ivchenko_bookreg

5.2 Emission Spectra of Localized Excitons

221

incident and excitonic electrons are nondistinguishable and make allowance for both direct and exchange processes.

We consider a collision process with a free quasi-2D electron and a localized exciton in the initial state, and a free (delocalized) exciton and an electron of lower kinetic energy in the final state. Schematically the process can be presented in the form

(k0, s0) + (loc. exc., s0, m) −→ (kf , sf ) + (free exc., K, sf , m).

Here k0, kf are the wave vectors of the initial and scattered electron waves, K is the free-exciton wave vector, and s0, s0, sf , sf is the spin component ±1/2 of the corresponding electron referred, say, to the structure principal axis z; m is the hole spin index. Hereafter we neglect the electron-hole exchange interaction which means that m is conserved in the collision process. The vectors k0, kf , K satisfy the energy conservation law

2k02	− Eloc =	2kf2	+	2K	2	(5.38)
				2M	,
2me		2me

where me, M are the in-plane electron and exciton e ective masses, and Eloc is the exciton localization energy.

Bearing in mind the electron nondistinguishability we write the wave function of the three-body system ‘electron + exciton’ as the Slater determinant

Ψ (r1, r2	, rh) = √2	&	ψj(exc);s m	(r1	, rh)
	1	&	ψk s(r1)

&

&

&

ψj(exc);s m	(r2	, rh) &		,	(5.39)
ψk s(r2)			&

&

&

&

where 3D vectors r1, r2, rh describe positions of the two electrons and the hole, ψk s(r) and ψj(exc);s m(re, rh) are, respectively, the wave functions of a free

electron in the state (k, s) and of an exciton in the localized state j = loc or the free state j = K. In the actual calculation the following approximate wave functions are used

1
ψk s(re) = √Ω eikρe ϕe(ze) |s ,	(5.40)

ψj(exc);s m(re, rh) = Fj (R)f (ρ) ϕe(ze) ϕh(zh) |s ; m .

Here |s and |s ; m are the Bloch functions at the center of the Brillouin zone, ϕe(ze) and ϕh(zh) are the envelopes for the lowest electron (e1) and hole (h1) quantum-confined states, the variational function f (ρ) = (2/π)1/2a˜−1 exp (−ρ/a˜) describes the relative electron-hole motion in the exciton, a˜ is the e ective Bohr radius of the quasi-2D exciton in the 1s ground state, ρ = |ρe − ρh|, ρe and ρh are the 2D vectors (xe, ye) and (xh, yh), R = (meρe + mhρh)/M is the exciton in-plane center-of-mass and Ω is the sample normalization area. The function Fj (R) describes the translational

222 5 Photoluminescence Spectroscopy

motion of the exciton as a whole. In the initial state it is approximated by the exponential function

Floc(R) =	2	e−R/L	(5.41)
	πL2

with L being the e ective localization radius. The final state is represented by the plane wave FK (R) = Ω−1/2 exp(iKR).

In order to calculate the matrix element of the electron-exciton interaction

M = k	, s	; K, s	, m Hˆ	k , s ; loc, s		, m
f	f	f	|	| 0 0	0

we start with the total three-body Hamiltonian V1-h + V2-h + U1 + U2 + Uh, where T stands for V is the Coulomb potential

H = T1 + T2 + Th + V1-2 + the kinetic energy operator,

	e2			e2
V1-2 =		,	Vi-h = −	æ|ri − rh| (i = 1, 2) ,
	æ|r1 − r2|

æ is the dielectric constant of the medium, U is the superstructural potential. It is worth to note that U contains as well the perturbation arising because of the QW-width fluctuation and leading to the exciton localization.

In the band-structure model under consideration the electron orbital motion is spin-independent and, therefore, it is possible to bring the matrix element M into the form

V =	A	f	0	+	B	(σsf s0	·	f	0	) ,	(5.42)
		δsf s0 δs	s					σs	s

where σ is the vector matrix with components being equal to the Pauli matrices σx, σy , σz . The coe cients A, B depend upon the wave vectors k0, kf , K but are independent on the spin indices. These coe cients can be expressed as

A = V↑↓,↑↓ +	1	V↑↓,↓↑ ,	B =	1		V↑↓,↓↑	(5.43)
	2				2

via matrix elements for the direct and exchange collision processes in the case where spins of the initial and excitonic electrons are antiparallel

V↑↓,↑↓ =	dr1dr2drhψkf (r1)ψK(exc) (r2, rh)V12hψk0 (r1)ψloc(exc)(r2, rh) ,
V↓↑,↑↓ = −	(5.44)
	dr1dr2drhψkf (r2)ψK(exc) (r1, rh)V12hψk0 (r1)ψloc(exc)(r2, rh) .

Here we use the notation V12h = V1-2 + V1-h. It follows from (5.42,5.43) that the similar matrix element, V↑↑,↑↑, for the parallel spin configuration is given by the sum V↑↓,↑↓ + V↓↑,↑↓.

In the Born approximation the probability rate for delocalization of the exciton (s0, m) by the s0-electron is expressed by

5.2 Emission Spectra of Localized Excitons

223

ws ,s (E0) =	2π	sf f	V 2 δ(Ef + Eexc		E0 + Eloc) .	(5.45)
0 0			f	−
			| |
		,s	k ,K

To simplify the writing we denote the kinetic energies 2k02/2me, 2kf2 /2me,2K2/2mexc by E0, Ef and Eexc, respectively. The total delocalization time, τ , can be written as 1/τ = W1/2,s0 + W−1/2,s0 , where

Ws0 ,s0 = Ω ws0 ,s0 (E0) fs0 (E0) g2D dE0 , (5.46)

g2sD = me/(2π 2) is the density of 2D-electron states with a fixed spin and fs(E) is their energy distribution function. We assume the Boltzmann character of the distribution: fs(E) = exp[(µs − E)/kB T ], where T is the e ective temperature which can be changed by microwave irradiation of the sample [5.20]. The chemical potential µs is related to the 2D concentration, Ns, of electrons with the spin s by the equation

	g2sD kB T
µs = kB T ln	Ns	.	(5.47)

In experiments the electron gas may be unpolarized (N1/2 = N−1/2) as well as completely spin-polarized (N1/2 N−1/2 or N1/2 N−1/2).

Figure 5.6 illustrates results of the numerical calculation of the probability rate ws0,s0 (E0) and the delocalization time τs0,s0 performed assuming the electron gas to be polarized and by applying (5.42, 5.45) for the configurations s0 = −s0 (curves 1) and s0 = s0 (curves 2). The used parameters correspond to the GaAs/AlGaAs QW structures: me = 0.067m0, mh = 0.15m0, the Bohr

˚ ˚

radius of the quasi-2D exciton a˜ = 100 A, the well width a = 75 A. A value of

˚

L = 180 A is chosen as the best fit to the envelope wave function calculated in the method of free relaxation [5.21] for a circular island of monolayerhigh fluctuation of the QW width (the localization energy Eloc = 2 meV is

˚

obtained for the island radius 80 A). The envelopes ϕe(z), ϕh(z) in (5.40) are approximated by the function (2/a)1/2 cos(πz/a).

The delocalization probability has a threshold in energy of the ingoing electron: ws0,s0 (E0) = 0 if E0 < Eloc. Near the threshold we can neglect the dependence of the coe cients A, B on E0 and obtain

ws0,s0 (E0) dEf dEexc δ(Ef + Eexc − E0 + Eloc) = E0 − Eloc . (5.48)

One can see from Fig. 5.6a that the linear dependence is a good approximation

up to E0 − Eloc = 0.5 meV. The curves ws0,s0 (E0) reach maximum values at E0 − Eloc ≈ 1.5 meV (k0L ≈ 1.4) and smoothly decrease with the further increase in E0. Note that the main contribution to w↑↑ and w↑↓ comes from the exchange processes. This can be understood taking into account that, in the Born approximation, for equal masses me = mh and for coinciding

224 5 Photoluminescence Spectroscopy

Fig. 5.6. (a) Relative probability of the exciton delocalization vs. the initial energy for antiparallel (curve 1) and parallel (curve 2) electron spin configurations. wmax is the maximum value of the function w↑↓(E0). Inset shows the same dependence in the region of small E0 − Eloc. Dashed curves 1 and 2 are calculated assuming the matrix element of the electron-exciton interaction to be constant. (b) Temperature dependence of the exciton delocalization time due to the 2D electron gas polarized antiparallel (curve 1) and parallel (curve 2) with respect to the electron spin orientation in the localized exciton. Inset shows the same dependence near the threshold; dashed curves are calculated for E0-independent matrix elements of the electron-exciton interaction. [5.16]

envelopes ϕe(z) = ϕh(z), the direct Coulomb interaction between an electron and an exciton vanishes, the matrix element V↑↓,↑↓ is equal to zero and, hence, the matrix elements V↑↑,↑↑, V↓↑,↑↓ coincide and are entirely determined by the exchange contribution to electron-exciton interaction. For me = mh, a value of V↑↓,↑↓ di ers from zero. However the estimation shows that, even for mh = 2me, it is small as compared to V↑↑,↑↑, V↓↑,↑↓. This explains why the probabilities w↑↑(E0), w↑↓(E0) di er only by 10 ÷ 20%.

For the electron gas of the density n = 109 cm−2 at the temperature T = 5 K the exciton delocalization time is 1 ns. With heating the electrons their average energy increases, the electron-exciton interaction is enhanced (Fig. 5.6a) and the time τ decreases. Figure 5.6b shows that, at low temperatures, this time is very sensitive to variation of T . In fact, it becomes shorter by an order of magnitude, as T increases from 5 K to 8 K. If the E0- dependence of the matrix elements V is ignored, i.e., if ws0,s0 E0 − Eloc, then the delocalization rate is proportional to T exp(−Eloc/kB T ) (dashed curves in the inset in Fig. 5.6b). It should be mentioned that the time τ corresponding to the electron density n = 109 cm−2 is obtained by multiplying the values in Fig. 5.6b by 109/n. To illustrate, the inverse time τ −1 for an unpolarized electron gas is given by (τ↑↑−1 + τ↑↓−1)/2.

It is to be noted in conclusion that above we considered only one of possible mechanisms of the interaction between an electron and a localized exciton.

5.3 Optical Spin Orientation of Free Carriers

225

The photoluminescence intensity and polarization can be e ected by other processes: the spin-flip elastic scattering of an electron or a hole by a localized exciton, the delocalization of an exciton with the free carrier captured in the final state by the localization site, free-carrier-induced hopping of an exciton between two di erent localized levels, etc.

5.3 Optical Spin Orientation of Free Carriers

Spin properties attract the great attention in recent years due to attempts to realize electronic devices based on the spin of electrons. Conduction electrons are obvious candidates for such devices, particularly in nanostructures where electron energy spectrum and shape of the envelope functions can e ectively be engineered by the growth design, application of electric or magnetic fields as well as by illumination with light. Thus, spin electronics, or spintronics, is aimed at the study of the role played by electron spin in solid-state physics and creation of possible devices that exploit spin properties in addition to or even instead of the charge degree of freedom. In this connection the behavior of hole and exciton spins in semiconductors and the role of nuclear spins are of great importance in spintronics as well. The key topics of this field to be considered are the methods used to polarize electronic spins, particularly optical orientation of spins (Sect. 5.3.1), relevant spin-relaxation mechanisms (Sect. 5.3.2), the Zeeman e ect and modification of the e ective g factors in nanostructures due to the quantum confinement (Sect. 5.3.3) and manifestations of non-equilibrium spin polarization in time-resolved PL spectroscopy (Sect. 5.3.4). In this section we focus upon the spin optical orientation of free carriers, mainly on the orientation of conduction-electron spins. PL of polarized excitons will be considered in Sect. 5.4.

5.3.1 Principles of Optical Orientation

Since the pioneer work by Lampel [5.22] the optical orientation of free-carrier spins has become an e ective method, contactless and nondestructive, for investigations of semiconductor band and kinetic parameters, see [5.23]. The essence of the method is to analyze the PL polarization as a function of the polarization of the initial light. We first formulate the basic principles of optical orientation of free-carrier spins.

Principle 1. Under optical interband excitation by circularly polarized light, the angular momentum component of σ+ (or σ−) photons is transferred to the spin (or angular momenta) of free carriers.

The electron-spin orientation under interband excitation follows immediately from the selection rules (2.154), see also Table 2.3. One can see that, for the σ+-induced transition from ±3/2 states the photoelectrons with s = −1/2 are generated. Similarly, the transitions from the states ±1/2 lead to the generation of electrons with s = 1/2. Note that, for simplicity, we assume that

226 5 Photoluminescence Spectroscopy

the light propagates along z and the PL is registered in the backscattering geometry.

The steady-state rate equations for the densities, n+ and n−, of electrons with the spin up (or 1/2) and down (or −1/2) have a simple form

n+	+	1		(n+ − n−) = g+ ,	(5.49)
τ0			2τs

n− + 1 (n− − n+) = g− . τ0 2τs

Here g± are the photogeneration rates, τ0 is the lifetime of photoelectrons in the conduction band, τs is the spin relaxation time. It is convenient to introduce the following notations: n = n+ + n− as the total density, g = g+ + g− as the total generation rate, sz = (n+ − n−)/2 as the total average electron spin, and

p =	n+ − n−	(5.50)
	n+ + n−

as the degree of electron spin polarization. Then the solution of the above two rate equations can be presented as n = gτ0 and p = p0(T /τ0), where

	p0	=	g+ − g−	,	1	=	1 +	1 ,	(5.51)
					T
			g+ + g−				τ0	τs

T being the lifetime of photoelectron oriented spins.

Hence, Principle 2 reads: If the photoelectron lifetime τ0 is not too long as compared to the spin relaxation time τs, then the photoelectrons retain spin polarization (at least partially).

Under steady-state excitation, one has
sz =	p0	T g =	p0	τsτ0	g .	(5.52)
	2		2
				τ0 + τs

The initial degree of spin polarization, p0, is proportional to the degree of circular polarization of the initial light: p0 = æPc0. The coe cient æ depends on the selection rules: for the interband transitions Γ8 → Γ6 in the bulk GaAstype semiconductors, æ = −1/2, while for the optical transitions hh1 → e1 in a GaAs-based QW one has æ = −1.

Principle 3. Due to the same selection rules for interband optical transitions, the radiative recombination of spin-polarized photocarriers gives rise to a (partial) circular polarization of the PL, or in an analytical form

Pc = æp = Pc0æ2	T	.	(5.53)
	τ0

This equation is valid for recombination with unpolarized carriers of another sort, e.g., for recombination of spin-polarized electrons with unpolarized holes. If both degrees of spin polarization, pe and ph, are nonzero the circular

5.3 Optical Spin Orientation of Free Carriers

227

polarization is given by a more complicated equation. For the recombination of e1 electrons and hh1 holes (æ = −1) it has the form

Pc =	ph − pe	,	(5.54)
	1 − peph

where ph = (nh+ − nh−)/(nh+ + nh−), and nh± is the density of heavy holes with the angular momentum ±3/2.

Principle 4. The transverse magnetic field, B z, leads to depolarization of the photoluminescence. The rate equations for n and s in the transverse

magnetic field can be written in the form
	n	= g ,	s	+ s × ΩL = s˙ ,	(5.55)
	τ0		T

where ΩL = geµB B/ is the Larmor frequency, ge is the electron’s e ective g factor, and s˙ is the spin generation rate. These equations are derived from

the equation for the so-called electron spin-density matrix							(5.56)
τ0		−	∂t		s.r.	+ [HB , ρˆ] = gˆ .
	ρˆ			∂ρˆ		i

The components of the spin-density matrix, ρ11 ≡ ρ1/2,1/2, ρ12 = ρ1/2,−1/2 etc., are expressed via the electron density n and components of the average electron spin s as

			ρˆ =	1		n + sz sx − isy .				(5.57)
				2
						sx + isy n − sz
The spin relaxation term is conveniently approximated by [5.24]
−	∂t		s.r.			2τs		2ρ21	ρ22 − ρ11
		∂ρˆ		=		1		ρ11 − ρ22	2ρ12	.

Let B x , then HB = ΩLσx/2 and

i

[ B , ρˆ] =

i

ΩL

ρ21 − ρ12 ρ22 − ρ11 .

(5.58)

H

2

ρ11

−

ρ22

ρ12

−

ρ21

The vector equation (5.55) for s is equivalent to three scalar equations

sx

sy

sz

= s˙x ,

+ ΩLsz

= s˙y ,

− ΩLsy = s˙z .

T

T

T

Under normal incidence s˙x = s˙y = 0 and

sz (B) =

sz (0)

, sy (B) =

−

ΩLT

sz (0) , sx = 0 ,

(5.59)

1 + (ΩLT )2

1 + (ΩLT )2

where sz (0) is given by (5.52). One can see that in the transverse magnetic field the average electron spin is rotated around B and depolarized. This is the so-called Hanle e ect.

Note that, in general, one has to introduce the tensor gij of g factors and write HB in the form of the sum (5.86) (Sect. 5.3.3). Only in the particular case of an isotropic g factor the Larmor frequency ΩL is directed parallel to the in-plane magnetic field B and (5.55, 5.59) are valid.

228 5 Photoluminescence Spectroscopy

5.3.2 Spin-Relaxation Mechanisms

For conduction-band electrons in bulk zinc-blende-lattice semiconductors, the following four mechanisms of spin relaxation, or spin decoherence, are relevant, see [5.25, 5.26] and references therein. The Elliot-Yafet mechanism is related to electron spin-flip scattering owing to a wave-vector dependent admixture of valence-band states to the conduction-band wave function. In the D’yakonov-Perel’ mechanism the spin coherence is lost not in an act of scattering as in the previous case but during the time between successive acts of scattering due to the wave-vector dependent spin splitting of electron subbands [5.27]. In p-doped samples the conduction electrons loose their spin polarization via scattering by holes (the Bir-Aronov-Pikus mechanism) [5.28]. In samples containing paramagnetic centers the spin relaxation is contributed by spin-flip scattering processes as a result of the exchange interaction between free electrons and electrons bound on paramagnetic centers. In bulk semiconductor crystals the Elliot-Yafet mechanism can be essential only in narrow-gap materials with a large spin-orbit valence-band splitting, e.g., in InSb. The Bir-Aronov-Pikus mechanism predominates at low temperatures and su ciently high hole densities [5.28]. The main spin relaxation at high temperatures is due to the D’yakonov-Perel’ mechanism.

With reducing the dimensionality from 3D to 2D, the importance of the D’yakonov-Perel’ mechanism is greatly enhanced and, in QWs with a moderate hole concentration, it is dominant. To consider this mechanism in more detail for (001)-grown QWs we take into account the linear-k Hamiltonian in the form (2.115) rewriting it as

	c1 =	1	(β		σx ky	−	β+σy kx ) ,	(5.60)
		2		−
	H
	where β± = 2(β2 ± β1) and the coordinate system x [110]¯ , y							[110], z
	[001] is used.

A simple physical picture of what happens is as follows. Let us first consider an electron gas of the density N occupying the lowest conduction subband e1 and assume that, at the initial moment t = 0, the electrons are spin-polarized in the same direction, say along the growth direction z, and somehow distributed in the k space. It is useful to present Hc1(k) as the scalar product ( /2)σ · Ωk, see (2.115). Then the linear-k spin-splitting of the electron subband plays the role of an e ective magnetic field and results in spin precession with the angular velocity Ωk during the time between collisions. The in-plane components of Ωk are nonzero and given by (2.117).

We start from analysis of the large splitting limit

1

|Ωk| τ ,

where τ is a microscopic time of electron scattering. According to (2.117) the Larmor frequency Ωk lies in the interface plane and is perpendicular to the

5.3 Optical Spin Orientation of Free Carriers

229

growth direction z. If s(t = 0) z then the spin component of an electron in the state k oscillates as

sz (t) = sz (0) cos Ωkt.

If Ωk = |Ωk| is angular independent, the spin polarized electrons occupying the circle of the fixed radius in the k space show the similar oscillatory behavior for sz component. However, if β−2 = β+2 or/and the electrons are occupying states with di erent values of |k|, the scatter in Ωk results in a decoherence of sz .

In particular case where β−2 = β+2 and the electrons occupying the lowest conduction subband e1 have the same spin s z at the initial moment t = 0 and are distributed equally within the energy interval 0 < E < E0 the average s¯z over the ensemble varies in the time domain 0 < t τ as

s¯z (t) = 2 sz (0)		Ω0t0	+	(Ω0t)−2
		sin Ω	t	cos Ω0t 1

where Ω0 equals to |Ωk| at k = 2mE0/ 2. One can see that s¯z (t) decays non-exponentially, and the decay time scale is given by Ω0−1.

D’yakonov and Perel’ [5.27] were the first to show that the electronmomentum scattering processes slow down the spin relaxation. In the collisiondominated limit

1

|Ωk| τ

the Larmor frequency direction changes randomly too fast, the spin rotation angle ∆ϕ ≈ Ωkτ between two successive acts of scattering is small and the time variation of sz is described by the exponential function

sz (t) = sz (0) exp −	t	,
	τs
where τs is given by		(5.61)
τs−1 Ωk2 τ

and the angle brackets mean averaging over the electron energy distribution. The dimensionless coe cient in (5.61) can be derived by solving the Boltzmann kinetic equation for the electron spin density matrix [5.27, 5.29].

In the frame of kinetic theory, the electron distribution in the wave vector and spin spaces is described by a 2 × 2 spin-density matrix

ρˆk = fk + sk · σ.

(5.62)

Here fk = Tr{ρˆk/2} is the average occupation of the two spin states with the wave vector k, or distribution function of electrons in the k-space, and the average spin in the k state is sk = Tr{ρˆk(σ/2)}. The macroand microscopic spin-density matrices (5.57) and (5.62) are related by

230 5 Photoluminescence Spectroscopy

ρˆ = ρˆk .	(5.63)

k

If we neglect the spin splitting then, for arbitrary degeneracy of an electron gas with non-equilibrium spin-state occupation but equilibrium energy distribution within each spin branch, the electron spin-density matrix can be presented as

ρˆ0	=	exp	Ek			µ¯ − µ˜ (σ · os)	+ 1		−1	,	(5.64)
					−
k						kB T

where Ek = 2k2/2m, kB is the Boltzmann constant, T is the temperature, os is the unit vector in the spin polarization direction, µ± = µ¯ ± µ˜ are the e ective Fermi energies for electrons with the spin component 1/2 or −1/2 along os so that the energy distribution functions of electrons with the spin

±1/2 are given, respectively, by

f 0	= exp		Ek − µ±	+ 1		−1 .	(5.65)
k,±			kB T

Note that (5.64) can be rewritten in the equivalent form [5.27]

ρˆ0k ≡ fk0 + s0k · σ = 12 fk,0 + + fk,0 − + (fk,0 + − fk,0 −) (σ · os) .

The densities n± of 2D electrons with a particular spin can be related with the e ective Fermi energies by

	n		=	m	k		T ln(1 + eµ±/kB T ) .	(5.66)
		±				B
				2π 2

If the spin splitting is non-zero but small compared to /τ , the distribution function Tr{ρk/2} = fk0 does not change, whereas the spin vector obtains a correction δsk = sk − s0k proportional to the spin splitting. Therefore, the spin-density matrix may be presented as

ρˆk = ρˆk0 + δsk · σ .

(5.67)

The quantum kinetic equation for the spin-density matrix has the form

∂ρˆk		i	ˆ
	+		[Hc1(k), ρˆk] + Qk{ρˆ} = 0 ,	(5.68)
∂t

where [P, R] = P R − RP and the third term in the left-hand side is the collision integral or the scattering rate, in this equation it is a 2×2 matrix. It follows from (5.68) that the kinetic equation for the pseudovector sk can be written as

dsk	+ sk × Ωk + Qk{s} = 0 ,	(5.69)
dt