e2.

3.3.2 Stark Ladder in a Superlattice

In 1960, Wannier [3.60] predicted that when subjected to a uniform electric field the continuous energy bands in perfect crystals would pass into discrete, evenly spaced subbands forming a ladder. Since then this phenomenon is known as the Wannier-Stark localization, and the ladder-like spectrum is called the Stark (or Wannier-Stark) ladder. However, the field-induced localization could be reliably observed and investigated only in semiconductor SLs with a period exceeding by far the microscopic lattice constant.

The formation of a Stark ladder in a SL is easier perceived in the framework of tight-binding description valid when the overlap of the single-QW wave functions in the neighboring wells is small and the electron wave function in the SL can be written in the form

1
ϕ(z) =	√		exp (iK · ρ) Cnϕn(z) .	(3.196)
		N S
			n

Here K is the in-plane electron wave vector, d is the period of the SL, n labels the wells, ϕn(z) = ϕ(z − nd), ϕ(z) is the envelope function of the quantum-confined state eν in a single QW centered at z = 0 (in the following ν = 1), S is the in-plane area and N is the total number of periods in the regular structure. The tight-binding coe cients satisfy the following set of linear equations

ICn−1 + E0Cn + ICn+1 = ECn ,

(3.197)

where E is the electron energy, E0 ≡ Ee1 is the quantum-confinement energy in a single QW, and I is the transfer integral between nearest neighbors. If we disregard the di erence of electron e ective masses in the well and barrier materials it has the form

	a/2
I = −V		ϕ(z)ϕ(z ± d)dz	(3.198)

−a/2

and is negative for the ν = 1 state. For the Bloch solutions in an unperturbed SL the coe cients Cn are given by Cn = exp (iKz dn) and the electron miniband has a cos-like dispersion

E(Kz ) = E0 + 2I cos Kz d .

(3.199)

148 3 Resonant Light Reflection and Transmission

Thus, in the tight-binding approximation the miniband width ∆ is equal to 4|I|. This approximation is applicable if the width is small compared to the energy di erence between the subbands ν = 2 and ν = 1. For small Kz d we can expand the cosine function and approximate E(Kz ) by E0 + ( 2Kz2/2M ) where E0 = E0 − 2|I| and the normal e ective mass M ≡ Mzz is connected with I by

M =	2	(3.200)
	2|I|d2 .

The application of the electric field F z contributes a linear term |e|F z to the electron Hamiltonian. In terms of the tight-binding description this means that the diagonal single-particle energy E0 is replaced by E0 + |e|F dn and (3.197) transform to

I(Cn−1 + Cn+1) + (E0 + |e|F dn − E)Cn = 0 ,				(3.201)
or
−2I	(Cn−1 + Cn+1) = 2(n +	E0 − E	)Cn ,
|e|F d
		|e|F d

or, in the dimensionless variables,

x(Cn−1 + Cn+1) = 2(n − λ)Cn ,

where x = 2|I|/(|e|F d) , λ = (E − E0)/|e|F d. Let us recall that the Bessel functions satisfy the recurrent relations

x[yµ−1(x) + yµ+1(x)] = 2 µ yµ(x) .

(3.202)

Hence, the solution for Cn can be presented as [3.61]

Cn = D1Jn−λ(x) + D2Nn−λ(x) .

(3.203)

Since the Neumann function Nµ(x) tends to ∞ for µ → ∞, we have to set D2 = 0. The Bessel function Jn−λ(x) is limited for n → −∞ if λ is an integer. Denoting this integer by n0 and the corresponding electron energy by En0 we obtain

Cn Jn−n0	|e||F|d and En0 = E0 − eF dn0 .		(3.204)
		2 I

Thus, each electron miniband in the SL splits into a series of subbands separated by |eF |d. The spread of the electron wave function along the growth direction is governed by the parameter

NF =	∆	(3.205)
	|eF |d .

The quantum number n0 indicates the QW where the probability to find an electron in the state (3.204) is the highest. As one moves away from this

3.3 Electro-Optics of Nanostructures

149

well the probability decreases but remains non-negligible within NF QWs, if NF 1, or within a single n0-th well, if NF < 1 (strong-localization limit). Another important parameter, NF , is the ratio between the spacing |eF |d and the energy uncertainty /τp, where τp is the electron relaxation time for available scattering mechanisms. In a SL, by definition, the free-path or coherence length l = vτp exceeds the period d. At small fields, NF 1, the Stark ladder can be ignored and the electric field causes the conventional dissipative transport of miniband electrons. In the opposite limit, NF 1, the Stark-ladder levels are well defined and can be observed in the optical

spectra.					˚	˚
In a GaAs/AlxGa1−xAs SL, for a = 40 A, b = 20 A, x = 0.35 the width
of the lowest electron	miniband ∆				0.07 meV, and the parameter t is of the
		5
order of unity at F		10	V/cm. Figure 3.10 presents experimental spectra

of the photocurrent induced in a GaAs/AlGaAs SL at various electric fields. Being proportional to the absorption spectra, the photocurrent spectra show strong oscillations periodic in the reciprocal of the electric field. Therefore the absorption spectra reveal not only one intra-well transition but rather a

series of transitions		− 0	| |
0	= E00 + (n0
En0n		n	) e F d	(3.206)

between conductionand valence-band Wannier-Stark states of indices n0 and n0, respectively. The interband optical transitions are illustrated in Fig. 3.11. At zero or small fields, NF 1, the absorption edges correspond to the di erence between the bottom of the conduction miniband and the top of the valence minibands, see the transitions T1 and T2 involving heavyand light-hole delocalized states. At moderate electric fields where NF 1 but yet NF 1, see sketch in Fig. 3.11b, the electronic wave functions are localized but extend over NF periods. Usually, in this regime the heavy holes are practically localized in one well. It follows then from (3.204) that the oscillator strength for the optical transition hh1, n0 → e1, n0 is proportional to J2n0−n0 (NF /2) and is remarkable up to |n0 − n0|d = L where L is the Wannier-Stark localization length

L = NF d =	∆	(3.207)
	|eF | .

In a high electric field the electron states in neighboring wells are so widely separated in energy (NF 1) that the electron is localized within one QW, the coe cients Cn for n = n0 are negligible, overlap integrals for the n0 = n0 interband transitions become very small, and only the intrawell transitions can be observed in the photocurrent spectrum, see the lower curve in Fig. 3.10 and Fig. 3.11c. Note that for very high fields the interminiband Zener tunneling becomes appreciable and gives rise to an additional broadening of the Wannier-Stark levels.

The Wannier-Stark localization is a phenomenon interpreted in terms of the energy levels and the electron packet spreading in real space. The

150 3 Resonant Light Reflection and Transmission

Fig. 3.10. Low-temperature spectra in the GaAs/Al0.35Ga0.65As SL for di erent electric fields F z. For convenience, the spectra are displaced vertically with respect to one another. The numbers refer to the peaks corresponding to hhn → e1n transitions (n − n = 0, ±1, ±2...) in the moderate fields. Analogous transitions from light holes are denoted by 0l and −1l. From [3.62].