ivchenko_bookreg

4.4 Infrared Reflection from Quantum Wells and Superlattices

201

Fig. 4.10. Theoretical (solid) and experimental (dotted) infrared reflectance spectra under oblique incidence with the θ0 = 45◦ for the s-polarized (a) and p-polarized (b) light. The strong peak at 270 cm−1 is the GaAs TO mode. From [4.18].

‘SL – substrate’ and interference of waves reflected from the external and internal SL boundaries. If the substrate is thick enough in order to neglect the reflection from its back face, we obtain for the reflection coe cients rs and rp [4.18]

rµ =		−	−		n¯b,µ	−		.		(4.72)
(1		n¯b,µ) cos Φµ		i	n¯b,µ		n¯µ	sin Φµ
					n¯µ
	(1 + n¯b,µ) cos Φµ − i n¯µ					+ n¯µ sin Φµ

Here

ΩL,2

ΓL,2

202 4 Intraband Optical Spectroscopy of Nanostructures

æb

sin2 θ0

1/2

æb − sin2 θ0

1/2

n¯b,s =

−cos θ0

, n¯b,p =

æb cos θ0

,

1/2

Φs = c L æ − sin

θ0

1/2

, Φp = c L

æ − sin

θ0

æ

2

2

,

ω

ω

æ

and L is the SL thickness. Figure 4.10 shows reflectance spectra of a GaAs/AlAs SL grown on a GaAs substrate measured and analyzed by Lou et al. [4.18] It should be noted that the solid-solution layer in a GaAs/AlxGa1−xAs SL with x = 0, 1 (or in any other similar zinc-blende- lattice SL) contains two optical modes associated with the vibrations of the Ga–As and Al–As atomic pair, and

æ	B	(ω) = æ∞	(ΩL,2 1	− ω2	− 2iΓL,1	ω)(ΩL,2 2	− ω2	− 2iΓL,2	ω)	.	(4.73)
		B (ΩT2 ,1 − ω2 − 2iΓT,1ω)(ΩT2 ,2 − ω2 − 2iΓT,2ω)

Here æ∞B is close to the average

(1 − x)æ∞(GaAs) + xæ∞(AlAs)

and ΩT,1, ΩT,2 are close to the transverse optical vibrational frequencies 268 cm−1 and 362 cm−1 in homogeneous GaAs and AlAs, respectively. For illustration, the parameters for the solid solution Al0.3Ga0.7As derived from comparison of experimental and theoretical normal-reflection spectra are as

follows [4.19]: ΩT,1 = 265.2 cm−1, ΩL,1 = 278.3 cm−1, ΩT,2 = 360.2 cm−1, = 379.1 cm−1, ΓT,1 = 4.32 cm−1, ΓL,1 = 3.07 cm−1, ΓT,2 = 6.05 cm−1, = 4.71 cm−1, æ∞B = 10.16.

The above approach can be readily extended to consider the vibrational spectra of wurtzite-lattice SLs grown along the hexagonal axis, for example, CdSe/CdS [4.20] or GaN/AlN [4.21] SLs. We recall that the wurtzite crystals belong to the symmetry group C6v . For this lattice, the group theory predicts one A1 and one E1 type optical phonon modes that are Raman and infrared active, two E2 modes which are Raman active and two B1 inactive modes. Within each layer, A or B, the dielectric tensor is characterized by two linearly

independent components
æ		(ω) = æ∞		ΩL,2 − ω2 − 2iΓL, ω		,
			ΩT2 , − ω2 − 2iΓT, ω
æ			(ω) = æ∞ ΩL,2 − ω2 − 2iΓL, ω
					ΩT2 , − ω2 − 2iΓT, ω

with the parameters dependent on A or B. The e ective dielectric tensor in a short-period SL is given by (4.68) where the components æxx = æyy are expressed in terms of æA, , æB, whereas æzz is related to æA, and æB, .

5 Photoluminescence Spectroscopy

Light is sweet,

and it pleases the eyes to see the sun.

Ecclesiastes 11: 7

The light of the righteous shines brightly, but the lamp of the wicked is snu ed out.

Proverbs 13: 9

Luminescence, or fluorescence, is an e cient tool to study the excited electronic states in solids. Since the luminescence intensity is determined both by the population of the excited states and the optical-transition probabilities, luminescence, in many cases, o ers an advantage in analyzing the fine structure of excited states, which does not show up in absorption or reflection spectra. Moreover, it provides the possibility to investigate the kinetics of population and relaxation of the excited states. By studying the luminescence for di erent populations of the electronic states, one can examine e ects of electron-electron interaction on electronic spectra.

In the present chapter we pay attention to PhotoLuminescence (PL), i.e., to emission of radiation induced by the optical excitation of a sample by using an external source of light. The conventional PL spectroscopy is based on measurements of secondary-emission spectrum at fixed parameters of the primary radiation. In this set-up the measured spectrum is mainly determined by the oscillator strength and lifetime of the radiative states with the energies lying close to the fundamental absorption edge and, indirectly, by processes of energy relaxation of ”hot” excited states. In another technique known as PhotoLuminescence Excitation (PLE) spectroscopy the spectrometer is set to detect emission of a particular photon energy from the sample. The intensity of this emission is recorded as a function of the excitation photon energy. PLE spectra give information about the oscillator strength and the density of states above the fundamental absorption edge but not about their lifetime. Photoluminescence of semiconductor nanostructures has its own specific features which will be successively considered in the following sections.

5.1 Mechanisms of Photoluminescence

In semiconductors, depending on the character of the radiative transition, one distinguishes between intrinsic, extrinsic and exciton luminescence. Intrinsic, or band-to-band, luminescence is connected with the recombination of free electrons and holes. Extrinsic or impurity luminescence originates from the

204 5 Photoluminescence Spectroscopy

radiative recombination of free electrons with holes bound to acceptors, or of free holes with bound-to-donor electrons, the so-called bound-to-free emission, as well as from radiative donor-acceptor recombination and optical transitions between the levels of the same impurity center. Exciton luminescence appears due to the recombination of free, impurity-bound or localized excitons. Other exciton-related mechanisms of light emission are exciton-polariton, biexciton (or multi-exciton) and trion luminescence. If the kinetic energy of the free carriers or excitons involved in recombination exceeds substantially the thermal energy, such light emission is called hot luminescence.

Low-temperature luminescence of undoped QW structures is attributed to radiative recombination of excitons localized by interface microroughness, substitutional alloy disorder, or both. The fluctuations in the well width and alloy composition modulate the local 2D potential giving rise to localized states forming the exciton-band tail. If the exciton hopping is ine ective the PL lineshape is governed by the density of localized states. In the multiple hopping regime to be considered in the next section, the population of the exciton-band tail and, hence, the PL spectrum are being formed as a result of competition between exciton recombination and acoustic-phonon-assisted transfer from higherto lower-energy localized sites. The same holds true for QWR structures even more so because in this case the width of a nanoobject fluctuates in two dimensions.

At higher temperatures excitons are delocalized and characterized by the quasi-equilibrium distribution function f (K) = C exp (− 2K2/2M kB T ), where K is the d-dimensional wave vector, d = 2 for QWs and d = 1 for QWRs, the normalization coe cient C is proportional to T d/2N , N is the exciton density. The PL decay time is given by

τP−L1 =	ddK τ −1	(K) f (K)
	r		,	(5.1)
	ddK f (K)

where τr(K) = [2Γ0(K)]−1 is the exciton radiative lifetime. Bearing in mind that only excitons with K < (ω0/c)nb can emit photons while those with K > (ω0/c)nb are “dark” (Sect. 3.1.1) one can derive the following equation for the inverse decay time

−1	f (0)		ddK τ −1	(K)		d/2
	≈		r		T			(5.2)
τP L							.
			ddK f (K)

Here we assumed that the exciton thermal wave vector KT = (2M kB T / 2)1/2 considerably exceeds a value of the light wave vector (ω0/c)nb.

Gershoni et al. used the temperature dependence of the PL decay time as an unambiguous signature of a 1Dor 2D-system [5.1]. They performed measurements of PL transients in GaAs/Al1−xGaxAs QWs and nanometerscale QWRs. The latter were prepared by cleaved-edge epitaxial overgrowth. In the temperature range 10–90 K the PL decay times followed roughly a

5.2 Emission Spectra of Localized Excitons

205

temperature power-law dependence, τP L(T ) = AT β . The best-fitted slope

˚	˚	˚
β obtained for the 80-A	QW was 1.05±0.1 whereas, for the 34-A	and 80-A

strain-induced QWRs, the slopes were 0.4±0.1 and 0.33±0.1 in reasonable agreement with theoretical prediction β = 1 for a QW and β = 0.5 for a QWR, see (5.2).

5.2 Emission Spectra of Localized Excitons

5.2.1 Stokes Shift of the Low-Temperature Photoluminescence

Peak

The PL spectral peak from undoped QW or QWR structures at low temperatures is usually red-shifted with respect to the peak in absorption or PLE spectra. This so-called Stokes shift is explained taking into account that the excitation spectrum is dominated by optical transitions to extended free-exciton states, while the low-temperature PL arises from radiative recombination of localized excitons. In this subsection we outline the main features of the theory describing the low-temperature multi-hopping relaxation of localized excitons and discuss experiments on cw and time-resolved PL spectroscopy. Theoretically, the key problem for the multi-hopping regime is a description of excitonic kinetics making allowance for a wide scatter in the intersite times because of the random spatial distribution of the sites.

Let us consider the temporal evolution of the PL peak under the pulsed optical excitation of a QW or QWR. Hereafter, to denote the localized states, we use the localization energy ε instead of the exciton excitation energy

E = E0 − ε ,

(5.3)

where E0 is the mobility edge separating the extended and localized states. To minimize the number of parameters entering into the theory we assume a simplified model with the following properties:

(1) The density of localized-exciton states, g(ε), decays exponentially with increasing the localization energy

g(ε) = g0 exp (−ε/ε0) ,

(5.4)

where g0 and ε0 are constants.

(2) Let w(ε, ε , r) be the exciton hopping transfer rate for the transition ε → ε between the localization sites separated by a distance r. The temperature is set to zero and the probability of such a hopping is nonzero only if

ε > ε (or E > E ). Therefore, one can write
w(ε, ε , r) = θ(ε − ε) wh(r) ,	(5.5)
where θ(x) is the Heaviside step function.

2065 Photoluminescence Spectroscopy

(3)The rate w(r) is independent of ε and ε , and decays exponentially with increasing the separation

wh(r) = ω0 e−2r/L ,

(5.6)

because, for a localized exciton, the envelope wave function exhibits an exponential asymptotic behavior, Ψ (R) → R(1−d)/2 exp (−R/L), where L is the localization length and d is the space dimensionality.

(4) The localized-exciton recombination time τ0 is independent of ε. The pulsed interband optical excitation of the structure is followed by

fast energy relaxation of photocreated electrons and holes to the bottom of conduction band and top of the valence band, binding of the photoelectrons and photoholes into free excitons and the free-exciton initial localization. In the multiple-trapping process, the excitons with small values of ε hop to the deeper states with larger localization energies and the PL peak red-shifts in time. We shall analyze the temporal evolution of the PL peak by using quite general physical considerations.

According to (5.4), the concentration of sites with the localization energy exceeding ε is given by

∞

ρ(ε) = dε g(ε ) = g0ε0e−ε/ε0 . (5.7)

ε

It follows then that the average distance, r¯ε, between the localization sites with ε > ε can be estimated from the condition to find in the circle of the radius r¯ε, on the average, one of such centers of localization. This is equivalent to the equation

Vd(¯rε)ρ(ε) = 1 ,

(5.8)

where Vd(r) is the volume of the sphere of radius r in the d-dimensional space

	V3(r) =	4π	r3	, V2(r) = πr2 , V1(r) = 2r .	(5.9)
		3

An exciton trapped onto the site ε hops within the distance r¯ε. The lifetime with respect to the hopping is given by

τε = wh−1(¯rε) = ω0−1e2¯rε /L .	(5.10)
The PL peak position Emax = E − εmax at the delay	time t follow-

ing the pulsed photoexcitation is found from the evident relation τεmax =
ω0−1 exp (2¯rε/L) = t, or
εmax(t) = ε0 ln !g0ε0Vd(L/2) lnd (ω0t)" = dε0 ln ln (ω0t) + A ,	(5.11)

where the constant A = ε0 ln [g0ε0Vd(L/2)]. Therefore, for localized excitons, the time-dependent Stokes shift is described by the double logarithmic law.

5.2 Emission Spectra of Localized Excitons

207

Using the same line of reasoning one can come to the conclusion that, under the cw photoexcitation above the mobility edge, the position of the PL peak is determined by (5.11) where the time t is replaced by the exciton recombination lifetime

εmaxcw = d ε0 ln ln (ω0τ0) + A .

(5.12)

Now we obtain the same result in the frame of a more rigorous kinetic theory. We proceed from the approximation of optimal hopping transfers. This means that an exciton localized at any site O is allowed to hop to a lower-energy site O characterized by the maximum value of w(ε, ε , r), see (5.5), for a given microscopic configuration near the site O. Then we can label every excitonic state by two parameters: the localization energy ε and distance r to the nearest site with the localization energy ε > ε. The acoustic-phonon-assisted hopping of localized excitons is described by the kinetic equation

∂	+		1	f (ε, r, t) + Iε,r{f } = Γ (0)(ε, t) .	(5.13)
∂t		τ0

Here f (ε, r) is the population, or occupancy, of the state (ε, r), Γ (0)(ε) is the generation rate at level ε due to phonon-assisted transitions from extended excitonic states (or under resonant optical excitation), and the intersite transfer term Iε,r{f } is an analogue of the collision integral in the conventional Boltzmann equation. In the approximation of optimal hopping it has the form

Iε,r{f } = wh(r) f (ε, r, t)				(5.14)
ε	∞
− dε1	dr1 ρ(ε1) Pε1 (r1) wh(r1) f (ε1, r1, t) .
		g(ε1)
0	0

The first term on the right-hand side describes the hopping down from the state (ε, r) while the second term accounts the rate for hopping to this particular state from the higher localized states with ε1 < ε. The function Pε(r) is the distribution of the optimal neighbors in energy and space. For an uncorrelated system of localized sites it can be presented as

Pε(r) = ρ(ε)Sd(r) exp [−Vd(r)ρ(ε)] .

(5.15)

Here Sd(r) = dVd(r)/dr is the area of the generalized sphere

S3 = 4πr2 , S2 = 2πr , S1 = 2 .

(5.16)

Note that, for the exponential density of states defined in (5.4), the ratio

g(ε1)/ρ(ε1) equals to ε−0 1.

An exciton localized in the state ε emits a photon of the energy

208 5 Photoluminescence Spectroscopy

ω = E0 − ε .

(5.17)

Therefore, the PL spectral intensity, I(ω), is proportional to the exciton energy distribution function, N (ε) = N (E0 − ω), related with f (ε, r) and

Pε(r) by

∞

N (ε) = g(ε) drPε(r)f (ε, r) .

0

Note that, as soon as τ0 is independent of ε, the integral intensity of PL radiation decays exponentially according to

J(t) = I(ω, t) dω = J(0) e−t/τ0 , (5.18)

where I(ω, t) is the spectral intensity at the time t.

Let us introduce the total generation rate Γ (ε) at the level ε including the income from both delocalized (extended) and localized states

ε∞

Γ (ε) = Γ (0) +	ε01		dr1 Pε1 (r1) wh(r1) f (ε1, r1) .	(5.19)
	dε
0		0

Using the same assumptions as in deriving (5.11, 5.12), we neglect the dependence of τ0, Γ (0) and wh(r) on ε. In the steady-state regime the time derivative in (5.13) vanishes and the occupancy f (ε, r) is time-independent. Then (5.13) can be rewritten as

τ0		+ wh(r) f (ε, r) = Γ (ε) ,					(5.20)
1
or					Γ (ε)
	f (ε, r) =						(5.21)
						.
			τ	−1	+ wh(r)
				0

Substituting the latter expression into (5.19) an integral equation containing only Γ (ε) is obtained

ε	ε01	Γ (ε1)Wh(ε1) ,	(5.22)
Γ (ε) = Γ (0) + 0
	dε

where Wh(ε) is the probability that the exciton in the state ε prefers to hop down rather than to recombine. It is defined by

∞

wh(r)

Wh(ε) = drPε(r) τ0−1 + wh(r) .

0

	5.2 Emission Spectra of Localized Excitons				209
Di erentiating (5.22) over ε we come to the equation
	d	Γ (ε) =	Wh(ε)	Γ (ε) .
dε			ε0

Its solution with the boundary condition Γ (0) = Γ (0), see (5.19), is given by

Γ (ε) = Γ (0) exp

ε

ε

1 Wh(ε1) .

(5.23)

0

dε

0

From this equation and (5.21) we finally arrive at

f (ε, r) = 1

Γ

(0)

exp

ε

ε

1 Wh(ε1) .

(5.24)

0

τ0− + wh(r)

dε

0

Further simplifications are possible following the approximate replace-

ment of the ratio

wh(r)

τ0−1 + wh(r)

by the step function θ(˜r −r), where the critical radius r˜ satisfies the condition

		1					L	ln (ω0τ0) ,
wh(˜r) =					or r˜ =					(5.25)
			τ0				2
and the ratio		τ0−1					wh(r)
					= 1 −
	τ −1	+ wh(r)				τ −1		+ wh(r)
0					0

by 1 − θ(˜r − r) = θ(r − r˜). Then we successively obtain for the probability to hop

	r˜
Wh(ε) ≈ 0		drPε(r) = 1 − eX(ε) ,
the total generation rate
Γ (ε) Γ (0) exp				ε	ε		1 1	−	eX(ε1)
≈						0
				0	dε

and the occupancy

f (ε, r) ≈ θ(r − r˜)τ0Γ (ε) ,

where

X(ε) = ρ(ε)Vd(˜r) . (5.26) As a result, the energy distribution function can be simplified to

210 5 Photoluminescence Spectroscopy

∞

ε

Wh(ε1)

N (ε) = Γ (0)τ0g(ε)

drPε(r) exp

1

ε

0

dε

r˜

0

ε

.

= Γ (0)τ0g0 exp

−

X(ε) +

1

1

−

eX(ε1)

ε

0

ε

0

−

ε

0

dε

The PL peak position, ωmax, is found from ωmax = E0 − εmax with εmax

being a root of the equation d ln N (ε)/dε = 0 or, in the explicit form,

1

dX(ε)

1

1 − eX(ε) = 0

−

−

+

ε0

dε

ε0

which reduces to the transcendental equation

X = e−X

(5.27)

¯

0.567. According to (5.26), the position of the

that has a single

solution X

≈

cw

¯

PL peak is given by ρ(εmax)Vd(˜r) = X or, explicitly,

cw

¯

¯

,

(5.28)

εmax = ε0 ln [g0ε0Vd(˜r)] + Xε0

= d ln ln (ω0τ0) + A + Xε0

where the constant A was introduced in (5.11). This result can formally be obtained from the approximate equation (5.12) just by adding the constant

¯ 0. It is worth to stress that the kinetic theory allows not only to find the

Xε

peak position but also to calculate the PL spectral shape.

Adding the time derivative to the kinetic equation one can solve various time-dependent problems. Golub et al. [5.2] numerically calculated the timeresolved PL spectra assuming that a nanostructure is excited by a short pulse and, at the initial moment t = 0, all the localized states (ε, r) are equally populated, f (ε, r, t = 0) = const. At very short times the spectral intensity I(ω) repeats the density of states, I(ω) g(E0 − ω), and is monotonic. For times

!"

t > ω0−1 exp ρ(0)Vd(L/2)−1/d

the PL maximum shifts from the mobility edge and moves continuously into the band gap with increasing t. The numerical results obtained for a QW structure (d = 2) are well described by the dependence

εmax(2	0	{	N0 ln (ω0t)	+ X } ,	(5.29)
t) = ε		ln	2	¯

¯ ¯ where N0 = g0ε0π(L/2) , X is a constant slightly di ering from X and

weakly dependent on N0 and ω0τ0. One can see that (5.29) di ers by a con-

stant term ε0 ¯ from (5.11) taken for d = 2.

X

Equations (5.28, 5.29) can be presented in the equivalent form
εmaxcw = ε0	ln N0 ln2(ω0τ0) + X¯ ,	(5.30)