ivchenko_bookreg

2.7 Excitons in Semiconductors	71
|ske, mkh = ψcske (re)ψmh kh (rh) ,	(2.158)

where ψcske (re) is the Bloch function for the electron in the conduction band, and ψmh kh (rh) is that in the hole representation obtained from the corresponding electron Bloch function for the valence band by the time inversion

operator ˆ defined in (2.40). With the correspondence between the valence

K

states in the electron and hole representations properly established, the use of (2.158) o ers a number of advantages, rather than presents any di culties.

By the envelope wave function of the exciton in the real-space representation, or the r-representation, one understands the function obtained by

inverse Fourier transform
ϕsm(re, rh) =	ei(ke ·re +kh ·rh )Cske ,mkh .	(2.159)
	k
	ke h

In terms of the two-particle wave function (2.158) the exciton wave function takes the form

		(r , r )ψ0		(r )ψh0		(r ) ,	(2.160)
Ψ exc = ϕ	sm
		e h	cs	e	m	h

sm

where ψcs0 , ψmh0 are the Bloch functions at the extremum point, here the Γ point.

In the e ective mass approximation, the envelopes ϕsm(re, rh) satisfy the two-particle (hydrogen-like) Schr¨odinger equation

s	ˆ ˆ
H	ke, kh ϕs m (re, rh) = Eϕsm(re, rh) .	(2.161)
sm,s m

m

Here E is the exciton excitation energy, i.e., the energy of the excited state (2.157) referred to the energy of the crystal ground state |0 in which the conduction band is empty and the valence band is completely filled. Other

	ˆ	−	ˆ	−		H
	notations are as follows: ke =		i∂/∂re, kh =		i∂/∂rh,
						sm,s m (ke, kh) is

the e ective Hamiltonian of the electron-hole pair, which for a homogeneous semiconductor has the form

Hsm,s m (ke, kh) = δmm Hsse (ke) + δss Hmmh (kh)						(2.162)
	2
+ δss δmm	Eg −	e			,
		æ|re − rh|
and which can be also written in a more compact and symbolic form as
H(ke, kh) = He(ke) + Hh(kh) + Eg −			e2			(2.163)
				.
			æ r
	| |

Here He and Hh are the e ective single-particle Hamiltonians for the electron and hole, r = re − rh, æ is the low-frequency dielectric constant whose dispersion is neglected. Equation (2.161) with the Hamiltonian (2.162) describes

72 2 Quantum Confinement in Low-Dimensional Systems

the state of the so-called mechanical exciton. The calculation of mechanical excitons does not include the exchange interaction between the electron and the hole in the exciton. This interaction as well as exciton-photon coupling will be discussed in Chaps. 3, 5 and 7.

For the simple conductionand valence-band structure, the exciton envelope functions are hydrogen-like. For example, the ground-state exciton, or 1s-exciton, is described by the envelope

Ψ exc(r

, r

) =

eiK·R

ϕ

(r) , ϕ

(r) =

e−r/aB

.

(2.164)

1s e

h

√V

1s

1s

πaB3

Here V is the crystal volume, R is the exciton center of mass,

R =	mere + mhrh	,	(2.165)
	me + mh

me,h is the electron or hole e ective mass, aB is the exciton Bohr radius

	aB =	æ 2	,	(2.166)
		µehe2

and the reduced e ective mass µ = memh/(me + mh). The exciton excitation energy is

E = Eg − EB +	2K2	(2.167)
	2M ,

where the exciton translational mass M = me + mh, and the binding energy for the 1s-exciton, or exciton Rydberg, is given by

	EB =		µehe4	.	(2.168)
		2æ2 2

2.7.2 Free Exciton in a Quantum Well

The two-particle Hamiltonian describing the exciton in a QW includes superstructure potentials (2.5) for the electron and the hole

H(ke, kh) = He(ke) + Hh(kh) + Eg −	e2	+ Ve(ze) + Vh(zh) . (2.169)
	æ r
	|	|

For the two-particle envelope ϕ(re, rh), the first Bastard boundary condition (2.10) takes the form

ϕ(re, rh)|ze =z− = ϕ(re, rh)|ze =z+ ,

ϕ(re, rh)|zh =z− = ϕ(re, rh)|zh =z+ ,

where z− and z+ are the leftand right-hand interface boundaries. The second boundary condition (2.10) taking into account the particle-flow continuity transforms to

2.7 Excitons in Semiconductors

73

! " ! "

e ˆ e ˆ

vz ke ϕ = vz ke ϕ ,

ze =z− ze =z+

! " ! "

h ˆ h ˆ

vz kh ϕ = vz kh ϕ .

zh =z− zh =z+

Here the components ϕsm(re, rh) are combined in one many-component vector ϕ(re, rh), vze,h is the projection of the electron or hole velocity operator

ve(k) = −1 kHe(k) , vh(k) = −1 kHh(k) .

(2.170)

For nondegenerate bands with e ective masses me, mh the exciton wave function can be presented in the factorized form

ϕ(re, rh) = χsmφ(re, rh) ,

(2.171)

where χ depends only on the spin indices s, m. In the equation for the spinindependent function φ the variables can be partially separated

eiK ·R

φ(re, rh) = √ ϕ(ρ, ze, zh) , (2.172)

S

where S is the sample area in the interface plane, ρ = ρe − ρh, ρe,h is the in-plane component of the 3D vector re,h, and the in-plane component of the exciton center of mass, R , is defined similarly to (2.165). The Schr¨odinger equation for ϕ(ρ, ze, zh) reduces to

H0e + H0h + Heh

φ(ρ, ze, zh) = E − Eg − 2M

ϕ(ρ, ze, zh) , (2.173)

2K2

2

∂2

2

∂2

H0e

= −

+ Ve(ze) , H0h

= −

+ Vh(zh) ,

2me

∂z2

2mh

∂z2

e

h

2

∂2

∂2

e2

Heh = −

+

−

.

2µeh

∂ρx2

∂ρy2

ær

The binding energy of the exciton formed by an electron in the lowest conduction subband e1 and of a hole in the upper valence subband h1 is determined by

ε = Ee1 + Eh1 − ϕ|H0e + H0h + Heh|ϕ ,

(2.174)

where Ee1, Eh1 are the electron and hole confinement energies at k = 0.

We first consider the case of thick QWs such as a aB. It corresponds to the weak confinement regime where the Coulomb interaction energy exceeds the intersubband separations ∆Ee, ∆Eh. In a thick well, one may neglect the distortion of internal motion of the electron-hole pair in the exciton and the

exciton is quantized as a single whole,
ϕ(ρ, ze, zh) = F (Z) ϕ(r) ,	(2.175)

74 2 Quantum Confinement in Low-Dimensional Systems

where ϕ(r) is the wave function of the electron and hole relative motion in a homogeneous material and Z is the exciton center of mass on the z axis. For the exciton ground state, ϕ(r) is given by (2.164) and F (Z) by

						cos (νπZ/a)	for odd ν ,
	F (Z) =			2				(2.176)
		a					for even ν ,
						sin (νπZ/a)

where, for simplicity, we assumed the boundary conditions F (±a/2) = 0. For the excitation energy of the 1s-exciton in the excitonic subband ν one has

E = Eg − EB + 2M		a	+ K2	.	(2.177)
2		πν	2

As the well width a decreases, the quantum confinement of carriers begins to predominate over the Coulomb interaction and one can solve equation (2.173) by the variational technique using the trial function in the following factorized form

ϕ(ρ, ze, zh) = f (ρ, z)ϕe1(ze)ϕh1(zh) ,

(2.178)

where z = ze − zh. The simplest trial function

f (ρ, z) =	2	e−ρ/a˜	(2.179)
	πa˜2

has one variational parameter, the e ective 2D Bohr radius a˜. A comparatively simple choice represents the trial function [2.106]

f (ρ, z) = C(1 + αz2)e−δ(ρ2+z2)1/2

with two variational parameters α and δ, C being a normalizing factor. The leftand right-hand barriers of the QW press the electron and the

hole to one another, with the result that the Coulomb interaction between them increases, as does also the binding energy, which in the case of infinitelyhigh barriers varies from EB in a thick well to the 2D Rydberg EB2D = 4EB

for a

a . At the same time, the e ective Bohr radius decreases from a to

2D

B

B

aB

= aB/2. As a result, in the 2D limit one has for the exciton ground state

ϕ(ρ, ze, zh) =

aaB cos

ae cos

ah e−2ρ/aB

(2.180)

π

2

4

πz

πz

for |ze,h| < a/2 and ϕ(ρ, ze, zh) = 0 otherwise.

e

for the

Now we consider the exciton

with the e ective Hamiltonians

H

h

simple conduction band Γ6, and H

for the degenerate valence band Γ8. Due

to the quantum confinement the degeneracy of the valence band is removed as illustrated in Fig. 2.2. In the strong confinement regime the o -diagonal terms in the Luttinger Hamiltonian are neglected and both the conduction

2.7 Excitons in Semiconductors

75

subbands eν and the valence subbands hhν and lhν are treated as isolated subbands from the point of view of exciton formation. This leads to the formation of two exciton systems, i.e., the heavy-hole and light-hole excitons associated with the hole angular momentum components m = ±3/2 and m = ±1/2, respectively. We rewrite the diagonal components of the 3D-hole Luttinger Hamiltonian (2.28) in the form

	F (k) =	2kz2	+	2k2	, G(k) =	2kz2	+	2k2	,	(2.181)
		2mhh		2mhh		2mlh		2mlh

where the masses mhh, mlh coincide with the e ective masses, mhh and mlh, of the heavy and light holes in the bulk material and determine the hole quantum-confinement energies Ehhν , Elhν in a single QW. Two other masses are defined by

2	=	A	+		|B|	,		2	= A	|B|	.
2mhh							2mlh
	|	|		2					| | −	2

They govern the corresponding reduced e ective masses in Heh, see (2.173), namely,

µe-hh = memhh/(me + mhh)		and µe-lh = memlh/(me + mlh) .

For the decoupled heavy and light holes, the wave functions of eν-hhν and eν-lhν excitons are written as

eiK ·R
Ψ exc = √S	ϕ(ρ, ze, zh) ψcs0 (re)ψmh0(rh) .	(2.182)

Here, ψcs0 are the spinor Bloch functions ↑S, ↓S and ψmh0 are the Bloch functions |Γ8, m in the hole representation. For the eν-hhν and eν-lhν excitons, one can use the trial functions

ϕe-hh(ρ, ze, zh) = fe-hh(ρ, z) ϕeν (ze) ϕhhν (zh) ,

(2.183)

ϕe-lh(ρ, ze, zh) = fe-lh(ρ, z) ϕeν (ze) ϕlhν (zh) .

Figs. 2.9a and 2.9b show the binding energies, εhh and εlh, of the heavyhole exciton e1-hh1(1s) and the light-hole exciton e1-lh1(1s) as a function of the thickness a of the GaAs/AlxGa1−xAs QW, for two Al concentrations x = 0.15, 0.30 and for the model QW with infinitely high barriers [2.106]. The parameters used in the calculation are as follows: me = 0.067m0, mhh = 0.45m0, mlh = 0.08m0, µe-hh = 0.04m0, µe-lh = 0.051m0, æ = 12.5. For the infinite potential well, the binding energy monotonously increases with decreasing a in accordance with the expected increase from EB at a aB to 4EB at a aB. For the finite barriers, the binding energies εhh, εlh depend nonmonotonously on a, namely, as the well thickness decreases, they reach a

76 2 Quantum Confinement in Low-Dimensional Systems

ε

		Å		Å

Fig. 2.9. Theoretical dependence of binding energy of e1-hh1 (a) and e1-lh1 (b) exciton on QW thickness in GaAs/AlxGa1−xAs heterostructure with x = 0.15 and 0.3, as well as in the infinitely high potential barrier model. From [2.106].

maximum and then fall o since at small a the electron and the hole bound in the exciton reside predominantly in the barrier regions. In the limit of very small thicknesses, this state of the 1s-exciton may be considered a 3D exciton in the barrier material attached to a thin layer with a potential well for the electron and the hole. The binding energy of such an exciton is close to that of a free exciton in the bulk barrier material.

We discuss the hybridization of heavyand light-hole states in the frame of combined description of strong-confinement regime for electrons and arbitrary regime for holes. In this case, for a stationary exciton with zero translational momentum, K = 0, the exciton wave function takes the form [2.107, 2.108]

1
Ψ exc = √		ϕeν (ze) Fm(ρ, zh)ψcs0 (re)ψmh0(rh) .	(2.184)
	S
		m

If the hole subband anisotropy is neglected and the QW is symmetrical, the envelopes Fm(ρ, z) are characterized by a particular parity with respect to the mirror reflection z → −z and a particular orbital angular-momentum component. As a result, they can be presented as

				2.7 Excitons in Semiconductors			77
Fp(ρ, z) =			F3/2,p(ρ, z)eilϕ			,	(2.185)
	F1/2,−p(ρ, z)ei(l+2)ϕ
					i(l+1)ϕ
	F−1/2,p(ρ, z)e i(l+3)ϕ
F		−	3/2,	−	p(ρ, z)e

where ϕ is the angle between the vector ρ = ρe − ρh and the x axis, the parity p = ± and l is an integer. The total angular momentum component of the exciton equals s + 3/2 + l. Comparing (2.185) with (2.182) we conclude that, for the e1-hh1(1s) exciton with m = 3/2 or m = −3/2, one has p = +, l = 0 or p = −, l = −3. Similarly, for the e1-lh1(1s) exciton with m = 1/2 or m = −1/2, one has p = −, l = −1 or p = +, l = −2. The four-component envelope (2.185) satisfies the equation

! "

˜ ˆ ˜

H(k) + Eg + Eeν + VC (ρ, z) F (ρ, z) = EF (ρ, z) .

Here ˜(k) di ers from the Luttinger Hamiltonian (2.24) by the substitution

H

˜	2k2	˜	2k2
F → F = F +	2me	, G → G = G +	2me

˜

and VC (ρ, z) is the Coulomb potential averaged over the electron-density distribution,

V˜C (ρ, z) =	e2	dze	ϕeν2 (ze)
				.
	æ		ρ2 + (ze − z)2

The calculation shows that the model of uncoupled heavyand light-hole states is a good approximation for the heavy-hole exciton e1-hh1 but can cause a significant error in calculating e1-lh1 excitons and, especially, excitons formed from holes in the higher hole subbands.

2.7.3 Excitons in Various Nanostructures

The physics of excitons has become a highly diversified field of science. The firm identification of exciton spectra were achieved for alkali halides in the 1930’s, for molecular crystals in the 1940’s and for semiconductors in the 1950’s. Excitons have been found in most non-metallic crystals and rare earth metals as well as in photosynthetics and other biological systems. At present it is well known that the excitonic e ects in bulk semiconductors dominate the optical properties near the fundamental band edge, at least at low temperatures and in undoped or moderately doped samples. In semiconductor nanostructures the electron-hole interaction is enhanced by confinement, which increases the overlap of the electron and hole wave functions. The development of new crystal-growth techniques and improvement in nanostructure fabrication allowed to study a significant enhancement of the excitonic binding energy, oscillator strength and exchange-induced splitting of exciton

78 2 Quantum Confinement in Low-Dimensional Systems

levels with transition from 3D excitons in bulk semiconductors to 2D excitons in QWs and similar 2D-1D and 1D-0D transitions, respectively from QWs to QWRs and from QWRs to QD. Omnipresent and many-sided excitons play a central role in the band-edge optical spectroscopy of semiconductor nanostructures even at room temperature. Optical spectroscopic measurements of excitonic parameters, such as the energies of excitonic transitions, exciton oscillator strengths and lifetimes, the fine-structure splittings etc., present e ective methods, contactless and nondestructive, for the characterization of nanostructures and give the information about the structure shape and size, the confining potential profile, interfacial quality, and so forth. In this subsection we briefly outline the theoretical description of excitons in nanostructures.

A simple extension of the intrawell exciton excited within a QW is the interwell exciton in a double QW structure. This indirect-exciton state is formed by an electron and a hole confined in di erent wells. In the presence of an electric field perpendicular to the interfaces the electron and hole ground states are localized in separate wells which allows the realization of a long-lived excitonic state and creates favorable conditions for Bose-Einstein condensation of excitons of high enough density [2.110–2.112]. Since the overlap of the electron and hole wave functions can be easily varied by changing the applied electric field, the lifetime of indirect excitons can be controlled externally. The interwell-exciton envelope function can be approximated by (2.178) where ϕe1(ze) and ϕh1(zh) are the single-particle envelopes of an electron and a hole spatially separated in the opposite wells.

In a single QW with a strong confining potential for conduction-band electrons and zero valence-band o set, the excitonic state can be considered as formed by a 2D electron and a 3D hole. The mixed 2D-3D exciton was realized in a CdTe/Cd1−xMnxTe QW by altering the barrier height in an external magnetic field [2.109]. In the variational approach one can take the envelope function (2.172) and choose a trial function in the form

ϕ(ρ, ze, zh) =	2			exp	−	a2		+ a2		ϕe1(ze) ,
	1						z2		ρ2
							h
			πa a

where a , a are variational parameters. Similar approach can be applied to a type-II QW structure where the conductionand valence-band o sets are of opposite signs. If the B layers neighboring the layer A are thick, then, as well as for the mentioned above 2D-3D exciton, the hole is confined only due to the attractive Coulomb potential. In a periodic type-II structure with strong confinement of electrons and holes, respectively, in A and B layers, one can use the approximation (2.178) for the exciton envelope function.

A special case is realized in GaAs/AlAs periodic multi-layered structures. With decreasing the GaAs-layer thickness, they present a transition from type-I (Γ electron and Γ hole) to type-II (X-valley electron and Γ hole) [2.113,2.114]. The exciton e1X-hh1Γ is doubly indirect, both in the real

2.7 Excitons in Semiconductors

79

space, because the envelope functions are spatially separated in the GaAs and AlAs layers, and in the k space, because the electron state is formed predominantly from the X-valley states. The oscillator strength for this exciton is much weaker than in type-I structures since no-phonon optical transitions between purely Γ -like valence and X-like conduction states are forbidden. This selection rule is removed due to the Γ -X mixing at interfaces [2.115–2.118].

In a MQW structure, excitons excited in individual QWs are coupled via electro-magnetic field to form exciton polaritons that are considered in Chap. 3. Exciton polaritons in quantum microcavities are discussed in Chap. 7. In a short-period SLs with 3D minibands, the exciton states are similar to those in a uniaxial 3D semiconductors (Sect. 3.1.3).

If the cross-sectional dimensions of a QWR or the sizes of a QD exceed the Bohr radius aB of a 3D exciton (weak confinement regime) then the exciton wave function is written in the form similar to (2.175) with the function F (X, Y ) or F (R) describing the translational motion of the exciton as a whole. The calculation of this function is performed analogously to sizequantization of single-particle motion in 1D and 0D nanostructures (Sects. 2.2 and 2.3).

In the strong confinement regime, the exciton wave function in a QWR is approximately presented as

ΨQWRexc =	exp (iKz Z)	f (z) ϕe1(xe, ye) ϕh1(xh, yh) ψcs0 (re) ψmh0(rh) , (2.186)
	√L

where ϕe1(xe, ye), ϕh1(xh, yh) are the single-particle envelopes defined in (2.55). The envelope function describing the relative motion along the principal axis z satisfies a second-order di erential equation

2 d2f (z)

˜

−

2µeh

dz2

+ VC

(z)f (z) = Ef (z)

with the e ective

1D Coulomb potential

V˜C (z) = −

e2

dxedyedxhdyh

ϕe21(xe, ye)ϕh21(xh, yh)

.

æ

(xe

−

xh)2

+ (ye

−

yh)2

+ z2

The simplest trial function is [2.119]

f (z) = √aw exp −a| w|

1

z

with a single variational parameter, aw. A slightly more complicated choice is to change f (z) in (2.186) by f (ρ, z), say, [2.120]

f (ρ, z) exp	−			,
			ρaw
			2 + z2

80 2 Quantum Confinement in Low-Dimensional Systems

where ρ = |ρe − ρh|.

Note that the 1D Coulomb potential VC (z) = −(e2/æ|z|) leads to an infinite binding energy of the 1D exciton ground state. Loudon [2.121] considered analytically the ground-state problem for a regularized 1D Coulomb potential VC (z) (|z| + R)−1 instead of |z|−1 and obtained for the ground state the asymptotic solution with the binding energy ε satisfying the equation

R	=	1	EB		exp	−	1	EB
							2			.
aB		2		ε					ε

One can see that as R → 0 the ground-state binding energy diverges.

The exciton wave function in a QD satisfying the condition of strong-

confinement regime is just a product
Ψ exc = ϕ	e1	(r ) ϕ	h1	(r	h	) ψ0	(r	e	) ψh0	(r )	(2.187)
		e				cs			m	h

of the single-particle wave functions. The excitation energy of the groundstate exciton is given by

˜

E = Eg + Ee1 + Eh1 + VC ,

where the last term is the Coulomb energy

V˜C = −	e2	dredrh	ϕe21(re)ϕh21(rh)
				.
	æ		|re − rh|

The exchange-interaction corrections to the 0D-exciton energy are analyzed in Chap. 5.

It is worth to mention that, in molecular crystals, the intermolecular interactions can be highly anisotropic. As a result, an e ective dimensionality d = 2 or d = 1 for the energy transport may exist. Particularly, the organic crystals 1,2,4,5-tetrachlorobenzene and 1,4-dibromonaphthalene possess 1Dexcitons [2.103].

Above we considered excitons in ideal nanostructures. Interface imperfections and composition fluctuations can give rise to localized exciton states which are quasi-0D states with a strong confinement along z and weak confinement in the interface plane (x, y). If the length of the exciton localization in a QW is much larger than the 2D Bohr radius, one can use the adiabatic approximation and represent the exciton envelope function in the form

Ψexc(re, rh) = F (X, Y ) f (ρ) ϕe1(ze) ϕh1(zh) ,

(2.188)

where the function F (X, Y ) describes the localization of the e1-h1(1s) exciton as a whole in the interface plane, X and Y are the in-plane coordinates of the exciton center of mass. The fine structure of the heavy-hole exciton localized on a rectangular island in an imperfect QW structure is considered in Sect. 5.5.1.