ivchenko_bookreg

2.5 Electrons, Photons and Phonons in Superlattices

61

In a structure with isotropic layers, there are two types of Bloch solutions,

or normal light waves, namely, TE waves (s-polarized waves) with	E	y
and B lying in the (x, z) plane, and TM, or p-polarized waves with	B	y.

Obviously, for TE waves, the electric field Ey plays the role of the generalized envelope ϕ(z). It follows from continuity of the tangential components Ey and Bx and from the relation Bx ∂Ey /∂z, that Ey |A =Ey |B and (∂Ey /∂z)|A = (∂Ey /∂z)|B. Therefore, for the TE-polarized light waves, we have CA = CB and N = kB/kA.

Similarly, for TM waves, the role of ϕ(z) is played by By , the first boundary condition is By |A = By |B, and from continuity of Ex æ−j 1(∂By /∂z) we obtain the second boundary condition

1		∂By		=	1	∂By	,
æA		∂z A			æB	∂z
							B

i.e., CA = æ−A1, CB = æ−B1, N = (kBæA)/(kAæB). The dispersion equation for both TEand TM-waves is given by the canonical equation (2.126).

In the long-wavelength approximation, when |kA|a, |kB|b, |K|d 1, the optical SL may be considered a homogeneous uniaxial medium with the effective dielectric tensor

	a + æ b		a		b	−1
æxx = æyy =	æA B	, æzz =		+		(a + b) . (2.132)
	a + b		æA		æB

This result can be obtained by expanding the trigonometrical functions in (2.126) in powers of kAa, kBb, Kd. However, it can be also derived in a simple way if we take into account that, in the long-wavelength limit, the fields E and B change very little within each layer. The electric field Ey is continuous at the interface and the di erence of its values in the neighboring layers may be ignored. The displacement Dy is approximately constant within each layer but its values jump at the interface from DA,y = æAEy to DB,y = æBEy . The electric displacement averaged over the period is given by

¯	a DA,y + b DB,y	=	æA a + æB b
Dy =	a + b			a + b		Ey ,

which agrees with the expression (2.132) for æyy . If we take into account the continuity of the normal component Dz of the displacement field and the relations EA,z = Dz /æA, EB,z = Dz /æB, we obtain for the average (macroscopic) field

¯	=	a EA,z + b EB,z	=	a æA−1 + b æB−1	D		,
		a + b		a + b
Ez						z

in agreement with the above expression for æzz .

Equation (2.126) for TM-modes suggests an important limiting case realized for

	ω			2
q2				|æA,B| .	(2.133)
		c

62 2 Quantum Confinement in Low-Dimensional Systems

In this case the retardation e ects are negligible, one can use the approxima-

tions

kA, kB → iqx , N → æA ,

æB

and obtains, instead of (2.126), the secular equation for the so-called interface modes [2.98, 2.99]

cos Kd = cosh (qxa) cosh (qxb)							(2.134)
+	1		æB	+	æA	sinh (qxa) sinh (qxb) .
	2		æA		æB

Taking into account the condition (2.133) the above equation can be derived from a scalar potential φ(r) which determines the electric field E = − φ if the retardation is neglected. In this approximation the magnetic field is zero. Within each A and B layer, the Laplace equation ∆φ(r) = 0 is fulfilled by the exponential functions exp (iqxx ± qxz). Thus, the generalized envelope ϕ(z) reduces to the potential φ and one indeed has kA = kB = iqx. The boundary conditions are φA = φb and æA(dφ/dz)A = æB(dφ/dz)B which leads to CA = æA, CB = æB, N = æB/æA and, finally, to (2.134).

For a SL with thick layers, qxa, qxb → ∞, this equation reduces to

1 +	1		æB	+	æA	=	(æA + æB)2	= 0 .
	2		æA		æB		2 æA æB

This is nothing more than the well-known equation æA(ω) + æB(ω) = 0 for surface modes at a single interface between semiconductors A and B. It has solutions only if there are frequencies for which the dielectric functions æA(ω) and æB(ω) have opposite signs. To illustrate we consider the surface optical phonon. Let ω lie in the region of optical-phonon resonance frequency of the host material A. Then, one has

	æ		(ω) = æ	ΩLO2 − ω2	,
		A		∞ ΩT2 O − ω2

where ΩT O, ΩLO are the resonance frequencies of transverse and longitudinal optical phonons. For simplicity, the dielectric function of the B material is assumed to be independent of ω and equal to the high-frequency dielectric constant of the A material, æ∞. Then, the frequency of the surface wave is

Ωs = (ΩLO2 + ΩT2 O)/2. If ΩLO − ΩT O ΩT O, the value Ωs lies at the midpoint between ΩLO and ΩT O.

For a SL with the thick B layers, qxb → ∞, and the A layer of finite

thickness, we obtain from (2.134)			+ æB
cosh qxa + 2	æA			sinh qxa = 0 .
1		æB	æA

2.5 Electrons, Photons and Phonons in Superlattices

63

These are mixed leftand right-hand-side surface modes coupled because of the finite thickness a [2.100]. For the above simple model, the eigenfrequencies are given by

ω2 = ωs2 ± 12 (ΩLO2 − ΩT2 O)e−qx a .

Particularly, if ΩLO − ΩT O ΩT O, we have

ω = Ωs ± 12 (ΩLO − ΩT O) e−qx a .

Now it is clear that the interface mode in a SL with finite a and b is a surface excitation transferred from one interface to another and propagating in that way along the SL principal axis.

The last problem involving wave propagation in SLs and related to the canonical equation (2.126) concerns folded acoustic phonons. Here we restrict ourselves to longitudinal acoustic phonons propagating along the principal axis z. The role of ϕ(z) is played by the lattice displacement uz (z) along z. In the elastic limit the dispersion relations of the constituents are linear, namely, ωA(q) = sAq and ωB(q) = sBq, where sA,B is the longitudinal sound velocity. Therefore, kA and kB in the generalized description equal ω/sA and ω/sB, respectively. The first boundary condition is the continuity of uz while the second condition follows from the continuity of the stress tensor component σzz related to the strain tensor component uzz = ∂uz /∂z by σzz = λuzz . Here λ is the elastic sti ness constant Czzzz which can be expressed via the bulk density ρ and the longitudinal sound velocity s: λ = ρs2. Since the second boundary condition is rewritten as

λAuzz |A = λBuzz |B ,

we have

CA = λA , CB = λB , N = λBkB = ρBsB , λAkA ρAsA

and equation (2.126) takes the form [2.99, 2.101]

cos Kd = cos		aω			bω
				cos						(2.135)
					sB
− 2		sA					sin	sA	sin	sB .
		ρAsA +			ρBsB
1			ρBsB		ρAsA			aω			bω

Usually, in semiconductor SLs the parameter

ρBsB − ρAsA ε = √ρAρBsAsB

is small (2.135) can be conveniently rewritten in the slightly modified form

	a		b		ε2	aω			bω
cos Kd = cos ω		+		−	2 sin			sin		.	(2.136)
	sA		sB				sA		sB

64 2 Quantum Confinement in Low-Dimensional Systems

Neglecting the term proportional to ε2 we obtain the dispersion relation

K = (kAa + kBb)/d = ω/s¯ ,

(2.137)

which is that of a homogeneous medium with the average sound velocity

s¯ = (a + b)	a	+	b	−1
				.
	sA		sB

In the reduced zone scheme the straight line ω = sK¯ is folded within the first Brillouin zone of the SL, −π/d < K ≤ π/d, as shown in Fig. 2.8 by the dashed curve. Allowance for nonzero value of ε results in the formation of acoustic allowed and forbidden minibands. For small values of the mismatch ε the width of the first forbidden miniband at K = 0 is found from the condition

1

ω

2

ε2

ωa

ωa

0 = −

d − 2π

−

sin

sin

2

s¯

2

sA

sB

or ∆ω = 2 (¯s/d) |ε sin (ωa/sA)|. Here we took into account that, at ω = s¯(2π/d), one has sin kBb ≈ sin (2π − kAa) = − sin kAa.

Fig. 2.8 schematically represents the dispersion of folded acoustic phonons in a SL. The solid curve shows the spectrum at a nonzero value of the mismatch ε. The dispersion branches, or acoustic minibands, are labelled by the index l running through the integers 0, ±1, ±2... The branch outgoing from the point ω = 0, K = 0 corresponds to l = 0, two next branches are labelled by l = ±1 etc. Under such labelling the dashed curve in Fig. 2.8 is described analytically by

ΩlK = s¯ |K|sign{l} +	2d |l| .	(2.138)
	π

Raman scattering of light by folded acoustic phonons is considered in Chap. 6.

2.6 Interband Optical Transitions

We shall now consider the optical transitions at the fundamental absorption edge in undoped semiconductors and semiconductor nanostructures. In a semi-classical approach the electromagnetic field is treated classically while the electronic subsystem is described quantum-mechanically. In the following we use the Coulomb gauge with a zero scalar potential. In this gauge the electric and magnetic fields, E and B, are expressed via the vector potential A(r, t) by

1			∂A	, B = curl A ,	(2.139)
E = −
	c		∂t

where c is the light velocity in vacuum. For a plane monochromatic light wave propagating in a homogeneous medium, one has

2.6 Interband Optical Transitions

65

+2		+2
-2		-2
+1		+1
-1		-1
0		0
-π/d	0	π/d

Fig. 2.8. Schematic representation of the dispersion of folded acoustic phonons in a SL. Dashed and solid lines show the dispersion, respectively, for zero and nonzero values of the parameter ε in (2.136).

A(r, t) = A0e−iωt+iq·r + A0eiωt−iq·r ,

(2.140)

where q is the light wave vector and A0 is the amplitude related with the energy flux I by

A2	=	2πc	I ,	(2.141)
0		nω ω2

nω being the refractive index of the medium at the frequency ω. It is convenient to present the vector A0 as a product A0e of the real (positive) scalar amplitude A0 and the polarization unit vector e satisfying the condition |e|2 = 1. We remind that for an elliptically polarized light wave the unit vector e is complex. In an interband optical absorption process, the radiation field transfers an electron from the occupied valence band to the unoccupied conduction band. In other words, the photon absorption is followed by the generation of an electron-hole pair. Here we treat the interband absorption in the single-particle approximation neglecting the interaction between the photogenerated electron and hole. In the following section and other chapters of the book we will go beyond this approximation and include the e ects of electron-hole Coulomb interaction, or excitonic e ects, into consideration.

[pˆ · A(r, t) + A(r, t) · pˆ] ,

66 2 Quantum Confinement in Low-Dimensional Systems

2.6.1 Transition Probability Rate

In linear optics, the perturbation operator of the electron-photon interaction

has the form

−e 1 c 2m0

where pˆ is the momentum operator −i . In the dipole approximation the coordinate dependence of A is neglected and the perturbation operator takes the form

−eA0 e e−iωt + e eiωt · pˆ . cm0

Using Fermi’s golden rule to second order perturbation one can write for the optical transition rate

W (d)	= 1	2π			M (eνs, vν m; k) 2 δ(Eeνsk		Evν mk	ω) , (2.142)
eν,hν	Vd			|	|	−		−
			smk

where the matrix element of the one-photon direct transition hν mk → eνsk is given by

M (eνs, vν m; k) =	eA0	e		p	eνs,vν m	(k) ,	(2.143)
−cm0			·

peνs,vν m(k) ≡ eνsk|pˆ|vν mk is the matrix element of the momentum operator taken between the single-electron conductionand valence-band states, and the indices ν, ν enumerate the quantum-confined electron states in QWs (d = 2), QWRs (d = 1) and QDs (d = 0). The valence-band energy Evν mk in the δ-function describing the energy conservation law is given in the electron representation. Note that it is related with the energy of the corresponding hole state by Evν mk = −Ehν m,¯ −k where the bar over m means the spin opposite to m. The transition probability rate (2.142) is related to unit volume of the d-dimensional space, Vd being the macroscopic volume of the system, namely, the 3D volume of a bulk semiconductor, the lateral area of a QW, the wire length L for a QWR, and Vd → 1 for a QD. The vector k in (2.142, 2.143) is a d-dimensional vector, if d = 1, 2, 3, and this quantum number is absent for 0D systems.

In what follows we focus the attention on direct optical transitions in the vicinity of the Γ -point. Neglecting k dependence of the matrix element peνs,vν m we can write it as

peνs,vν m = iνν pcs,vm .

(2.144)

The first multiplier called the overlap integral is defined for a QW as

iνν = ϕeν (z)ϕhν (z)dz (2.145)

and similarly for QWRs and QDs. The second multiplier is the matrix element of the momentum operator calculated between the Γ -point Bloch functions,

pcs,vm = Ω0		Ω0	2.6 Interband Optical Transitions	67
			ucs(r) pˆ uvm(r) dr ,	(2.146)
	1

where Ω0 is the unit-cell volume.

If the spin splitting of electronic states is neglected the probability rate

is rewritten as
Weν,hνQW =	e2		I
					geν,hν ( ω) iνν2	|e · pcs,vm|2 ,	(2.147)
	cnω		m2	ω2
		0				sm

where geν,hν (E) is the reduced density of states

1

geν,hν (E) = Vd δ(Eeνk − Evν k − E) . (2.148)

k

To illustrate we consider the parabolic dispersion of the electron and hole bands, or subbands, assuming

	Eeνk = Ec0 + Eeν +		2k2	, Evνk = Ev0 − Ehν −	2k2
						,
			2me		2mh
where E0	, E0	are the band edges in the well host material, E and E					hν	are
c	v				eν

the electron and hole quantum-confinement energies. For this simple band structure, we obtain the reduced densities of states

1

3/2

g3(E) =

2µeh

E − Eνν0 θ(E − Eνν0 ) ,

(2.149a)

4π2

2

g2(E) =

µeh

θ(E − Eνν0

) ,

(2.149b)

2π 2

1

2µeh

1/2

g1(E) =

θ(E − Eνν0 ) ,

(2.149c)

2π

2(E − Eνν0 )

g0(E) =

δ(E − Eνν0 ) ,

(2.149d)

where µeh = memh/(me +mh) is the reduced e ective mass, Eνν0

is the band

gap attached to the pair of subbands eν and hν ,

Eνν0 = Eg + Eeν + Ehν ,

(2.150)

and Eg = Ec0 − Ev0. The fundamental absorption edge

EgQW ≡ E110 = Eg + Ee1 + Eh1

(2.151)

corresponds to the absolute valence-band maximum and conduction-band minimum.

It follows from (2.149) that, in a QW, the density of states g2(E) looks like a horizontal step. In a QWR, the dependence g(E) has a square-root peak and is similar to that for electron states in a bulk semiconductor subject to

68 2 Quantum Confinement in Low-Dimensional Systems

a quantizing magnetic field. In a QD, the energy level structure is discrete, the isolated levels are somewhat broadened only due to finite lifetimes of electrons and holes. The density of states for single electrons and holes is defined by expressions similar to (2.148). Note however that, as compared with (2.148), the single-particle density of states usually has an additional factor of 2 to take into account the spin degeneracy.

2.6.2 Selection Rules

In QW, QWR or QD structures with isotropic, both electron and hole, e ective masses in the well host material and infinitely high barriers, the envelope functions ϕeν and ϕhν are identical and independent of the e ective masses. Since, by definition, each of the sets is orthonormalized, we come to the selection rule

iνν = δνν .

(2.152)

Thus, the optical transitions take place only between conduction and valence subbands (or levels in QDs) with the coinciding quantum numbers.

For finite barriers, the shape of envelope wave function depends on the e ective masses and the sets of envelopes for electrons and holes are di erent. Nevertheless, in many nanostructures, this di erence is not dramatic and one can use the following soft selection rules

iνν	≈	1 and	|	iνν	|		(2.153)
						1 for ν = ν .

If the confining potential has a center of symmetry, then the interband transitions between the states with envelopes ϕeν , ϕhν of opposite parity are of course forbidden.

Now we turn to the selection rules imposed by the second multiplier in (2.144). Table 2.2 presents the interband matrix elements e · pcs,vm calculated between the conduction-band states |Γ6, s = αsS and the valence-band states |Γ8, m in the basis (2.22). The initial and final states, |vm and |cs ,

are both taken in the electron representation. The values presented in the

√

table should be multiplied by pcv / 2, where the interband matrix element pcv is defined in (2.46).

Table 2.2. The interband matrix elements of the momentum operator,

√

e · pcs,vm, related to pcv / 2. Here e is the light polarization unit vector.

	v, + 3/2	v, + 1/2				v, – 1/2					v, – 3/2
		√						√
c,+ 1/2	– (ex + iey )	2ez /	3		√	(ex −√ie	y )/		3		0
c, – 1/2	0	−(ex + iey )/			3	2ez / 3					ex − iey

We remind that, in the electron and hole representations, the angular momenta of the valence states di er in sign and, therefore, the matrix elements in these two representation are related by M (eνs, hν m; k) =

2.6 Interband Optical Transitions

69

M (eνs, vν , m¯ ; −k). Now we take into account that, in QW structures, heavyand light-holes at kx = ky = 0 are quantized independently. By using Table 2.2 and changing the sign of m we obtain the probability rates for the generation of electron-hole pairs |eνs ; hhν, m and |eνs ; lhν, m in QWs

|M (eν, ±1/2	; hhν, 3/2)|2	|ex ± iey |2 ,					(2.154)
|M (eν, ±1/2	; hhν, ±3/2)|2	= 0 ,
|M (eν, ±1/2	; lhν, ±1/2, )|2	1				|ex iey |2 ,
				3
		4
|M (eν, ±1/2	; lhν, 1/2)|2				|ez |2 .
			3

Similar equations for the interband transitions from the spin-orbit (so) split valence band Γ7, in the basis (2.23), are as follows

|M (eν, ±1/2	; h, so, ν, 1/2)|2	2		|ex iey |2 ,		(2.155)
		3
		2
|M (eν, ±1/2	; h, so, ν, 1/2, )|2				|ez |2 ,
			3

where the symbol h reminds that here we use the hole representation for valence band states.

For the σ+ circularly-polarized light propagating along z, the following equations hold

ox + ioy

, ez = 0 and

ex + iey

2

= 0 ,

iey

2

= 2 ,

e = √

ex

|

−

|

2

|

|

whereas, for the σ−-polarized light, one has

e =	ox − ioy			,	e	+ ie	2 = 2 , e	ie	2	= 0 ,
									y |
	√2				| x	y |	| x −

where ox, oy are the unit vectors pointing in the directions x and y, respectively. The selection rules for the interband photoexcitation of an electron (eν, s) and a hole (hhν, m) in a QW for both linear and circular polarizations are presented in Table 2.3. One can see that, under interband transitions, the angular momentum component along z is conserved

s + m = M ,

(2.156)

where M = ±1 for the σ± polarization and M = 0 for the linear polarization along z.

Table 2.3. Selection rules for interband optical transitions in QW structures. Numbers give values of squared matrix elements |M (eν, s ; jν, m)|2 (j = hh, lh) in relative units. |jν, m is the valence band state in the hole representation.

70 2 Quantum Confinement in Low-Dimensional Systems

		linear	linear	circular	circular
s, m		e z	e z	σ+	σ−
				ey = iex	ey = −iex
1/2, −3/2		1	0	0	2
−1/2, 3/2		1	0	2	0
1/2, 3/2		0	0	0	0
−1/2, −3/2		0	0	0	0
1/2, 1/2		1/3	0	2/3	0
−1/2, −1/2		1/3	0	0	2/3
1/2, −1/2		0	4/3	0	0
−1/2, 1/2		0	4/3	0	0

2.7 Excitons in Semiconductors

2.7.1 Free Excitons in Bulk Crystals

Up to now we followed the single-particle Bloch scheme describing the independent motion of charge-carrying electrons and holes. The concept of excitons proposed by Frenkel in 1931 [2.102] goes beyond the scope of this scheme. The free exciton is an electron excitation which involves correlated motion of electrons and holes, does not carry current, but does carry energy. Frenkels’s original model is appropriate for molecular crystals. Due to intermolecular interaction, an excited molecule can induce an upward transition in the neighboring molecule de-exciting itself. Thus, the molecular exciton can move and an exciton band is formed. In addition to Frenkel excitons, there are two other basic types of excitons [2.103]. In charge-transfer excitons, the excited state is formed by an electron and a hole lying on neighboring atoms. In the limit opposite to the Frenkel case and applicable for most semiconductors, the electron and hole are separated by many inter-atomic spacings and one can use the e ective mass approximation to calculate the exciton energy levels and wave functions. This is the so-called Wannier-Mott exciton first considered in [2.104, 2.105].

The wave function of the Wannier-Mott exciton can be expanded in the states of noninteracting electron-hole pairs as follows

ske	h	|ske, mkh .
Ψ exc =	Cske ,mkh		(2.157)

,mk

Here s, m are the indices identifying the conductionand valence-band spin branches, ke,h is the electron or hole wave vector, |ske, mkh is the excited state of the crystal in which only one conduction-band state |ske is occupied

and only one valence state ˆ mkh is empty. In fact, ske, mkh is a many-

K| |

particle state. Sometimes it is presented in the e ective form of a two-particle wave function