ivchenko_bookreg

2.1 Charge Carriers in Quantum Wells

21

Fig. 2.1. Band diagram of a single QW and the electron energy dispersion in the 2D conduction and valence subbands. The arrows indicate possible optical transitions, namely, (1) direct intraband intersubband transitions, see Sect. 4.1, (2) indirect intrasubband transitions (Sect. 4.3), (3) transitions from quantum-confined to continuum states, and (4),(5) direct interband transitions (Sect. 2.6).

2.1.2 Luttinger Hamiltonian. Heavyand Light-Hole Subbands

If we ignore the spin and spin-orbit interaction (the nonrelativistic approximation), the bottom conduction and top valence Γ -states in GaAs-like semiconductors are characterized by s- and p-type symmetries. The corresponding orbital Bloch functions are designated S(r) ≡ S and X, Y, Z. They form bases of the irreducible representations A1 (Γ1) and F2 (Γ15) of the point group Td (see Appendix). However, one has to bear in mind that these functions are pe-

22 2 Quantum Confinement in Low-Dimensional Systems

riodical with the periodicity of zinc-blende structure, say, X(r + ai) = X(r) with ai being the primitive lattice vectors. The inclusion of the spin degeneracy doubles number of states at the Γ point: ↑ S, ↓ S (the spinor irreducible representation Γ6) for the conduction band and ↑ X, ↑ Y, ↑ Z, ↓ X, ↓ Y, ↓ Z for the valence band. The spin-orbit interaction leads to splitting of the six degenerate valence states into the quartet

+3/2 = − ↑

X + iY

(2.22)

|Γ8,

√

,

2

X + iY

|Γ8, +1/2

=

2

↑ Z− ↓

√

,

3

6

iY

|Γ8, −1/2

=

2

↓ Z+ ↑

X

√−

,

3

6

Γ

,

3/2

=

X − iY

,

−

↓

| 8

√2

that transforms according to the spinor representation Γ8, and the doublet

|

1

↑

↓

3

Γ7

, +1/2

= √

[ Z+

(X + iY )] ,

(2.23)

|

−

1

− ↓

↑

−

3

Γ7

,

1/2

= √

[

Z+

(X

iY )]

that forms the spinor representation Γ7. We use the canonical bases for the representations Γ8, Γ7, they transform under operations of the Td group as

the spherical functions Y3/2,m (m = ±3/2, ±1/2) and Y1/2,m (m = ±1/2). By analogy with spin-orbit splitting of the p-level of a hydrogen atom, the states

Γ8, Γ7 are ascribed to those with the total angular momentum J = 3/2, 1/2 and its z-components m = ±3/2, ±1/2 and m = ±1/2, respectively. This explains the notations of the corresponding basic functions in (2.22, 2.23).

The e ective Hamiltonian for electrons in the valence band Γ8 is a 4×4

matrix and, in the basis (2.22), has the form										,
	(Γ8) = H G						0	I			(2.24)
					F	H	I	0
H					I0	I0	H	−F
							G	H
where							−
						A + 2		(kx2 + ky2) ,			(2.25)
F = (A − B) kz2 −
							B
G = (A + B)kz2 −						A − 2		(kx2 + ky2) ,
							B
	√
			3					D		kxky ,
I = −					B(kx2 − ky2) − 2 i √
		2
									3

H = −Dkz (kx − iky ) ,

2.1 Charge Carriers in Quantum Wells

23

and i , i = x, y, z are the principal axes [100], [010] and [001]. This matrix, called the Luttinger Hamiltonian and depending on three band parameters, A, B and C, is derived in the second-order k ·p perturbation theory or in the method of invariants. It describes the energy spectrum and electron (hole) states in the vicinity of the point k = 0 in the band Γ8 of zinc-blende- lattice semiconductors (e.g., GaAs, InP, CdTe, ZnSe) and in the band Γ8+ of centrosymmetric diamond-lattice semiconductors (Ge, Si). Notice that the symmetry forbids odd terms in the expansion of the Hamiltonian in powers of k in centrosymmetric crystals but allows such terms in Td crystal classes. The linearand cubic-in-k contributions to the e ective Hamiltonian are discussed in Sect. 2.6.

The dispersion equation Det||Hm(Γ8m) − Eδm m|| = 0 is reduced to

(E − F )(E − G)− | H |2 − | I |2 2 = 0 .

Its solutions are
E		=	F + G	±		F − G	2	+	H	2 + I 2	(2.26)
			2			2
	hh,lh							|	|	| |

= Ak2 ± B2k4 + (D2 − 3B2) kx2 ky2 + ky2kz2 + kz2kx2 .

For most semiconductors with a zinc-blende lattice the band parameter A is negative. As a result, the e ective masses 2/(2Ehh,lh) are negative as well. Sometimes, instead of working in the electron representation with negative masses, it is more convenient to use the hole representation with positive masses and replace H(Γ8) by −H(Γ8). The energy in the latter representation will be labelled by the superscript ‘h’. Thus for hole dispersion we have instead of (2.26)

Eh

=

2k2

, m

=

2

,

(2.27)

hh

hh

k

h

2mhh

2(|A| − B2 + C2f )

2

2

2

Elh =

, mlh =

,

2mlh

2(|A| +

)

where

B2 + C2f

k2 k2

+ k2k2

+ k2k2

C2 = D2

− 3B2 , f =

x y

y z

z x

.

k4

√

Usually values of A, B and D/ 3 are close to each other because the k · p coupling of the Γ8 valence band with the lowest conduction band Γ6 is the strongest. As a result, the hole branch labelled ‘hh’ has a small dispersion and hence large mass, it is referred to as the heavy-hole subband. The other branch labelled ‘lh’ is known as the light-hole subband. Note that the heavyand light-hole subbands are 3D bands and have nothing in common with the size-quantized subbands in QWs and QWRs.

24 2 Quantum Confinement in Low-Dimensional Systems

The Luttinger Hamiltonian (2.24) can be conveniently presented in the following invariant form [2.8]

H(Γ8)(k) = A + 4 B				k2 − B			i	Ji2ki2 − √3 i =i{Ji Ji}ski ki			(2.28)
			5					D
= 2m					+ 2 γ2
				γ1			k2 + 2γ2		Ji2ki2 + 2γ3 Ji Ji ski ki .
2	0		−		5			i	i =i{	}

ˆ

Here Ji are the matrices of the angular-momentum operators Ji in the basis Y3/2,m, the curly brackets with the subscript s mean the antisymmetrization: {M N }s = (M N +N M )/2 for any operators or matrices M and N . The complete set of 4×4 matrices constructed from products of Ji is presented in Table 2.1. In the method of invariants, these matrices can be considered as objects transforming according to representations of the Td group. Particularly, the matrices Ji and Ji3 (i = x, y, z) form equivalent bases of the representation F1 (pseudovector) and the matrices Vi and {Ji+1Ji+2}s transform according to the representation F2 (vector). The three matrices Ji2 form a basis of the reducible representation which is reduced to A1 + E. The invariant matrix is the sum Jx2 + Jy2 + Jz2 which is equal J(J + 1)I, where J = 3/2 and I is the

4×4 unit matrix with the components I = δ . As a basis of the represen-

√ i i i i

tation E one can use the pair 3(Jx2 −Jy2), 2Jz2 −Jx2 −Jy2. Finally, the matrix

{JxJy Jz }s is attributed to the representation A2 (pseudoscalar). The matrices Ji, Ji3, Vi and {JxJy Jz }s are odd and the matrices Ji2, {Ji+1Ji+2}s are even under time inversion operation. The dimensionless constants γj called the Luttinger (or Kohn-Luttinger) band-structure parameters are connected with the constants A, B, D by

	2		2		2	D			(2.29)
		γ1 = −A ,		γ2 = −B ,		γ3 = −√		.
	2m0		m0		m0
							3

Table 2.1. Matrices Ji and their products for the representation Γ8 in the basis Y3/2,m (m = 3/2, 1/2, −1/2, −3/2). The used notations are Vi = {Ji(Ji2+1 −Ji2+2)}s,

{JxJy Jz }s = {Jx{Jy Jz }s}s.

Jx = √3/2 0					1			0			, Jy =	i√3/2 −				0		−i			0			,
	0	√3/2 0						0					0			i√3/2 0					0
					√0			√		/2								√0			−i√		/2
	0	1							3				0			i						3
	0	0			3/2 0								0			0		i 3/2			0

		3/2	0		0		0
Jz =		0	1/2	−	0		0		,
		0	0		0		3/2
		0	0		1/2		0
						−

													2.1 Charge Carriers in Quantum Wells															25
Jx2 =	0			7/4						√3/2			, Jy2 =		0				7/4						−√3/2 ,
							0															0
	3/4			0			√3/2				0				3/4				0			−√3/2			0
	√		/2 √0												−√		/2 √0
		3					7/4			0						3						7/4			0
	0			3/2 0						3/4					0				3/2			0			3/4
																		−

																9/4									0	0			0
												Jz2 =							0 1/4 0 0																,
																			0						0	0			9/4
																			0						0	1/4			0
			i√3				−																																												,
2 Jy Jz s =								0						0					0						, 2 Jz Jx s									=		√3 0 0													0
					0		i√3 0												0																			0			√3 0								0
{	}				0			0						√		3 0										{					}															√ −
					0			0						i																						0 0										3 0
														0				i√3																				0			0					0			√3
													−																																−
							2 JxJy s =														0 0						−		0 −i√3 ,
																					0				0			i√3							0
																		i					3		√
							{						}						√						0	3			0						0
																					0				i				0						0
Jx3 = 7			√				5/2							0							, Jy3 = i7												3/8					√							−i5/2						0			,
	3/8 0																																					0
	√0	7	3/8				0							3/4															√0							−i7					3/8					0					i3/4
							√0						7√				/8																													√0			−i7√				/8
	0		5/2												3																0							i5/2														3
	3/4		0				7 3/8							0																	i3/4							0							i7 3/8						0
																													−

√0

− 3/4

Vx =

0

−3/4

					Jz3 =								0 1/8											0					0							,
										27/8									0					0					0
													0						0				−	0					27/8
													0						0					1/8					0
																												−
−	0				3/4						−0								, Vy =									i√3/4								−			0		i3/4			0			,
√3/4					0								3/4																	0							i√3/4 0							i3/4
					√0					−√						/4																				−i3/4 √0 −i√										/4
3/4														3																0															3
	0			−									0																−	i3/4									0					0
					3/4																																				i 3/4
				Vz =						0									0							0 −√3/2 ,
										0									0				√3/2							0
									√			/2						√0
											3													0						0
																−
										0									3/2					0						0
																												−
2 JxJy Jz s =													0										0						0							i√3/2 .
													0										0						i√3/2								0
	{				}					i
													3/2										√
													0									i		3/2					0								0
													√						−				0						0								0

The method of invariants readily allows to write down a deformationinduced contribution to (2.28), the so-called Bir-Pikus Hamiltonian, as

Hu(Γ8) =	a + 4 b Tr{uˆ} − b	i	Ji2uii − √3 i =i{Ji Ji}sui i , (2.30)
	5		d

26 2 Quantum Confinement in Low-Dimensional Systems

where uˆ is the strain tensor with the components ui i and a, b, d are the deformation potential constants. The Zeeman Hamiltonian in the band Γ8 reads

HB(Γ8) = 2µB KJ · B + q(Jx3Bx + Jy3By + Jz3Bz )

(2.31)

and is characterized by two dimensionless coe cients K and q. Here B is the magnetic field and µB is the Bohr magneton. Note that the di erence between γ2 and γ3 as well as nonzero value of q are due to the cubic symmetry of the zinc-blende lattice. The warping of the valence band, or the angular dependence of mhh, mlh, is directly related to the di erence between γ2 and γ3. In the frequently used isotropic approximation, γ3 is identified with γ2 in which case the warping vanishes. In the e ective Hamiltonian approach, the electron wave function is expanded in the Bloch states near the extremum point, here the Γ point, and written as a sum

Ψ (r) = Φm(r) |3/2, m	(2.32)
m

of products of the smoothly-varying envelope functions Φm and the Γ -point Bloch functions |3/2, m , see (2.22).

In QW structures, the envelope functions Φm presented as a four-component

column Φ satisfy the matrix Schr¨odinger equation
HΦ(r) = EΦ(r) ,					(2.33)
where the e ective Hamiltonian H is the Luttinger Hamiltonian H					(Γ8)	ˆ
						(k)
				ˆ
with the superstructure potential V (z) included into the diagonal, k = −i .
The simplest boundary conditions for the envelopes Φm(r) are
ΦA = ΦB , (ˆvz Φ)A = (ˆvz Φ)B ,					(2.34)
where vˆz is the normal component of the velocity operator
vˆz =	1		∂H	.	(2.35)
∂kz

In this subsection the consideration is restricted to zinc-blende-lattice QW structures grown along the direction [001]. QWs grown along low-symmetry directions are treated in Sect. 3.1.6. We start from quantum-confined valence states with zero in-plane wave vector, kx = ky = 0. Since in this case the non-diagonal elements of the Luttinger Hamiltonian vanish, heavyand light-hole states are size-quantized independently and form two series of subbands, heavy-hole (hhν) and light-hole (lhν) subbands. At the same time the boundary conditions (2.34) reduce to (2.10). The hole quantum-confinement energies Ehhν and Elhν are found from (2.12, 2.14) by substituting into these equations the hole masses and the valence band o set. For each state only one envelope Φi(z) is nonzero so that the heavy-hole states are characterized

2.1 Charge Carriers in Quantum Wells

27

by the angular momentum component Jz = ±3/2 whereas, for the lh states, Jz = ±1/2. Therefore, one has

Ψhhν,±3/2	(r) = ϕhhν (z) |3/2, ±3/2 ,	(2.36)
Ψlhν,±1/2	(r) = ϕlhν (z) |3/2, ±1/2 ,

where the envelopes ϕhhν (z), ϕlhν (z) are given by (2.8, 2.13).

For nonzero values of the in-plane wave vector k , the o -diagonal terms of the Luttinger Hamiltonian mix the heavyand light-hole states and give rise to large nonparabolicities of the hole subbands [2.9–2.14]. In QWs with perfect interfaces, the in-plane motion may be factorized out of the envelope function

Φ(r) = ei(kx x+ky y)F (z) ,

(2.37)

where F is a four-component column, z and k -dependent. With no magnetic field, the solutions of (2.33) at each subband hhν, lhν and in-plane wave vector k are twofold degenerate. It is accepted to label the hole subbands as Ehhν,k , Elhν,k referring to the subband index hhν or lhν at k = 0.

In symmetrical QWs, it is possible to classify the two degenerate states by a parity number p = ± under the mirror reflection z → −z. It follows then that, for the hole state with parity p, the four-component envelope F (z) can

be presented as [2.11, 2.13]	F1/2,−p						.
Fp(z) =					(z)			(2.38)
		F3/2,p(z)
	F			−1/2,p		(z)
			−3/2,−p
		F			(z)

Particularly, this means that, if the component F3/2,p is even (p = +), then the component F1/2,−p is odd. In other words, the mth component of the column Fp(z) has parity pm given by

pm = (−1)3/2−m p .

The fact that the Luttinger Hamiltonian is invariant under both time and space inversion allows to relate the even and odd solutions with the same k by

ˆ ˆ	(2.39)
Ψ−(r) = JKΨ+(r) .

Here it is assumed that the time inversion operator ˆ acts both on the en-

K

velopes Φm(r) and the Bloch functions |Γ8, m whereas the space inversion

ˆ ˆ ˆ

operator J acts only on the envelopes, JΦm(r) = Φm(−r). The operator K is defined by

ˆu =	−	iσ	u ,	(2.40)
K		y

where u is an arbitrary spinor function and σy is the Pauli matrix. In particular,

28 2 Quantum Confinement in Low-Dimensional Systems

ˆ	ˆ	, 1/2 = |Γ8, −1/2 ,	(2.41)
K |Γ8	, 3/2 = − |Γ8, −3/2 , K |Γ8
ˆ	ˆ	, −3/2 = |Γ8, 3/2 .
K |Γ8	, −1/2 = − |Γ8, 1/2 , K |Γ8

A typical calculated hole energy spectrum is illustrated in Fig. 2.2 for a

˚

GaAs/Al0.5Ga0.5As QW of the width a = 200 A. Neglecting the o -diagonal terms in the Luttinger Hamiltonian, i.e., neglecting the heavy-light-hole hybridization, the hhν and lhν hole subbands are parabolas, see (2.25),

Ehhνk	= Ehhν0 + |A| +	|	2	|	k2 and Elhνk	= Elhν0 + |A| −	|	2	|		k2 ,
			B					B

where Ejν0 = Ejν,k =0 (j = hh, lh). The valence-subband coupling leads to anticrossing of these parabolic subbands and even to nonmonotonical dispersion for some of them. The nonparabolicity in the hh2, lh1 subbands is considerably larger than that in the hh1 subband because of the mutual repulsion of the hh2, lh1 subbands arising from the small energy separation between them.

The boundary conditions (2.34) ignore the anisotropy of chemical bonds at an interface. In fact, the point symmetry of a perfect GaAs/AlAs(001) interface, or of a CA/C A (001) interface between the zinc-blende-lattice semiconductors CA and C A , is C2v . For this point group, the heavy-hole (hh) and light-hole (lh) states at k = 0 transform according to equivalent spinor representations and, hence, interface-induced lh-hh mixing is allowed even for zero in-plane wave vector. In [2.16, 2.17] this kind of mixing was postulated by including additional terms in the boundary conditions for the hole envelope functions. In a matrix form, the proposed boundary conditions for the envelopes Fm(z) at k = 0 are written as

2
FA = FB , (ˆvz F )A = (ˆvz F )B − i √				tl-h{JxJy }sFB ,	(2.42)
			a0m0
	3

where x [100], y [010] The prefactor with Planck’s constant , free electron mass m0 and the lattice constant a0 (assumed to be the same for CA and C A ) is introduced to characterize the lh-hh mixing by the dimensionless real parameter tl-h. The mixing under consideration has a nonrelativistic nature and is due to the interface-induced mixing of |X and |Y orbital Bloch functions under normal hole incidence. In [2.18] a tight-binding model has been used to relate the microscopic parameters with the coe cient tl-h. The tight-binding estimation of tl-h = 0.44 is in reasonable agreement with experiment on anisotropic exchange splitting of excitonic levels in type-II GaAs/AlAs SLs (Sect. 5.5.1). The additional term in the boundary conditions (2.42) plays a decisive role in the formation of interface optical anisotropy of heterostructures with no common atom (Sect. 3.1.7) and a large role in the quantum-confined Pockels e ect (Sect. 3.4.3). Di erent theoretical aspects of the heavy-light-hole mixing at zinc-blende (001) interfaces under normal hole incidence were also analyzed in [2.19–2.26].

2.1 Charge Carriers in Quantum Wells

29

Fig. 2.2. Calculated in-plane energy dispersion of holes for the top-most valence

˚

subbands in a GaAs/Al0.5Ga0.5As QW of the thickness a = 200 A. The horizontal dashed-and-dotted line indicates the Fermi level position, EF , corresponding to the 2D hole density ps = 2 × 1011 cm−2. The vertical arrows illustrate direct optical transitions accompanied by photon absorption (solid) or emission (dashed). The shaded areas demonstrate the anisotropy of the hole energy spectrum: the curves bounding these areas correspond to the dispersion along the [100] and [110] in-plane directions. From [2.15].

30 2 Quantum Confinement in Low-Dimensional Systems

2.1.3 E ect of Nonparabolicity on the Confinement Energy

E ects of energy dispersion in bulk constituent semiconductors on quantum confinement of electrons and holes in nanostructures are naturally taken into account in multiband envelope-function theories. In a N -band model, the electron wave function is presented, similarly to (2.32), as a sum

N
Ψ (r) = Φn(r) |n ,	(2.43)

n=1

where |n are the Bloch functions at the extremum point, here the Γ -point k = 0, and N equals the number of involved basic states |n . The model e ective Hamiltonian H(k) is a N × N matrix divided into three parts: the diagonal matrix H(0) containing the energies En0 at k = 0, the matrix H(1) describing the k ·p and spin-orbit coupling between the chosen set |n that is treated exactly, and the matrix H(2) which takes into account by perturbation theory the coupling between the chosen set and remaining (remote) Bloch states. Among di erent k · p models used for description of electronic states, it is worth to mention (i) the 6×6 Luttinger model, Γ8v +Γ7v , of the coupled Γ8 and Γ7 valence bands [2.8, 2.27, 2.28], (ii) the Kane model [2.29–2.32] with a 8×8 Hamiltonian acting in the eightfold space Γ6c +Γ8v +Γ7v of the conduction states Γ6 and the valence states Γ8, Γ7, and (iii) the 14-band model [2.33–2.35] with a Hamiltonian H14×14 acting in the space of states in the Γ6, Γ8, Γ7 conduction bands and Γ8, Γ7 valence bands. Here we will apply the Kane model to demonstrate strengths of the multicomponent envelope-function theories and go beyond the parabolic-band approximation in calculations of conduction-band states.

The 8-band Kane model applied here explicitly takes into consideration the k · p mixing between the lowest conduction band Γ6c and the highest valence-band Γ8v , Γ7v states and ignores coupling with other bands, H(2) = 0. Following Suris [2.36] we present the electron wave function in the form

Ψ = u S + vx X + vy Y + vz Z ,

(2.44)

where u(r) and v(r) are scalar and vector spinor envelope functions. In terms of u and v, the Schr¨odinger equation is conveniently written as follows

		ˆ		(2.45)
	Eu = −iP k · v ,
E + Eg + 3		v = iP kˆu + i	3	σ × v .
	∆		∆

Here E is the electron energy referred to the conduction band bottom Γ6,

ˆ	is the band gap, ∆ is the spin-orbit splitting of the valence
k = −i , Eg
band and		pcv
	P = i		, pcv = S|pˆz |Z .	(2.46)
		m0