Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics

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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 239

10 20 30 40 50

2.4 2.0 1.6 1.2 0.8 0.4

Diameter (nm)

10 20 30 40 50 60

Diameter (nm)(a) (b)

⊥

Height (nm)

Figure 7.8 Averaged value of the in-plane strain ε

⬜

(in %) inside the QD as a function of its height and diameter for

(a) an isolated QD and (b) the central dot of a stack of 21, separated by an 8 nm AlN spacer. With permission from

Within the volume of the central dot, the compressive in-plane deformation ε

⬜

is reduced when

stacking an increasing number of dots, whereas ε

experiences a change of sign (from positive to

negative). In the matrix region in between the dots, the in-plane expansion given by ε

⬜

increases,

whereas the matrix becomes more and more compressed along the z -direction. It can be seen that

the above trends saturate quickly as N increases: when going from N  11 to N  25 the strain

varies very little, and can be taken as rather constant for N  25, meaning that the interaction

between the dots separated by more than 10 periods can be neglected. This is already apparent in

Fig. 7.7 when looking at the extension into the barrier of the strain ﬁ eld created by a single dot.

Figure 7.8 shows a comparison of the volume averaged in-plane strain ε

⬜

 in an isolated

dot, on the left, and in the central dot of a stack of 21, on the right, as a function of the dot dimen-

sions. These results have been employed in the interpretation of the effects of stacking on the

Raman scattering [56] (see section 7.3). Whereas the strain in the isolated QD is independent of

the dimensions for a given ratio, as determined by the straight lines in Fig. 7.8a , it presents notice-

able differences in the QD of the stack. First, the strain is almost insensitive to the QD diameter

provided it is larger than 20 nm. On the other hand, the values of strain change considerably with

height. Even more pronounced changes were found for the strain component ε

(not shown).

Calculations of the elastic strain ﬁ eld of the above type have been successfully employed in the

analysis of various experiments of X-ray diffraction [57, 35] , TEM [58] and ion scattering [59] .

To illustrate the relevance of the theoretical simulations we present in Fig. 7.9 a comparison

2.70

2.65

2.60

2.55

2.50

2.45

2.40

20246

c/2 (

)

z (nm)

Figure 7.9 Comparison between the values of the strained lattice parameter c for a single GaN/AlN dot obtained

after processing of the HRTEM images [60] (points), and the theoretically calculated (in-plane averaged) values. The

different lines correspond to the different volumes employed for the strain average as indicated in the inset. The calcu-

lations correspond to a vertical QD superlattice of period 10 nm, the dot dimensions being R  1 5 n m a n d h  4 n m ,

with a WL of thickness d  0.5 nm, as determined from the TEM images.

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240 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

between the (0001) interlayer distances in a QD as extracted from HRTEM experiments [60] and

those obtained from a calculation similar to that shown in Fig. 7.6 . The very good agreement

between the experimental and theoretical results highlights the adequacy of the continuum elas-

tic model presented here for the description of the strain in self-assembled wurtzite QDs.

Although the above considerations have been made taking as example GaN/AlN(0001) QDs,

a similar methodology has been applied for the study of dots grown along a non-polar axis [61] .

Some related results will be presented in the following sections.

7.3 Raman scattering

In this section we will analyse the characteristics of the vibrational modes in GaN/AlN quantum

dots and the structural information that can be obtained from their study. The section starts with

a summary of the particularities of phonon modes in GaN and AlN wurtzite crystals, giving spe-

cial attention to the dependence of their frequency with strain. Concerning the characteristics of

the vibrational modes in the dots, we will perform an analysis of the inﬂ uence of strain on non-

polar and polar vibrations of (0001) GaN/AlN QDs considering results on resonant and non-

resonant Raman scattering. The last part of this section is devoted to non-polar quantum dots.

7.3.1 Vibrational modes in bulk GaN and AlN

The symmetry of wurtzite GaN, AlN and their alloys belongs to the C

(P63mc) space group,

with two formula units per primitive cell and all atoms occupying C

sites. A group theory

analysis [62, 63] of the lattice vibrations at the Γ point predicts six optical modes which decom-

pose into the following representations of the C

(6mmm) point group: A

 E

 2 E

 2 B

The polar modes, A

and E

, are both infrared and Raman active. The A

mode is polarized along

the wurtzite c -axis ( z -direction), while the E

mode is polarized in the xy plane. The non-polar

modes are Raman active and the B

modes are silent. The uniaxial anisotropy of wurtzite

crystals, together with the high ionic character of the crystal bonding, has a strong inﬂ uence

on the dynamic properties of these semiconductors and their nanostructures. The anisotropy of

the short range atomic forces is responsible for the A

–E

splitting while the long-range Coulomb

ﬁ eld causes the longitudinal–transverse (LO–TO) splitting of the optical polar modes. In crystals

with the wurtzite symmetry, the long-range forces prevail over the short-range ones. As a con-

sequence, there is a dependence of the energy and polarization properties of the polar phonon

modes on the direction of the phonon wavevector near the Γ point. These characteristics lead to

phonon modes in wurtzite QDs that are strongly different from those of the more common zinc-

blende semiconductors. Table 7.3 summarizes the symmetry, selection rules and frequencies of

the Raman active phonon modes of bulk GaN and AlN.

The frequency of the vibrational modes is affected by strain, a fact that has been used to charac-

terize strain in semiconductor nanostructures [65] . The relation between the phonon modes and

Table 7.3 Symmetry, selection rules and wavenumber of the zone-

centre phonon modes of bulk GaN and AlN [64] .

Mode Selection rules

GaN (cm

 1

) AlN (cm

 1

)

(LO) z ( x , x ) z 734.0 890.0

(TO) x ( z , z ) x , x ( y , y ) x 531.8 611.0

(LO) x ( y , z ) y 741.0 912.0

(TO) x ( y , z ) y , x ( y , z ) x 558.8 670.8

z ( x , x ) z , z ( x , y ) z , x ( y , y ) x 144.0 248.6

z ( x , x ) z , z ( x , y ) z , x ( y , y ) x 567.6 657.4

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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 241

strain is established within the framework of the linear deformation potential theory. This approx-

imation has been developed in detail for crystals with wurtzite symmetry by Briggs et al. [66] .

The model describes the shift of the phonon modes, Δ ω

 ω

 ω

, with respect to the

unstrained bulk value, ω

, by means of the phonon deformation potentials (PDP) of each mode λ .

Under deformation, the energy of a mode belonging to the A

representation will shift by:

Δωεεε

AA A

11 1

ab

xx yy zz

()

(7.3)

where a and b are the corresponding PDPs. The modes belonging to the E

and E

representations

are doubly degenerate. The application of strain may break this degeneracy, giving rise to two

separate modes. The resulting frequencies are given as a function of strain by:

Δωεε ε εε ε

λλ λ λ

 abc

xx yy zz xx yy xy

() [()]

/2212

(7.4)

with λ  E

, E

. The values of the deformation potentials a

, b

and c

have been reported for

various modes. They are displayed in Table 7.4 .

Table 7.4 Deformation potentials for the different GaN and AlN phonon modes, expressed

in cm

 1

Mode λ a

GaN b

GaN c

GaN [69] a

AlN [70] b

AlN [70]

115  25 [67]  80  35 [67]

– – –

(LO)

 782  174 [68]  1181  245 [68]

 643  84  1157  136

(TO)

 630  40 [67]  1290  80 [67]

 930  94  904  163

(TO)

 820  25 [67]  680  50 [67] 379  43  982  83  901  145

(LO) – –

678  49

– –

 850  25 [67]  920  60 [67] 379  107  1092  91  965  161

7.3.2 Vibrational modes in GaN QDs

The interpretation of the Raman spectra of GaN QDs would require a theoretical model taking

into account the peculiarities of the wurtzite nanostructures. These include not only the anisot-

ropy characteristic of the wurtzite structure, but also the effect of strain on the vibrations, as well

as the inﬂ uence of the piezoelectric and pyroelectric properties of the constituting materials on

the energy of the vibrational modes. A model incorporating all these effects has not been devel-

oped yet. However, calculations performed on GaN/AlN QWs [71] lead to the conclusion that the

piezoelectric and pyroelectric effects may be considered as second-order corrections to the vibra-

tional frequencies when compared with the inﬂ uence of strain. Even so, few works have addressed

the problem of the effect of strain on GaN/AlN quantum dot vibrations. Garro et al. [72]

followed a two-step procedure for the quantiﬁ cation of the effect of strain on the frequencies

of the QD vibrational modes. First, the atoms in the dot are considered to vibrate in the same

manner as in a bulk sample, but under the inﬂ uence of an average strain ﬁ eld. The changes in

the vibration frequency due to strain can then be obtained by Eqs 7.3 and 7.4 , both for the QD

and the AlN barrier. The vibrational energies obtained by this calculation can be directly com-

pared to experimental values for the non-polar E

mode, since conﬁ nement effects for this mode

are very small. The polar vibrations, however, besides being affected by strain, change energy

due to the presence of the QD interfaces. For the comparison of the energy of these modes with

experimental values, a second step is necessary: the energy of the polar modes, modiﬁ ed by

strain, must be recalculated within the anisotropic dielectric continuum model and the Loudon

model of uniaxial crystals [73–75] . With this respect, Fonoberov et al. [74] developed a tech-

nique that allows the calculation of the polar phonon modes in wurtzite QDs of arbitrary shape,

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242 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

with the advantage of giving analytical solutions for QDs of ellipsoidal shape. The spectrum of

polar optical phonons obtained from this model, however, does not depend on the absolute size of

the QD, but only on its aspect ratio. This feature arises from the fact that the bulk optical phonon

dispersion near the Γ point is neglected. Given the small phonon dispersion in GaN, this approxi-

mation is good for QDs with sizes over 3 nm. In smaller QDs, the energy of the vibrational modes

will experience a small red shift due to the bending of the dispersion relations to lower frequen-

cies for ﬁ nite values of q .

Within the dielectric continuum model, the vibrational modes in a GaN quantum dot may be

classiﬁ ed as quasi-conﬁ ned TO, quasi-conﬁ ned LO, and interface modes (TO, LO and mixed LO–TO

modes). The frequency intervals corresponding to the modes of each character are summarized

in Table 7.5 . The preﬁ x “ quasi ” used in reference to the conﬁ nement characteristics of the pho-

non modes arises from the fact that, unlike in zinc-blende nanostructures, in the wurtzite system

the modes are not completely conﬁ ned in the dot volume, but may penetrate to some extent into

the surrounding material. Furthermore, all modes are dispersive and do not necessarily coincide

with those of the bulk constituents. These two characteristics are peculiar to wurtzite semicon-

ductors and arise as a consequence of the anisotropy of the dielectric function. On the other

hand, the fulﬁ lment of the boundary conditions imposed over the electric ﬁ eld associated to polar

vibrations, combined with the anisotropy of the wurtzite dielectric function, originates a strong

dependence of the energy of the QD vibrational modes on its aspect ratio, h/D , where h is the dot

height and D its diameter.

Based on the model developed by Fonoberov et al. [74] , it is possible to calculate analytically

the quasi-conﬁ ned and interface phonon modes of an oblate ellipsoidal dot of GaN embedded in

an AlN matrix as a function of the QD aspect ratio [72] . The resulting vibrations can be classiﬁ ed

Table 7.5 Classiﬁ cation of the nature of the different vibrational

modes in GaN/AlN QDs according to their frequency [72] .

Interval Character

GaN

(TO)  ω  E

GaN

(TO)

Quasi-conﬁ ned (TO)

GaN

(LO)  ω  E

GaN

(LO)

Quasi-conﬁ ned (LO)

GaN

(TO)  ω  A

AlN

(TO)

Interface (TO)

GaN

(LO)  ω  A

AlN

(LO)

Interface (LO)

AlN

(TO)  ω  A

GaN

(LO)

Interface (LO–TO)

according to the angular and magnetic quantum numbers l and m and are shown in Fig. 7.10

for values of the QD aspect ratio ranging from 0 to 0.35. Unlike the results found in zinc-blende

quantum dots, in the GaN/AlN system the interface modes cannot be ascribed to a particular

material forming the heterostructure, and present in general mixed GaN–AlN character.

7.3.3 Non-resonant Raman scattering

The ﬁ rst micro-Raman studies performed on multilayers of GaN/AlN QDs were reported by Gleize

et al. [76, 77] . In [77] the authors analysed the Raman spectra of a sample grown by MBE on

an Si (111) substrate that consisted of 85 layers of GaN quantum dots. The features observed

in the spectra were assigned to the GaN QDs and AlN spacer by a study of the dependence of

their intensity with focus depth, their polarization selection rules and a detailed comparison of

the spectra with that of an AlGaN alloy of similar composition. The authors identiﬁ ed the peak

located at 606 cm

 1

as that of the E

of the GaN QDs. Two phonon modes were assigned to the

AlN spacer: one with E

symmetry, at 645 cm

 1

, and a second one with E

(TO) symmetry, at

657 cm



. A comparison of these values with those displayed in Table 7.3 for bulk GaN and AlN

indicates a considerable blue shift of the QD related mode (almost 40 cm

 1

), as a clue that the

dots are under compressive stress. On the other hand, the modes corresponding to the spacer

are red shifted by circa 13 cm



, as a consequence of the effect of tensile strain. Figure 7.11

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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 243

shows the Raman spectra recorded in a sample consisting of 200 GaN/AlN QD layers grown on

6H-SiC, obtained for various polarization conﬁ gurations. The peaks are identiﬁ ed by their sym-

metry and assigned to the QDs (GaN) or the spacer (AlN). Modes of A

(TO), E

(TO) and E

symmetries can be identiﬁ ed for both QDs and spacer. All the peaks related to the QDs are con-

siderably blue shifted with respect to their bulk values, while those corresponding to AlN are red

shifted. However, these shifts are found to be somewhat smaller than that reported for samples

grown on (111) Si.

The identiﬁ cation of the phonon modes corresponding to the GaN/AlN dots paved the way

for the analysis of structural changes in the QDs by means of Raman scattering. In this respect,

it was found that the strain state of dots and spacer depended on the material constituting the

substrate [56, 62, 76–78] , its orientation [61, 79] and on the number of QD stacks [56, 78] .

A systematic study of the evolution of strain with the number of dot layers was performed by

Cros et al. [56] . The samples analysed were grown by MBE on 6H-SiC on a thin AlN buffer layer.

Each period consisted of QDs formed after the deposition of 6 ML of GaN separated by 8 nm of

AlN. The evolution of the E

phonon mode of QDs and spacer in stacks with 7, 10, 20, 50 and

200 QD layers is shown in Fig. 7.12 . The linewidth of both spacer and QDs decreases, indicating

increasing strain homogeneity in the samples. The compressive strain in the dots was found to

remain almost constant in the different samples. The tensile strain of the spacer, on the other

hand, increased with the number of layers. These changes seem to saturate for samples with

more than 20 layers. This evolution of the phonon modes has been related to the homogeniza-

tion of the QD sizes with stacking due to the vertical ordering of the dots. The detailed analysis

of the strain can be performed by comparison of the Raman data concerning the non-polar E

mode with the result of a calculation of strain [56] , in combination with Eq. 7.4 .

880

840

800

760

600

580

560

540

Phonon frequency (cm

1

)

l  1

l  3

l  4

l  2

GaN E

(LO)

GaN E

(TO)

GaN A

(TO)

GaN A

(LO)

AlN A

(TO)

AlN A

(LO)

0.0 0.1 0.2 0.3

Aspect ratio

Figure 7.10 Frequencies of the polar optical vibrations for an oblate spheroid GaN QD embedded in an AlN matrix

as a function of the QD aspect ratio. Each vibration is labelled by its corresponding value of l . Only modes with

m  0 have been plotted. The dotted lines indicate the energy of the bulk phonon modes of GaN and AlN [72] . With

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244 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

Intensity (arb. Units)

560 580 600 620 640 660 680

Raman shift (cm

1

)

(TO)

GaN

AIN

(TO)

AIN

(TO) AIN

n  200

515 nm

x(zz)x

x(zy)x

x(yy)x

z(xx)z

Figure 7.11 Micro-Raman spectra of a GaN/AlN QD sample (200 periods) recorded in various polarization con-

ﬁ gurations. The dots were obtained after the deposition of 6 ML GaN, and were separated by an 8 nm spacer. The

spectra were excited at 514.5 nm. The dashed line indicates the frequency of the E

GaN mode.

The three curves included in Fig. 7.12 are the results of the calculations and differ in the

way the values of the strain are obtained. For the solid line, strain was calculated considering

stacks of QDs with variable diameter, with a distribution of sizes according to the AFM char-

acterization of the samples [37] . In this calculation, the value of strain has been obtained by

averaging to the QD material (shown in (a)) or to the AlN spacer (b) along the axis of the

stacked structure. This is indicated schematically with the arrow in the picture beside the ﬁ gure.

The dashed lines, on the other hand, have been obtained considering stacks of QDs with equal

diameter (45 nm), but averaging the value of strain to the volume of the QD (a) or spacer (b)

located at the centre of the stack. In the calculations, the mean value of shear was found to be

negligible. The dotted line in the ﬁ gure corresponds to the shift calculated only with the in-plane

strain (ﬁ rst term in Eq. 7.4 ). Concerning the QDs, the comparison of the calculations with the

5

10

Shift (cm

1

)Shift (cm

1

)

QDs

AIN

(a)

(b)

10 100

Periods

Figure 7.12 Strain induced shift of the E

phonon mode of the GaN QDs (a) and AlN spacer (b) as a function of

the number of periods. The curves are the result of a theoretical model based on the calculation of strain in the QD

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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 245

experimental Raman shift revealed a good agreement, reproducing the large positive shift of the

phonon mode. The results conﬁ rmed the small dependence of strain with QD diameter, on

the one hand, and its small variation with the number of stacks, in agreement with the results

obtained by the model discussed in section 7.2.2 (see Fig. 7.8 ). It was found that the in-plane

strain (dotted line) is responsible for 80% of the Raman shift observed in the GaN QDs. Regarding

AlN, the fact that the best agreement with theory is obtained when only the in-plane strain com-

ponent is taken into account (dotted line) suggests the existence of a relaxation mechanism

along the growth direction that affects mainly the barrier. Other studies on Raman scattering

in GaN/AlN multilayers grown on (111) Si [78] have found small shifts in the energy of the E

mode when comparing samples containing 40 and 85 dot periods. The dots in the thicker sample

showed a slightly larger compressive stress, a tendency that was conﬁ rmed by power-dependent

cathode luminescence (CL) experiments.

7.3.4 Resonant Raman scattering

As we have discussed before, it is possible to obtain information about the vibrational modes in

QD multi-stacks by means of non-resonant Raman scattering. However, when the amount of

material is very small, the Raman signal is too weak. The most efﬁ cient way to obtain information

on GaN quantum dots is to work under resonant conditions. Resonant Raman scattering occurs

when either the incident or the scattered photon energy approaches an electronic transition

of the system under study [80] . Besides allowing the observation of the Raman signal from very

small scattering volumes, the resonant enhancement enables the selective detection of different

materials whenever their characteristic electronic energies can be independently tuned. Several

experimental problems have to be confronted when measuring the Raman signal under reso-

nance. First, the intense emission from the QDs may disguise the resonant Raman signal. This

problem might be overcome by taking advantage of the quantum conﬁ ned Stark effect, which

shifts the fundamental transition of GaN QDs towards the visible. Under these conditions, the

resonant excitation of the Raman signal with higher electronic states of the QD is possible far

from the inconvenient inﬂ uence of the PL signal. Second, the lack of tunable lasers in the UV

may prevent the achievement of the resonant conditions in small quantum dots.

Gleize et al. [81] used various laser lines in the UV range to investigate resonance effects and

multiphonon scattering in a GaN/AlN QD sample. The sample considered was grown along the

[0001] direction on [111] Si and consisted of a stacking of 39 periods of GaN QDs. In Fig. 7.13

the spectra excited at 3.81 eV are displayed. The QD spectrum corresponds to that labelled as

sample (b) in the ﬁ gure, while sample (a) refers to the spectra obtained on a thick GaN layer

under the same experimental conditions and has been included for comparison. The enhance-

ment of the polar A

(LO) mode due to Fröhlich electron–phonon interaction is visible in the

spectra. Peaks corresponding to scattering by 1, 2, 3 and 4 LO-phonon modes are observed in

the QD sample. The inset expands the region of the spectra corresponding to scattering by one

phonon. The features observed at 602 cm



and 744 cm



are ascribed to the E

and A

(LO),

602cm

1

744cm

1

(b)

(a)

(b)

(a)

600 700 800

 3.81 eV

3 LO

4 LO

5 LO

GaN PL

0 2000 3000 40001000

Raman shift (cm

1

)

Figure 7.13 Raman spectra of a GaN layer (a) and a QD sample (b) excited at 3.81 eV. The multiphonon scat-

tering by the A

(LO) mode is indicated. The inset shows in detail the region corresponding to one phonon scattering

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246 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

Intensity (arb. units)

2.41 eV

3.52 eV

3.72 eV

(a)

(b)

z(x, x)z

x(z, z)x

600 700 800 900

Raman shift (cm

1

)

Figure 7.14 Raman spectra of a GaN/AlN quantum dot stack measured for different excitation energies and polari-

respectively. This assignment is supported by the detection of the E

mode under off-resonance

excitation and by the presence of the A

(LO) multiphonon peaks at higher energies.

The characterization of QD stacks by means of resonant Raman scattering probed at various

excitation energies in the UV and with different polarization conﬁ gurations led to the identiﬁ ca-

tion in [72] of three longitudinal optical modes that were attributed to quasi-conﬁ ned modes

of the QDs (see Fig. 7.14 ). As explained before, the effect of strain in the QD vibrational modes

is usually estimated by calculations of the average strain in the dot and the use of Eq. 7.4 . In

addition, conﬁ nement effects in polar phonon modes can be accounted for by means of a theo-

retical model based on the anisotropic dielectric continuum model [74] . This theoretical analy-

sis, in combination with the experimentally observed selection rules, allowed the assignation of

peak A at 582 cm



to a TO quasiconﬁ ned mode. On the other hand, peaks C (780 cm



) and

D (802 cm



) were attributed to LO interface modes of the GaN type. Finally, the peak labelled

as B (751 cm

 1

) showed better agreement with the quasiconﬁ ned LO mode characteristic of a

very ﬂ at (aspect ratio smaller than 0.02) quantum dot, and it is assigned to an interface mode

localized at the ﬂ at bottom surface of the quantum dots. This last, more intense peak is the one

observed in Fig. 7.13 [81] and identiﬁ ed on uncapped GaN QDs grown on AlGaN using Si as

antisurfactant [82] .

By means of Raman scattering at various excitation energies in the UV, it is also possible to

probe selectively the phonon modes of QDs of various sizes in single layers of GaN QDs [82] .

Smaller dots resonate at higher energies, and the vibrational frequencies of their polar modes

may red shift as a consequence of conﬁ nement, reﬂ ecting the shape of the phonon dispersion

curves. Cros et al. [83] took advantage of the resonance enhancement of the Raman signal to

study quantitatively the evolution of ε

and ε

strain components in only one layer of GaN/

AlN quantum dots as a function of AlN coverage. The QDs studied in this work were grown on

6H-SiC. The thickness of the AlN capping layer was varied from sample to sample, ranging from

0 to 14 ML. The resonance energy of the QDs was investigated by excitation with various UV

laser lines. A resonant mode around 740 cm



was found in all the samples when exciting at

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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 247

3.71 eV. The peak was identiﬁ ed as the quasi-conﬁ ned A

(LO)-phonon mode labelled as mode B

in Fig. 7.6 [72] . Its energy increases with AlN coverage, indicating an increasing compressive

strain with increasing AlN deposition. To obtain quantitative results of strain from the experi-

mental data, the authors took advantage of the small aspect ratio of the investigated quantum

dots (0.08) and combined Eq. 7.3 with the biaxial strain approximation. As discussed in section

7.2, this approximation strictly holds for 2D heterostructures (QWs and superlattices), but may

be applied in ﬂ at isolated quantum dots. The samples were investigated as well by X-ray tech-

niques (multiwavelength anomalous diffraction and diffraction anomalous ﬁ ne structure) using

grazing incidence geometry. These techniques are extensively described in chapter 6. The analy-

sis of the diffracted intensities allowed the extraction of a and c lattice parameters of the GaN

quantum dots as a function of AlN coverage.

Figure 7.15 [83] displays the comparison of the lattice parameters obtained by X-rays (trian-

gles) and resonant Raman scattering (dots) as a function of coverage. It can be seen from the

ﬁ gure that the in-plane compression and axial expansion increase monotonically with capping

thickness. It was found that the values of the in-plane lattice parameter obtained by both meth-

ods compare very well over the whole range of AlN capping investigated, in spite of the use of

the biaxial approximation for the interpretation of the Raman data. Regarding the c parameter,

the agreement was found to be good for small coverage (less than 4 ML). For thicker coverage the

interpretation of the Raman results by means of the biaxial approximation underestimates the

value of c by 0.4%. In spite of very good agreement found between the X-ray and Raman data,

the quantitative determination of strain by a combination of resonant Raman measurements

and the application of the biaxial approximation must be used with caution in other samples.

It must be kept in mind that strain in dots with larger aspect ratios will be very far from the biax-

ial value, and that this approximation fails completely for correlated QD stacks, as was already

pointed out in section 7.2.2.

3.170

3.165

3.160

3.155

3.150

3.145

3.140

AIN coverage (MLs)

2 0 2 4 6 8 10 12 14 16 18 20

5.18

5.20

5.26

5.24

5.22

GaN

(A)

GaN

(A)

Figure 7.15 Comparison of a and c lattice parameters of GaN QDs obtained by X-ray measurements (triangles)

and micro-Raman scattering (dots). The dashed lines are guides to the eyes [83] . With permission from Wiley.

7.3.5 QDs grown along non-polar directions

Raman scattering has been found to be very useful for the characterization of GaN/AlN quan-

tum dots grown perpendicular to the wurtzite c -axis. As can be deduced from Table 7.3 , one of

the advantages found in the samples with this orientation is that the usual Raman experiments

performed in backscattering geometry along the growth direction allow peaks related to the E

(TO) and A

(TO) modes to be obtained simultaneously.

Figure 7.16 shows the Raman spectra obtained from an a -plane QD sample at various polari-

zation conﬁ gurations [79] , obtained with visible excitation. The sample was grown on a -plane

6H-SiC, and consisted of 18 layers of GaN quantum dots separated by 5.5 nm AlN. In the ﬁ gure,

the vibrational modes of GaN QDs and AlN spacer are classiﬁ ed according to their symmetry.

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248 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

GaN QD

(TO)

GaN QD

(TO)

GaN QD

AIN

(TO)

AIN

(TO)

AIN

(TO)

x(z, z)x

x(y, y)x

x(z, y)x

560 600 640 680

Raman shift (cm

1

)

Intensity (arb. units)

Figure 7.16 Raman spectra obtained for three polarization conﬁ gurations under 514.5 nm excitation. The z -direction

coincides with the c -axis of the wurtzite structure, while x corresponds to the growth direction. The vibrational

modes of the QD and the AlN spacer have been labelled according to their symmetry [79] . With permission from

A comparison with the selection rules of the phonon modes of the SiC substrate conﬁ rmed that

the c -axis of the QDs, spacer and substrate grow aligned [61] .

Concerning strain in the sample, if x denotes the growth axis, and z is parallel to the wurtzite

c -axis, it is expected that

εεε

xx yy zz

≠≠

. Inspection of Eq. 7.4 predicts under these conditions the

splitting of the E

and E

vibrational modes. This splitting was not observed in the experiment, due

to the small value of the c

deformation potential, reported already in Table 7.4 . Values of strain

may be extracted from the data combining the Raman shift measured for modes with different

symmetry, and using Eqs 7.3 and 7.4 . Values of ε

 0.015 and ε

 ε

  0.009 are obtained.

Note that from the experiment it was not possible to obtain ε

and ε

independently. Strain in

AlN was found to be in good agreement with that expected for the epitaxial growth of AlN on

a -plane SiC, which for these samples can be related to the elastic constants and the SiC and AlN c

lattice parameters through the expressions:

εε

yy zz







12 13

1and

SiC

AlN

(7.5)

Here, the growth direction is identiﬁ ed with the x -axis, while y and z lie on the sample plane. For

the GaN QDs the best agreement with the experimental data was obtained from the shift of the

and A

modes, giving ε

  0.023 and ε

 ε

  0.008. However, this value was found

to overestimate the strain induced frequency shift of the E

mode by 30%. The strain state of the

QDs is also compatible with that expected for a biaxially strained layer grown on SiC (Eq. 7.5

applied to GaN), indicating that the deformation is largely imposed by the chosen substrate.

7.4 Luminescence

This section analyses the main characteristics of the interband electronic transitions in self-

assembled GaN QDs as manifested in optical experiments (continuous wave and time-resolved

photoluminescence (PL), cathode luminescence (CL), photoluminescence excitation (PLE)) and

theoretical simulations.

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