ivchenko_bookreg

2.3 Size Quantization in Quantum Dots

51

band model for both electrons and holes. A systematic theoretical investigation of electronic properties of InAs pyramid QDs in the frame of the eightband k · p theory is presented in [2.62]. It is shown that the electron ground state is mainly represented by s-type Bloch functions with the envelope u in (2.44) while the hole ground state is almost completely described by p-type Bloch functions with the envelopes vx, vy , vz . The first and second electron excited levels are two slightly split states with p-like envelopes, the energy separation between them is approximately proportional to the pyramid base length, b, in agreement with the proportionality between the piezoelectric potential and b. No remarkable intermixing of the confined-electron states with wetting-layer states is found. In contrast to the e ective-mass calculation using the same strain distribution, the eight-band theory leads to the

¯

hole ground-state functions pronouncedly elongated along [110]. Also the hole excited states in this theory are very di erent from previous e ective mass results with respect to both the wave-function shapes and the level separations. The holes are generally confined to the bottom of the pyramid and do not reach the tip. However, experiments on Stark e ect spectroscopy on InAs/GaAs self-assembled QDs [2.73] show that the hole is localized towards the top of the dot, above the electron, an alignment that is inverted relative to the predictions of previous calculations. This unexpected finding can be understood assuming that the nominally InAs dots contain significant concentration of Ga and have a graded In1−xGaxAs composition, with x decreasing from base to apex. Sheng and Leburton [2.74] address the issue of electron-hole alignment inversion in pyramidal QDs. They use the eightband strain-dependent k · p e ective Hamiltonian and consider three types of QDs, namely, pyramidal, truncated pyramidal and lens-shaped. Di erent models of Ga concentration profiles are analyzed, including dots containing five In1−xGaxAs monomolecular layers at the bottom with intervals 10% and dots with variable number of In0.75Ga0.25As monomolecular layers. For structures with sizes studied experimentally, the theory allows to obtain inverted electron-hole alignments that are consistent with the experiment for several Ga di usion profiles. However, the inverted alignment is not a universal property of self-organized QDs, but ultimately depends on di usion profile and dot size that are imposed by fabrication processes, including the nominal composition of the dot layer, the growth temperature, composition changes during capping, post-growth annealing, etc. A microscopic pseudopotential description [2.75] also strongly suggests that to obtain accurate agreement between theory and experiment one needs to adopt a model that includes some Ga in-di usion within the QD.

In [2.76] experimental and theoretical tools were used interactively to determine the geometric and compositional parameters and the resulting electronic properties of InGaAs/GaAs self-organized alloy QDs, grown by Metal-Organic Chemical Vapor Deposition (MOCVD) with nominal 50% In content and a deposition of four monolayers. Theoretically, an atomistic em-

52 2 Quantum Confinement in Low-Dimensional Systems

pirical pseudopotential model was used as an alternative to the commonly used e ective-mass and k·p envelope-function approximations. The QD structural model with best agreement between measured and predicted electronic

	˚
properties of the studied samples is a truncated cone with height 35 A, base
˚	˚

diameter 200 A, and top diameter 160 A, having a nonuniform, peaked composition profile with average 60% In content.

In contrast to InAs and GaAs crystals, GaN and related nitride alloys generally have a hexagonal (wurtzite) structure, which leads to strong build-in piezoelectric fields in heterostructures, of the order of MV/cm. Electronic structure of nitride-based QDs, namely, GaN/AlN and InxGa1−xN/GaN QDs, was calculated for hexagonal prisms (prismatic QDs) and truncated hexagonal pyramids [2.77,2.78], respectively by using a multiband k ·p model and a tight-binding method.

2.4 Spin Splitting of Electron Subbands: Bulkand

Structure-Inversion Asymmetry

It is well-known that, in centrosymmetric crystals, the electronic state at each band n and wave vector k is at least twofold degenerate (the so-called Kramers degeneracy, or spin degeneracy). In crystals lacking a center of inversion, the Kramers degeneracy of the Bloch states is lifted except for special points or lines in the Brillouin zone. Particularly, in zinc-blende-lattice semiconductors like GaAs, InAs, ZnSe, CdTe, etc. (the class Td) the conduction band Γ6 and the valence band Γ8 are, respectively, twofold and fourfold degenerate at the Γ point. However, away from this point the conduction and valence bands are split into nondegenerate spin branches, even at zero magnetic field. The spin-dependent Hamiltonian can be constructed by expanding the e ective Hamiltonian in powers of k and applying the method of invariants mentioned above in connection with the invariant form of the Luttinger Hamiltonian (2.28). For the Td-symmetry crystals, one can show that the spin-dependent term in the conduction-band Hamiltonian appears starting with k3 [2.79]

Hc3 = γc σxkx(ky2 − kz2) + σy ky (kz2 − kx2 ) + σz kz (kx2 − ky2) , (2.111)

where σi are the Pauli matrices. For the band Γ8 the main contribution is given by a similar expression

Hv3 = γv Jiκi , (2.112)

i

where Ji are the 4×4 matrices defined in Table 2.1 and we use the notation

κx = kx(ky2 − kz2) , κy = ky (kz2 − kx2 ) , κz = kz (kx2 − ky2) .

2.4 Spin Splitting of Electron Subbands: Bulkand Structure-Inversion Asymmetry

53

In contrast to the conduction band, the constant γv has a nonrelativistic nature. Indeed, this constant equals 2/3 of the similar constant in the 3×3 Hamiltonian for the nonrelativistic valence band Γ15. Besides (2.112), the valence-band Hamiltonian expanded up to third order in k contains four relativistic terms, a linear-in-k term

Hv1 = γv1 i

Viki

and three cubic terms

δHv3

= γv3

i

Ji3

κi + γv3k2

i

Viki

+ γv3

i

Viki ki2

−

k2

3 .

In heterostructures, QWs or SLs, grown from zinc-blende-lattice semiconductors, the spin-dependent Hamiltonians contain both linear and cubic terms. Particularly, in (001)-grown QWs with symmetrical interfaces (D2d point-group symmetry) the linear-k spin-dependent term in the conduction subband e1 has the form [2.80]

HBIA = β1(σy ky − σxkx),

(2.113)

where x [100], y [010] and β1 is a constant. This term can be obtained from the cubic-k term (2.111) describing the removal of spin degeneracy of the conduction-band states in a bulk semiconductor. Really, taking into account the quantum confinement e ect we can replace in (2.111) kz and kz2 by the average values kz = 0 and kz2 = 0, respectively, and arrive at (2.113) with β1 = γc kz2 . Here the symbol kzl means the expectation value

ˆl	ϕe1(z)dz, ϕe1	(z) is the electron envelope function (2.8) at the low-
ϕe1(z)kz
	ˆ	= −i∂/∂z. Since the term HBIA is due to the lack
est subband e1 and kz

of inversion symmetry in the bulk material it is called the Bulk Inversion Asymmetry (BIA) term which explains the subscript BIA. Sometimes it is also called the Dresselhaus term [2.81].

In heterostructures with asymmetrical superstructural potential (the C2v

point group) there exists another spin-dependent contribution
HSIA = β2(σxky − σy kx)	(2.114)

which is called the Structure-Inversion Asymmetry (SIA) term, or the Rashba term. The structure asymmetry can be related with non-equivalent normal and inverted interfaces, external or built-in electric fields, compositionally stepped QWs etc. The spin-orbit interaction term in the form of (2.114) was first predicted in [2.82, 2.83] for bulk polar hexagonal crystals with the wurtzite structure (the C6v point symmetry). Nature of the similar term in an asymmetrical 2D system has been analyzed by di erent authors ( [2.84–2.90], see also [2.91] and references therein). Note that anisotropic orientation of

54 2 Quantum Confinement in Low-Dimensional Systems

chemical bonds at the interfaces gives rise to an additional contribution to the linear-k Hamiltonian, even in symmetrical QWs [2.92, 2.93]. This is the so-called Interface Inversion Asymmetry (IIA) term. Since it has the same structure as the contribution (2.113), they can be described together by one inseparable Dresselhaus term with the common parameter β1.

¯

It is convenient to introduce the Cartesian coordinates x [110], y [110], z [001] which allow to write a sum of the BIA and SIA terms in the form

H	2		−		−
c1(k) =	1	(β		σx ky		β+σy kx ) ,	(2.115)

where β± = 2(β2 ± β1) and from now on k ≡ k . Introducing the e ective Larmor frequency Ωk (at zero magnetic field) by

Hc1 =		σ · Ωk	(2.116)
	2

and comparing with (2.115) we obtain

Ωk,x = β−ky / , Ωk,y = β+kx / , Ωk,z = 0.

(2.117)

Thus, in the parabolic approximation the resulting energy dispersion is

Ee1k = E0 + 2k2 ± 1 Ωk , e1 2m 2

with the spin splitting given by

∆E = Ωk = β+2 kx2 + β−2 ky2 . (2.118)

If only one of the constants β1, β2 is nonzero then β−2 = β+2 ≡ β2, and the splitting Ωk = βk is angular independent. Figure 2.7 illustrates the electron dispersion Ee1k (a,d) and contours of constant energy in the kx-ky plane for di erent relations between β1 and β2 (b,c,e). For β1 = 0, β2 = 0 and β1 = 0, β2 = 0 the dispersion has the same shape and consists of the well known shifted parabolas in all directions, displayed in Fig. 2.7a. However, the Rashbaand Dresselhaus spin-orbit coupling, or coupling between the spin and wave vector, results in di erent spin orientation of the eigenstates.

The spin orientation of split states at di erent k can be visualized by indicating the direction of the e ective Larmor frequency Ωk which provides the relevant quantization axis for spin at given k. The spin orientations for purely Rashba and purely Dresselhaus coupling are schematically shown in Fig. 2.7b and 2.7c by arrows. It is assumed that β1 is negative and β2 is positive. For the Rashba contribution the e ective Larmor frequency and, hence, the spin is always perpendicular to the corresponding k-vector (Fig. 2.7b). If the Dresselhaus term is dominant, the angle between k-vector and spin depends on the crystallographic orientation of k.

2.4 Spin Splitting of Electron Subbands: Bulkand Structure-Inversion Asymmetry						55

1

2

1

2

Fig. 2.7. Schematic 2D electron band structure with k-linear terms for the C2v symmetry for di erent strengths of the SIA and BIA contributions and the distribution of spin orientations at the 2D Fermi energy. (a) shows the case of only Rashba (b) or Dresselhaus (c) spin-orbit coupling, and (d, e) represent the case of simultaneous presence of both contributions. Arrows indicate the orientation of spins.

In the presence of both Rashba and Dresselhaus spin-orbit terms, rele-

	¯
vant for the C2v symmetry, the [110] and [110] axes become strongly non-
equivalent. For k	[110], the eigenvalues of	the	Hamiltonian are then
given by Ee1 =	2k2/2m ± (β1 + β2)k and,	for	k	[110],¯ by Ee1 =

2k2/2m ± (β2 − β1)k. This anisotropic dispersion is sketched in Fig. 2.7d. The corresponding contours of constant energy together with the spin orientation for some k-vectors are shown in Fig. 2.7e.

At low electron energies the spin splitting due to the linear-in-k term dominates. At higher energies relevant to high temperatures or large concen-

56 2 Quantum Confinement in Low-Dimensional Systems

trations, the cubic-in-k term can be important as well. For the (001) QW structures the two Dresselhaus contributions to Ωk,i, linear and cubic, can be written as

Ωk,x = −Ω1 cos ϕ−Ω3 cos 3ϕ, Ωk,y = Ω1 sin ϕ−Ω3 sin 3ϕ, Ωk,z = 0, (2.119)

where

Ω1 =

k kz2 − 4 k2

, Ω3 =

2 k3 ,

2γc

1

γc

k2 = k2

+ k2 and ϕ is the angle between k and the axis [100]. For the [111]

x

y

structures, one has

Ωk,x = Ω1 sin ϕ , Ωk,y = −Ω1 cos ϕ , Ωk,z = −Ω3 sin 3ϕ ,

where x

¯

¯

¯

[112], y

[110], z

[111], ϕ is the angle between k and [112], and

Ω1 = √3 k

kz2 − 4 k2

, Ω3 =

k3 .

3

4γc

1

2 γc

For the [110] structures, one has

Ωk,x = Ωk,y = 0 , Ωk,z = Ω1 cos ϕ − Ω3 cos 3ϕ ,

where x

¯

[001], z

¯

[110], y

[110], ϕ is the angle between k and [110], and

Ω1

= k kz2 − 4 k2

, Ω3 =

4 k3 .

γc

1

3 γc

The Rashba term can be presented in an invariant form as

HSIA = β2(σ × k) · N ,

(2.120)

where N is the unit vector directed along the normal to the interface. In a symmetrical QW subject to a homogeneous electric field F z, the constant β2 can be crudely estimated as [2.90]

β2 =

P 2

∆(2Eg + ∆)

|e|F

3

Eg2(Eg + ∆)2

+

∆

∆

ϕe21

(a/2) − ϕe21

(−a/2)

,

Eg (Eg + ∆)

− Eg (Eg + ∆ )

where the band parameters Eg , ∆ and Eg , ∆ refer to the well and barrier bulk materials, respectively, the parameter P is defined in (2.46), ϕe1(z) is the envelope calculated in the presence of electric field, and ±a/2 are the interface coordinates.

The BIA contribution to the valence band e ective Hamiltonian responsible for removal of spin degeneracy of the hole subbands can be obtained by

2.5 Electrons, Photons and Phonons in Superlattices

57

averaging the linear and cubic terms, H1v and Hv3 +δH3v , over the quantumconfined states hhν or lhν calculated neglecting the spin splitting. Note that this procedure applied to the nonrelativistic term (2.112) does not lead to linear-k splitting of heavy-hole states because the o -diagonal components Ji;±3/2, 3/2 equal zero. The Rashba spin splitting in 2D hole systems was analyzed by Winkler [2.95]. At small values of k, the heavy-hole spin splitting is of third order, in qualitative di erence with the conduction-band and light-hole states in QWs.

Entin and Magarill [2.96] have derived the e ective Hamiltonian for electrons confined near the curved surface and showed that the curved system has an additional spin-dependent term. For example, in a spherical QW where electrons are confined in a thin spherical layer of the radius R, the spin-orbit term is proportional to

σ ·	R2 × pˆ			,
		R
where			i ∂		i ∂
pˆ = oθpˆθ + oϕpˆϕ , pˆθ = −R ∂θ				, pˆϕ = −				,
					R sin θ		∂ϕ

and the vectors oθ, oϕ, oR = R/R compose the orthogonal basis in the spherical coordinates.

Linear-in-k spin splitting of electron subbands in QWRs is described by a Hamiltonian

H1 = (σ · β) kz ,

(2.121)

where z is the principal axis of the wire and β is a constant vector, its nonzero components are determined by symmetry of the structure. The e ective Larmor frequency defined according to (2.116) is equal to 2βkz / . Depending on the sign of kz it is directed parallel or antiparallel to the fixed direction of β. This is the main di erence with QWs where the direction of Ωk is independent of k only in asymmetrical QWs in the particular case β1 = ±β2 when either β+ or β− vanish. In [2.97] a simple 3D model of an asymmetric QWR is introduced in which the Rashba spin-orbit coupling (2.121) is derived from a realistic description of the bulk semiconductor electronic structure.

2.5 Electrons, Photons and Phonons in Superlattices

The e ective-mass method, the formalism of macroscopic electrodynamics for photons, the dielectric model for optical phonons and the approximation of a homogeneous elastic medium for acoustic phonons allow a generalized description of the quasi-particle propagation in a semiconductor SL. We consider an infinite SL made up of alternating layers A and B with thicknesses a and b, respectively, and hence with the period d = a + b. The state of a generalized quasi-particle is described by the generalized envelope

58 2 Quantum Confinement in Low-Dimensional Systems

e−iωt+iq·ρϕ(z) ,

where z is the SL principal axis, ρ is the in-plane component of the 3D radius-vector r, q is the in-plane component of the quasi-particle wave vector. Within each homogeneous layer the z-dependent envelope ϕ(z) is a linear combination of two exponential functions,

ϕ(z) = F+eikAz + F−e−ikAz	within the layers A
and
ϕ(z) = G+eikBz + G−e−ikBz	within the layers B .

The coe cients F±, G± may depend on the layer number. At the interfaces the function ϕ(z) satisfies the boundary conditions

	ϕA = ϕB , CA	dϕ	= CB	dϕ	.	(2.122)
		dz		dz
			A		B

Physical meaning of the function ϕ(z) and the coe cients CA,B as well as the relation between the wave vectors kA, kB and the frequency ω, or the energy E = ω, will be discussed with reference to electrons, photons and phonons after deriving the dispersion equation for a generalized quasi-particle propagating in a SL. Let us present the function ϕ(z) and its derivative as a two-component column

ϕˆ(z) =	ϕ˙	, ϕ˙ j ≡ CA kA dz (j = A, B) .		(2.123)
	ϕ		Cj 1 dϕ

The transfer matrix through the layer (z0, z) is a 2×2 matrix that relates the columns (2.123) at the points z and z0, namely,

ˆ

ϕˆ(z) = t(z, z0)ϕˆ(z0) .

The transfer within a homogeneous layer A or B is realized by the matrix

tˆ(z, z0) =	cos kl	N¯ −1 sin kl	,	(2.124)
	−N¯ sin kl	cos kl
¯		¯

where l = z − z0, N = 1 in the layer A and N = (CBkB)/(CAkA) ≡ N in the

ˆ

layer B. Note that the transfer matrix is unimodular: Det t = 1.

According to the Bloch theorem one can seek eigensolutions satisfying the condition

ˆ	iKd	ϕˆ(0) ,
ϕˆ(d) = T ϕˆ(0) = e

where K is the wave vector characterizing the propagation of a quasi-particle

ˆ

along the principal axis z, T is the transfer matrix through a SL period, it is

ˆ ˆ

a product tAtB of the transfer matrices through the layers A and B. As the first step in deriving the dispersion equation relating K and ω, we rewrite the above condition in terms of the components Tij as follows

2.5 Electrons, Photons and Phonons in Superlattices

59

Det	T11 − eiKd	T12	= 0 .
	T21	T22 − eiKd

ˆ

Taking into account the unimodularity of T we can present this equation in the form

	cos Kd =		T11 + T22		.		(2.125)
				2
By direct matrix multiplication we obtain
	cos kAa	sin kAa		cos kBb		N −1 sin kBb	=
	Tˆ = tˆAtˆB = − sin kAa cos kAa			−N sin kBb		cos kBb
	........	cos kAa cos kBb − N −1 sin kAa sin kBb
	cos kAa cos kBb − N sin kAa sin kBb			........				.

For the sake of brevity the o -diagonal components which do not enter into (2.125) are replaced by omission points. Substituting the expressions for T11 and T22 into (2.125) we finally arrive at

cos Kd = cos kAa cos kBb − 2	N +	N sin kAa sin kBb .	(2.126)
1		1

Now we pass on to particular cases and start with electrons in a shortperiod SL. In this case ϕ is the electron envelope function in the e ectivemass approximation, ω = E/ , the coe cients in the boundary conditions are given by

CA =	1	, CB =	1	, N = mA		kB	.
	mA		mB		mB kA
For the states with the	energy E lying				below the barrier V , one has

kA = k, kB = iæ, see (2.7), and the dispersion equation can be conveniently rewritten in the form

cos Kd = cos ka cosh æb +	1	η −		1	sin ka sinh æb ,	(2.127)
	2		η

where η is introduced in (2.12). In the reduced zone scheme values of K lie within the limits from −π/d to π/d. At a fixed value of q, for instance at q = 0, the energy spectrum E(q, K) consists of alternating allowed and forbidden minibands. For SLs with thick potential barriers, the lower allowed minibands with E < V are narrow and lie near the electron size-quantized levels in a QW of the same thickness a. As the barrier thickness b decreases the allowed minibands widen at the expense of the convergent forbidden minibands. In the limit b → 0 the miniband character of the electron energy spectrum transforms into the parabolic dispersion of an electron in the host material A.

In order to analyze the dispersion at small values of K we can rewrite (2.127) in the following equivalent form

60

2 Quantum Confinement in Low-Dimensional Systems

1

sin ka sinh æb f1f2

1 − cos Kd =

≡ F ,

(2.128)

2

a

− η tanh æ

b

1

cot k

a

+ coth æ

b

f1 = tan k

, f2 =

.

2

2

η

2

2

For |K| π/d one has

E ≈ Eeν +

d2

K2 ,

(2.129)

2F

where ν is the miniband index, Eeν is the energy at K = 0 defined by the equations f1 = 0 or f2 = 0, F is the derivative dF (E)/dE at E = Eeν . Notice that, in the limit b → ∞, the equations f1 = 0, f2 = 0 coincide with (2.12) and (2.14) for the confinement energy in a single QW. It follows from

(2.129) that the electron e ective mass for the motion along z is related to F by

Mzz = F 2/d2 .

(2.130)

For SLs with barriers thick enough so that exp(æb) 1, the allowed minibands can be described by

E(K) ≈ Eeν + (1/F )(1 − cos Kd) .

As regards the dependence of E on q, it is parabolic if mA = mB, as in the bulk host materials. For mA = mB, the in-plane dispersion has a quasiparabolic character with the e ective mass slowly varying within the interval between mA and mB.

Next we consider the normal light waves in an optical SL with the layers A and B characterized by the local isotropic dielectric functions æA(ω) and æB(ω). For completeness we write down the Maxwell macroscopic equations

	1		∂D		1		∂B
curl B =					, curl E = −			,
		c		∂t		c	∂t

divD = 0 , divB = 0 .

Here E and D are the electric-field and electric-displacement vectors related by D = æj (ω)E, and we use the form of material relation where the magnetic induction B coincides with the magnetic field H. The space dispersion, or the dependence of æA,B(ω) on the light wave vector q, is neglected. As a rule, in what follows we omit the variable ω and use the notation æA,B. For a monochromatic light wave the operator ∂/∂t may be replaced by the factor −iω. Within each separate layer A or B the solution is a linear combination of plane waves exp [i(q · ρ ± kA,Bz)], where

kA = [(ω/c)2æA − q2]1/2 , kB = [(ω/c)2æB − q2]1/2 .

(2.131)

We fix the coordinate system x, y with x q in which case the	3D wave
vectors (qx, qy , ±kA,B) lie in the plane perpendicular to the y axis.