ivchenko_bookreg

1.3 Nanocrystals and Quantum Dot Structures

11

Fig. 1.8. Geometry of an InAs pyramid on an InAs wetting layer (WL) deposited on a GaAs(001) surface.

of islands, the elastic interaction between them via the strained substrate may be neglected. As the total InAs coverage increases and the system of islands transforms from dilute to dense, the elastic interaction between islands becomes essential, and a quasi-periodic array of self-organized islands with a narrow dot size distribution is formed. Therefore, the dense array of QDs is ordered both in shape, orientation, size and lateral arrangement.

An array of coherently strained InAs islands being covered and buried by the deposited GaAs creates a strain field in the surrounding GaAs matrix. When the next layer of InAs is grown, the modulated strain field pushes In ad-atoms on the surface towards buried islands resulting in vertical stacking of InAs islands in multisheet structures (vertically coupled QDs). With increasing separation layer thickness the vertical correlation of InAs islands is lost because the surface strain field due to the underlying dots becomes too weak to influence the growth kinetics.

Figure 1.9 shows a TEM image of a CdSe nanocrystal in a glassy dielectric matrix. The sample was prepared by di usion-controlled phase decomposition of an oversaturated solid solution. This three-stage technique makes it possible to vary the size of the grown nanocrystals in a controlled manner from some tens to thousands of Angstroms, but they are too small to form continuous bands of electronic state and represent a special class of QDs. Moreover, it is established that CdS and CdSe nanocrystals prepared by this method have nearly spherical form, small dispersion in size and retain the wurtzite crystalline structure.

As predicted in [1.25, 1.26], the cleaved edge overgrowth can be also employed to produce QDs at the juncture of three orthogonal QWs. The growth of the initial QW, the first cleave and overgrowth on the (110) plane used to obtain a T -shaped QWR are followed by the second cleave and overgrowth

12 1 Kingdom of Nanostructures

Fig. 1.9. High-resolution TEM image of a CdSe nanocrystal. The nanocrystal shape is close to spherical, the framework for Cd and Se atoms is a wurtzite lattice. From [1.24].

¯

on top of the (110) plane. Such twofold cleaved edge overgrowth QDs have been realized in the GaAs/AlGaAs system using MBE [1.27].

1.4 Structure of the Book

The book is organized as follows: Chapter 1 which maps out the continuously expanding kingdom of semiconductor nanostructures is followed by Chapter 2 describing methods to calculate electron, exciton, phonon and photon states in various nanostructures. The consideration is based on the envelopefunction approximation which provides a satisfactory accuracy and, on the other hand, allows a transparent physical interpretation. Other chapters are focused on particular fields of optical spectroscopy of nanostructures.

Chapter 3 deals with the reflection, propagation, absorption and transmission of light waves in nanostructures near the fundamental absorption edge. We focus the attention on coupling of excitons with electro-magnetic field and show that the concept of exciton polariton undergoes a substantial

1.4 Structure of the Book

13

modification in nanostructures. We calculate successively the reflection and transmission coe cients from a single quantum well, multiple quantum wells, periodic arrays of quantum dots and two-dimensional superlattices. Two final sections of Chapter 3 are devoted to electroand magneto-optics.

Chapter 4 presents results on intraband optical spectroscopy including intraand intersubband optical transitions in n- and p-doped periodic heterostructures and infrared dielectric response of undoped superlattices.

In Chapter 5 we review di erent aspects of photoluminescence spectroscopy emphasizing the role of excitons in the secondary emission of semiconductor nanostructures. We start with the description of macroand microphotoluminescence spectra of localized excitons in undoped and modulationdoped quantum wells. Then we give a comprehensive introduction into optical spin orientation of free carriers and polarized photoluminescence of excitons under polarized photoexcitation. The description of polarized photoluminescence necessitates a detailed information about the electron and exciton spinrelaxation mechanisms, the g factor of free carriers and the fine structure of excitonic levels in low-dimensional systems. The chapter ends by describing the giant lateral optical anisotropy of type-II heterostructures.

Chapter 6 surveys various mechanisms of light scattering. We shall see that the semiconductor nanostructures o er new possibilities and enrich the Raman e ect by scattering from intersubband excitations, confined and interface optical phonons as well as folded acoustic phonons.

Chapter 7 gives an account of nonlinear optical properties of nanostructures summarizing the studies of two-photon absorption, four-wave mixing and second-harmonic generation. Next, we describe optical properties of quantum microcavities. We discuss the energy dispersion and the Rabi splitting of low-dimensional exciton polaritons and then turn to nonlinear optical response of the microcavities. The chapter concludes with the description of angle-resolved polarization-dependent stimulated polariton-polariton scattering in quantum microcavities.

In Chapter 8 we give a consistent introduction to the physics of photogalvanic e ects in noncentrosymmetric systems and summarize the results of recent theoretical and experimental studies on the circular and linear photogalvanic e ects, the spin-galvanic e ect and the photon drag e ect in twoand one-dimensional structures.

In Conclusion (Chap. 9) we estimate perspective for low-dimensional physics of semiconductors, as a whole, and for the optical spectroscopy, in particular. The tables of characters of irreducible representations for some relevant point groups are presented in Appendix.

2 Quantum Confinement in Low-Dimensional

Systems

So when Joseph came to his brothers,

they ... took him and threw him into the cistern. Now the cistern was empty; there was

no water in it.

Genesis 37: 23,24

Nanostructures such as QWs, SLs, QWRs and QDs today constitute the main platforms for electronic structure engineering, or quantum-mechanical engineering. This calls for an e cient description and prediction of the electron energy spectra. At present di erent sophisticated microscopical approaches and atomistic theories have been proposed and developed to compute the free-carrier states in bulk semiconductors and semiconductor heteronanostructures. They are mostly based on empirical or first-principles pseudopotential and tight-binding methods. However, up to now the latter methods are not omnipotent and all-powerful. As in the past development of the physics of bulk semiconductors, the approximate continuum (not-atomistic) theories, by virtue of their simplicity and ease of interpretation, are by far the most popular methods for calculating the properties of electrons in semiconductor nanostructures. They are in the order (i) the e ective mass approximation in case of a simple electronic band structure, (ii) the e ective Hamiltonian approach for degenerate bands, and (iii) the envelope-function theory in the multi-band k · p models (e.g., the Kane model). In these methods of calculation, first, within each homogeneous layer of a multilayered structure or within a homogeneous region of lower dimensionality in QWR and QD structures, the solution is written as a linear combination of bulk wave functions. Then the sets of envelope functions in di erent regions are matched at the heterointerfaces by applying appropriate boundary conditions. In this chapter we will consider quantum confinement of electronic and excitonic states and modification of vibrational spectra in various nanostructures of di erent dimensionalities. The consideration is based on the approximate continuum methods but, from time to time, if it is pertinent we will refer to theoretical results obtained by using the refined atomistic theories.

16 2 Quantum Confinement in Low-Dimensional Systems

2.1 Charge Carriers in Quantum Wells

2.1.1 Size-Quantization of Electrons with Simple Parabolic Energy Spectrum

Calculation of electron states in semiconductor nanostructures performed in the e ective-mass approximation often looks like practical training in the quantum mechanics. Let us consider an A/B QW structure consisting of the well layer A sandwiched between two barrier half-subspaces B. For example, the materials A and B can be GaAs and AlxGa1−xAs. For a simple isotropic and parabolic conduction band, the electron envelope function is separable and can be written as

ψ(r) =	1		eik ·ρϕ(z) .	(2.1)
	√S

Here z is the structure growth axis, k = (kx, ky ) is the 2D wave vector characterizing the electron’s free motion in the interface plane (x, y) and S is the structure in-plane area. Notice that since translational invariance in the plane (x, y) is preserved, the space-group theorems concerning the k conservation apply to kx and ky but not to the z-component kz . In the e ective-mass method the z-dependent envelope satisfies the following Schr¨odinger equation inside the well

	2 d2ϕ(z)				2k2
−				+		ϕ(z) = Eϕ(z) ,	(2.2)
	2mA		dz2		2mA

where mA is the electron’s e ective mass in the material A. It is instructive to start from the electron states and energy spectrum in an idealized case of infinitely high (impenetrable) barriers. In this approximation, the envelope ϕ(z) vanishes outside the QW and the boundary conditions are given by as

ϕ(±a/2) = 0 ,

(2.3)

where a is the well thickness and, since the origin of the coordinate system is chosen at the QW center, the points ±a/2 are located at the interfaces. The system under consideration is invariant under the mirror reflection z → −z. Therefore the totality of solutions of the Schr¨odinger equation splits into sets of even and odd functions of z, which can be written as C cos kz and C sin kz, respectively, where C is the normalization factor. Taking into account the boundary conditions (2.3) we obtain that the electron wavevector z-component, kz ≡ k, is size-quantized and the allowed values of k and E are as follows

k =	νπ	, E(k ) =	2		νπ		2	,
							+ k2		(2.4)
	a		2mA		a

where ν = 1, 3, ...2n + 1... for even solutions and ν = 2, 4...2n... for odd solutions. The corresponding electron quantum-confined states are labelled as eν.

2.1 Charge Carriers in Quantum Wells

17

The energy spectrum E(k ) consists of the parabolic branches Eeνk , called subbands, which are shifted vertically with respect to each other. The electron total energy is the sum of the confinement energy Ez = ( 2/2mA)(νπ/a)2 and the kinetic energy Exy = ( 2/2mA)(kx2 + ky2) of the electron free motion in the (x, y) plane.

Now we turn to a finite barrier height V = EcB − EcA. Then the super-

structure potential of a square QW B/A/B is
V (z) =	V	for	z−< −a/2	or z > a/2 .	(2.5)
	0	for	a/2 < z < a/2 ,

In QW structures with finite values of V , the envelope ψ(r) is nonzero both in the A and B layers. Due to the translational symmetry, the components kx, ky are the same inside and outside the well. The equation for ϕ(z) in the barrier layers is

−2mB			dz2	+		2mB + V		ϕ(z) = Eϕ(z) .	(2.6)
2			d2ϕ(z)			2k2

Except for the fact that the masses mA and mB may be di erent, the equations (2.2, 2.6) are identical to those solved in quantum-mechanics textbooks for a particle confined in a 1D square-potential well. In general, there are two kinds of solutions to (2.2, 2.6). When E − V − ( 2k2/2mB) is positive, the solutions within each layer are linear combinations of two plane waves and their energy spectrum is continuous, even for a fixed in-plane wave vector k . In case of negative values of E − V − ( 2k2/2mB), which is considered below, the function ϕ(z) is a linear combination of plane waves exp (±ikz) inside the well, and exponentially decays as exp (±æz) in the left and right barriers, respectively, where

k =	2	− k2	1/2	, æ =	2	−		+ k2	1/2	(2.7)
									.
	2mAE				2mB(V		E)

Since the potential (2.5) possesses reflection symmetry, the functions ϕ(z) have a certain parity. For even solutions, one can write

	D exp [−æ(|z| − a/2)] ,	if	||z||	≥ a/2,
ϕ(z) =	C cos kz ,	if	z	≤ a/2,	(2.8)
The coe cients C and D are found from the normalization condition
	ϕ2(z) dz = 1				(2.9)

and boundary conditions that relate the envelopes ϕA, ϕB and their derivatives (dϕ/dz)A, (dϕ/dz)B on both sides of the interface between the materials A and B. The most popular are the so-called Bastard boundary conditions [2.1, 2.2]

18 2 Quantum Confinement in Low-Dimensional Systems

ϕA = ϕB ,	1		dϕ	A =	1	dϕ			(2.10)
								B .
	mA		dz		mB		dz

They ensure continuity of the envelope ϕ(z) and the particle flux across the interface. For the solution (2.8) the boundary conditions lead to the following two linear homogeneous equations for C and D

	a		k		a	æ
C cos k		= D , −		C sin k		= −		D .	(2.11)
	2		mA		2		mB

After some algebra we obtain the transcendental equation for the energy of

even solutions

a

mA æ

tan k

= η ≡

.

(2.12)

2

mB

k

For odd solutions, the envelope ϕ(z) has the form

ϕ(z) =

C sin kz ,

if

z ≤ a/2

,

(2.13)

D sign{z} exp [−æ(|z| − a/2)] , if

||z|| ≥ a/2

,

and the energy is found from

cot k

a

= −η .

(2.14)

2

The coe cient C is derived from the normalization condition (2.9) and can be presented as

					sin ka		1		−1
	2
C =				1 ±			+ æa (1 ± cos ka)		,	(2.15)
		a			ka

where the sign ± corresponds to the even and odd solutions. By using (2.12) and (2.14) one can reduce the above expressions for C to

								1	−1
	2			1 +	(1 +2η2)a	η	+			(2.16)
C = a						k		æ	,

which is valid for solutions of both parities. It is known that in a symmetrical 1D well there is always at least one quantum-confined state. Therefore, for a finite value of V , the electron energy spectrum contains a finite number of subbands eν (ν = 1...N ) and continuum of states with E −( 2k2/2mB) > V . For the case of coinciding e ective masses, mA = mB, the dispersion Eeνk is parabolic with the same mass as in the bulk compositional materials. If the di erence between mA and mB is relatively small, the subband dispersion is close to parabolic.

It is worthwhile to analyze the transition from a finite to an infinite barrier height. For this purpose we take kx = ky = 0 and assume V to be high enough and satisfy the condition

2.1 Charge Carriers in Quantum Wells

19

2

π

2

V

.

(2.17)

2mA

a

Then, for the ground state e1, we can replace æ by æ0 =

, rewrite

2mAV / 2

(2.12) in the form cot (ka/2) = (mBk/mAæ) ≈ (mB

k/m

æ

) and consider

A

0

the ratio k/æ0 as a small parameter. For the state e1, in the zero-order approximation we obtain ka/2 = π/2 or k = π/a which coincides with (2.4) derived in the limit V → ∞ for ν = 1. In the first-order approximation we

find

k ≈ a 1 − mA æ0a

π

mB 2

and

2

π

2

1 −

mB 4

Ee1 ≈

(2.18)

æ0a .

2mA

a

mA

If the masses mA and mB are comparable in the order of magnitude the criterion for validity of (2.18) is the inequality æ0a 4. It follows then that the concepts “high barrier”, “low barrier” are relative and, for a wide well, equation (2.18) is valid even in heterostructures with relatively small band o sets.

One should bear in mind that, for semiconductors with the simple band structure, the most general boundary conditions are (see, e.g., [2.3–2.7])

ϕA = t11 ϕB + t12 ϕ˙ B , ϕ˙ A = t21 ϕB + t22 ϕ˙ B ,

(2.19)

where

ϕ˙ A = l	∂ϕ		, ϕ˙ B = l	mA	∂ϕ		,
	∂z A			mB		∂z
							B

l is an arbitrarily chosen microscopic length introduced in order to get the

ˆ

matrix elements tij dimensionless. The matrix t is unimodular, i.e., it satisfies the condition t11t22 − t12t21 = 1 (to within an overall complex phase factor), to insure that the electron flux is continuous. For a sharp heterointer-

ˆ

face A/B, the determination of the matrix t is beyond the competence of the envelope-function theory. Therefore the chosen set of components tij is either postulated or carried out by means of comparison with experiment or results of calculation performed in the frame of some microscopic model. Clearly, the Bastard boundary conditions (2.10) is a special case of the general conditions (2.19) with t11 = t22 = 1, t12 = t21 = 0. They are in a satisfactory agreement with the microscopical calculations in linear-chain tight-binding and empirical pseudopotential models [2.5]. Note that instead of solving the Schr¨odinger equation within each layer and sewing the solutions at the interfaces by using the boundary conditions in the form (2.10) one can equivalently use the Hamiltonian

∂ 2 ∂

+

2

∂2

∂2

H = −

+

+ V (z) ,

∂z

2m(z)

∂z

2m(z)

∂x2

∂y2

20 2 Quantum Confinement in Low-Dimensional Systems

defined in the whole space. Here m(z) is a discontinuous function jumping at the A/B interface from mA to mB.

If the carrier spectrum is anisotropic but the normal, z, to the interface is directed along one of the principal axes 1, 2, 3 of the e ective-mass tensor, then the above equations for the electron energy are valid with minor modifications. Particularly, for z 3, the quantum-confinement energy Ez is governed by the e ective masses mA3 , mB3 while the kinetic energy Exy is governed by the masses mAi , mBi with i = 1, 2. If the carrier spectrum is anisotropic and the normal z is directed in an arbitrary way with respect to the principal axes 1, 2, 3, then in the coordinate frame x, y, z the electron

e ective Hamiltonian within the well,
			2		ˆ ˆ		ˆ	∂
H =					kikj
						with	kj = −i		,
	ij	2			mijA			∂xj

contains nondiagonal components of the reciprocal e ective-mass tensor mAij −1. In this case, the solution of Schr¨odinger’s equation can be also represented in the form (2.1). However, here,

ϕ(z) = exp	−i	mxz kx + myz ky z				C1eikz + C2e−ikz .	(2.20)
			mzz		mzz

For simplicity, we omit the superscript A indicating the e ective mass in the layer A. The electron energy is given by

E =

2k2

+ Exy ,

(2.21)

2mzz

where

Exy =

2k2

m m

2k2

1 −

m m

2mxx 1 −

mxz2

zz

+ 2myy

myz2

x

xx

y

yy zz

+

2kxky

1 −

mxy mzz

.

2mxy

mxz myz

In a QW structure, a value of k2 is quantized. Just as in (2.4), for infinitely high barriers, k = νπ/a.

The space confinement, or size-quantization, of hole states for the simple valence-band structure is treated in the same way, the hole subbands are labelled hν. Fig. 2.1 illustrates the size-quantized electron subbands e1, e2 in the conduction band and subbands h1, h2 in the valence band. For semiconductors with degenerate valence bands, the calculation of hole energy spectrum presents a more complicated procedure discussed in the next subsection.