of GaN based, doped
semiconductor heterostructures
Angelika Pretorius
of GaN based, doped
semiconductor heterostructures
Vom Fachbereich f¨ur Physik und Elektrotechnik der Universit¨at Bremen
zur Erlangung des akademischen Grades eines
Doktor der Naturwissenschaften (Dr. rer. nat.)
genehmigte Dissertation
von
Dipl. Phys. Angelika Pretorius aus Dorsten
1. Gutachter Prof. Dr. rer. nat. A. Rosenauer
2. Gutachter Prof. Dr. rer. nat. D. Hommel
Eingereicht am: 26.07.2006
Abbreviations v 1 Introduction 1 2 Basics 5 2.1 The material . . . 5 2.1.1 Crystal structure . . . 5 2.1.2 InxGa1−xN phase diagram . . . 8 2.1.3 Doping . . . 10 2.1.4 Optical properties . . . 11 2.2 Epitaxial growth . . . 14
2.2.1 Metalorganic vapour phase epitaxy . . . 14
2.2.2 Molecular beam epitaxy . . . 15
2.3 Growth modes . . . 17
3 TEM theory and evaluation 19 3.1 Electron diffraction . . . 20
3.1.1 Kinematic theory . . . 20
3.1.2 Dynamic theory . . . 26
3.2 Image formation . . . 30
3.2.1 Effect of incoherence . . . 33
3.3 Evaluation of In concentration in InxGa1−xN structures . . . 35
3.3.1 Derivation of lattice distances . . . 36
3.3.2 Strain state analysis . . . 39 i
3.3.3 Sources of error in derivation of the In concentration x . . . 42
3.4 Polarity measurement . . . 55
4 Pyramidal defects in Mg doped GaN 59 4.1 Experimental . . . 62 4.2 Results . . . 63 4.3 Discussion . . . 69 5 InxGa1−xN islands 71 5.1 Literature survey . . . 71 5.1.1 Patterning . . . 72 5.1.2 Antisurfactants . . . 73
5.1.3 Stranski-Krastanow growth mode . . . 74
5.1.4 Growth interruptions . . . 75
5.1.5 Droplet epitaxy . . . 76
5.1.6 Fluctuation of QW width . . . 77
5.1.7 Phase separation . . . 77
5.1.8 Segregation . . . 79
5.1.9 Other effects concerning island formation . . . 81
5.2 Recognition of In droplets . . . 82
5.3 MBE grown InxGa1−xN islands . . . 85
5.3.1 Low temperature growth of InxGa1−xN island samples . . . 86
5.3.2 High temperature growth of InxGa1−xN island samples . . . 96
5.4 MOVPE grown InxGa1−xN islands . . . 103
5.4.1 Geometric InxGa1−xN islands . . . 103
5.4.2 InxGa1−xN sample series with QD like PL . . . 118 5.5 Concentration profile of geometric InxGa1−xN islands grown by MOVPE . 124
A General notes 137
A.1 Physical constants . . . 137
A.2 Vacuum classification . . . 137
A.3 Brackets for directions, planes, and reflections . . . 138
A.4 Special planes in crystals of hexagonal structure . . . 139
A.5 Centre of Laue circle . . . 140
B Fourier transform, δ-function, and convolution 141
C Fit functions 143
AFM atomic force microscopy BF bright field
CBED convergent beam electron diffraction CCD charge coupled device
CD compact disc
COLC centre of Laue circle CTF coherent transfer function
DALI digital analysis of lattice images DFT density functional theory
DP diffraction pattern DVD digital video disc
EDS energy dispersive x-ray spectroscopy EELS electron energy loss spectroscopy EMS electron microscopy image simulation fcc face centred cubic
FE finite element FEG field emission gun FIB focused ion beam FOLZ first order Laue zone
FVIR focus variation image reconstruction FWHM full width at half maximum HAADF high angle annular dark field HOLZ higher order Laue zone
HRTEM high resolution transmission electron microscopy ID inversion domain
IDB inversion domain boundary
laser light amplification by stimulated emission of radiation LD laser diode
LED light emitting diode
LEEBI low energy electron beam irradiation MAL maximum likelihood
MBE molecular beam epitaxy ML monolayer
MOVPE metalorganic vapour phase epitaxy MQW multiple quantum well
PD pyramidal defect PL photo luminescence QD quantum dot QW quantum well
relrod reciprocal lattice rod
RHEED reflection high energy electron diffraction RT room temperature
SAD selected area diffraction
SEM scanning electron microscopy SK Stranski-Krastanow
STEM scanning transmission electron microscopy STM scanning tunnelling microscopy
TD threading dislocation
TEM transmission electron microscopy TMA trimethylaluminium
TMG trimethylgallium TMI trimethylindium UHV ultra high vacuum
VCSELs vertical cavity surface emitting lasers WBDF weak beam dark field
WL wetting layer XRD x-ray diffraction ZA zone axis
Introduction
Light emitting devices such as light emitting diodes (LEDs) and laser diodes (LDs)1 are of wide interest [1]. LEDs have a longer life time and lower energy consumption than conventional light bulbs. They can be used e. g. as light sources for traffic lights, displays, etc. LDs are e. g. used for compact disc (CD) or digital video disc (DVD) players and recorders. CD recorders usually contain a laser diode (LD) with an AlGaAs active region, which emits light with a wave length of 780 nm. Digital video discs (DVDs) with a storage capacity of 4.7 GB contain LDs emitting at 650 nm. A new development is the HD-DVD and “Blu-Ray Disc” with a storage capacity of 16 GB and 25 GB, respectively. This is mainly (disregarding other technical details) accomplished by usage of a LD emitting a shorter wave length in the blue violet spectral range (405 nm). Due to the smaller wave length, the emitted light can be focused more precisely and the storage capacity can be increased. Another application for LDs are laser projectors or laser displays, where red, green, and blue light is needed. While LDs emitting red light are widely spread, green and especially blue light emitting LDs are still challenging. In order to realise a high efficiency of the LDs, semiconductors with a direct bandgap are desirable. To cover the whole visible spectrum from red over green to blue emission, the material systems (Mg,Zn,Cd)(S,Se) and (Al,Ga,In)N are suitable. The II-VI materials crystallise in the zincblende structure, while the III-nitrides crystallise in the wurtzite structure. The drawback of the II-VI materials is their rapid degradation. In contrast, GaN shows a high physical hardness and thermal as well as chemical stability. Furthermore - in comparison to other II-VI or III-V compound materials - GaN is not toxic for humans. Therefore, the III-nitrides are now of great commercial interest. In the present work, the analysed material system is restricted to (Ga,In)N, as the variation of the In concentration already provides the possibility to adjust the wave length of the emitted light.
To manufacture optical devices, doping of the heterostructures is indispensable. While Si can be used as donator and sufficiently high n-type doping levels are reached, p-type
1
Laser is the abbreviation for light amplification by stimulated emission of radiation.
doping is more problematic. Mg was found to be the most suitable acceptor. Still, for metalorganic vapour phase epitaxy grown GaN based, Mg doped structures, extended defects are formed for doping levels exceeding 1019cm−3. Because of their geometry, these defects are called pyramidal defects in the literature. The pyramidal defects are discussed to be responsible for an observed decrease of the free hole concentration at high p-type doping levels. Nevertheless, the nature of the pyramidal defects is still under debate. One aim of this work is to illucidate the nature of the pyramidal defects in metalorganic vapour phase epitaxy grown Mg doped GaN by means of high resolution transmission electron microscopy (HRTEM) and image simulation.
Another challenge is the fabrication of quantum dots. A quantum dot has discrete en-ergy levels comparable to an atom. This leads to a narrow line width of emitted light, i. e. sharp emission lines are obtained. LDs with quantum dots in their active region exhibit improved optical properties such as a reduced and temperature independent threshold cur-rent density. A homogeneous size, shape, and composition of the quantum dots is necessary to obtain sharp emission lines also for quantum dot ensembles. For some material systems the heteroepitaxial growth of quantum dots is quite successfull, and also for InGaN many reports of quantum dots can be found in the literature. Nevertheless, for an understanding of the optical properties of InGaN islands, their structural properties such as size, den-sity, or composition need to be known. In this work, HRTEM is the main technique used to determine these structural properties, with main focus on the composition of InGaN islands.
Synopsis of this work
This thesis addresses the analysis of GaN based heterostructures with transmission electron microscopy (TEM). Basic properties of the material of interest are introduced in chapter 2. These include the structural and optical properties as well as an introduction to the growth methods used for the samples analysed in this work.
In chapter 3 a brief theoretical treatment of TEM is given. The calculation of the object exit wave function by two different methods and the image formation in the transmission electron microscope is derived, including the multislice approach, which is used for HRTEM image simulation of basal plane inversion domain boundaries. As one main topic of this work is the determination of the In concentration in InGaN islands using strain state analysis, a detailed description of the method is given. This is followed by a discussion of the errors of this method, where also imaging conditions for TEM analysis are given which minimise the errors. Usually, the c lattice parameter is of interest and used to calculate the In concentration in InxGa1−xN structures using Vegard’s rule and elasticity theory. In these cases, the imaging conditions require the knowledge of the polarity of the wurtzite material. Hence, also a method is given to derive the polarity at the beginning of each TEM session.
Chapter 4 describes the results obtained for pyramidal defects present in metalorganic vapour phase epitaxy grown GaN:Mg with high dopant concentration. Based on the exper-imental results and the well established knowledge that GaN of inverted polarity is present inside the pyramidal defects, a variety of basal plane inversion domain boundary models was set up. From these models, HRTEM images were simulated using the multislice ap-proach, followed by a quantitative comparison to experimentally obtained HRTEM images. Those models which lead to a good fit between simulated images and the experimental im-ages of a basal plane inversion domain boundary are presented, and possible sources for deviations are discussed.
Another focus of this work is the analysis of InxGa1−xN islands grown on GaN presented in chapter 5. Following a literature survey which describes different methods used to obtain InxGa1−xN islands, the first topic is the distinction of InxGa1−xN islands and metal droplets, which can form during growth. This is followed by the experimental results of molecular beam epitaxy and metalorganic vapour phase epitaxy grown InxGa1−xN island and quantum dot samples. For an overview on the high number of analysed island samples, a detailed list is provided in appendix D (p. 145), containing the sample numbers and important growth parameters. For the molecular beam epitaxy grown samples, uncapped islands as well as a detailed study of island dissolution during cap layer growth is presented in chapter 5.3. In case of the metalorganic vapour phase epitaxy samples, the dissolution of islands due to cap layer growth is less pronounced. For free standing, elastically relaxed islands indications are presented that the In concentration gradually increases to the top of islands, as is described in chapter 5.4.1. metalorganic vapour phase epitaxy samples showing quantum dot like luminescence could be realised using a new growth approach. The TEM results of this sample series are presented in chapter 5.4.2. The apparent increase of the In concentration in free standing, uncapped islands is analysed further in chapter 5.5. Two different methods are employed, which are the evaluation of the object exit wave function using focus variation image reconstruction and the “superposition method”, which was developed for this work. The results obtained with these methods support that the In concentration increases in growth direction within the islands.
Basics
In this chapter some basic properties of GaN and InN are discussed. In the first part the structural and optical characteristics of the materials are given. The second part gives an introduction to metalorganic vapour phase epitaxy (MOVPE) and molecular beam epitaxy (MBE), two methods for epitaxial growth of GaN based structures, followed by a description of different growth modes observed for epitaxial growth.
2.1
The material
2.1.1
Crystal structure
Three different modifications of GaN and InN are reported [2], these are the wurtzite, sphalerite, and rocksalt structure. As the rocksalt structure is only found at a pressure of ten or several tens of GPa [2–4], it is not relevant for the material studied in this work.
GaN usually crystallises in the hexagonal wurtzite phase which is stable under ambient conditions [2]. The space group of the wurtzite structure is P63mc (no. 186, Pearson symbol hP4, Strukturbericht designation B4). The hexagonal unit cell has the axes a1, a2, and c , with angles ( a1, a2) = 120◦ and ( a1, c ) = ( a2, c ) = 90◦. The Ga or In atoms in this unit cell are located at the position vectors r0i of [1/3,2/3, 0] and [2/3,1/3,1/2], the N atoms in GaN at [1/3,2/3, 0.377] and [2/3,1/3, 0.877] and in InN at [1/3,2/3, 0.375] and [2/3,1/3, 0.875] [5]. The wurtzite GaN unit cell is shown in figure 2.1 a in [1210] projection (Miller-Bravais indexing).
The sphalerite or zinc blende structure has the space group F43m (no. 216, Pearson symbol cF8, Strukturbericht designation B3). The cubic unit cell shown in figure 2.1 b and c contains eight atoms. Four cations are at [0, 0, 0], [1/2,1/2, 0], [1/2, 0,1/2], and [0,1/2,1/2]. The nitrogen atoms are located at [1/4,1/4,1/4], [3/4,3/4,1/4], [3/4,1/4,3/4], and [1/4,3/4,3/4]. The
a) b) c)
Figure 2.1: a Wurtzite and b, c sphalerite unit cell of GaN. The sphalerite unit cell in b is rotated to visualise the epitaxial relationship with the wurtzite structure. The ABAB stacking sequence of the wurtzite material in [0001] direction and the ABC stacking sequence of the sphalerite material in h111i direction is indicated. The crystal directions are given using Miller-Bravais and Miller indexing for the hexagonal and cubic cell, respectively. structure of the unit cell can be seen in figure 2.1 b in h101i projection and in figure 2.1 c in [010] projection (Miller indexing).
The ABAB stacking sequence along the polar [0001] direction of the wurtzite material is indicated in figure 2.1 a. Wurtzite GaN with [0001] growth direction is also called Ga polar; analogously [0001] is called N polar1. The sphalerite crystal shows ABC stacking along the polar h111i direction (figure 2.1 b). If both modifications are deposited during epitaxial growth of GaN or InN, the preferential epitaxial relationship between both phases is h0001ikh111i, h1120ikh101i, and h1100ikh112i.
The lattice parameters for ambient temperature and elastic constants used in this work for both modifications of GaN and InN are given in table 2.1. The given lattice parameters may vary depending on the substrate or growth technique [8]. If a ternary compound InxGa1−xN is grown, the cation position is statistically occupied by In and Ga atoms and
1
As pointed out by Liliental-Weber et al. [6] “this nomenclature comes from the fact that, for the case of Ga growth polarity, the Ga atom is connected with the N atom by only one bond aligned along the c axis. This single bond can break easily (in comparison with the three bonds lying on inclined planes), removing the N and leaving Ga on the surface. The frequently used nomenclature of “Ga terminated surface” (or N terminated surface) is not strictly applicable, since Ga adatoms can be present not only on the Ga polar surface but also on the N polar surface, as shown by Northrup et al. [7].”
GaN, wurtzite InN, wurtzite a [nm] 0.3189 [2], 300 K 0.35374 [10], 296 K c [nm] 0.5185 [2], 300 K 0.57027 [10], 296 K C11 [GPa] 367 [11] 223 [11] C12 [GPa] 135 [11] 115 [11] C13 [GPa] 103 [11] 92 [11] C33 [GPa] 405 [11] 224 [11] C44 [GPa] 95 [11] 48 [11]
GaN, sphalerite InN, sphalerite
a [nm] 0.45 [2] 0.498 [12], room temperature (RT)
C11 [GPa] 274.2 [13] 206.57 [13]
C12 [GPa] 166.1 [13] 117.73 [13]
C44 [GPa] 199 [13] 112.71 [13]
Table 2.1: Material parameters for wurtzite and sphalerite GaN and InN. The given lattice constants are experimental values for approximately 300 K. The elastic constants Cij are obtained from density functional theory (DFT) calculations.
the lattice parameters are given by Vegard’s rule [9]
a(x) = (1 − x)a(0) + xa(1) and c(x) = (1 − x)c(0) + xc(1) , (2.1)
i. e. the lattice parameter can be linearly interpolated between the lattice constants of GaN (a(0), c(0)) and InN (a(1), c(1)). Hence the concentration x can be derived straightforward from measured lattice parameters. In case of epitaxially grown InxGa1−xN on GaN, the lattice is distorted. The composition evaluation from the lattice parameters needs elasticity theory and is described in section 3.3.2. If InN is grown on h0001i oriented GaN, the misfit
f =(a(1)−a(0))/a(0)is about 0.11.
The substrate for the GaN films and InGaN/GaN heterostructures analysed in this work was in all cases h0001i oriented sapphire (Al2O3, space group R3c, no. 167, Pearson symbol hR10, Strukturbericht designation D51). The structure can be described by a hexagonal unit cell with lattice constants a of 0.4765 nm and c of 1.2982 nm [14]. The epitaxial relationship between sapphire and GaN is h0001iAl2O3 k h0001iGaN, h1120iAl2O3kh1100iGaN,
and h1100iAl2O3kh1120iGaN, i. e. the GaN unit cell is rotated by 30◦ around h0001i in
comparison to the sapphire unit cell. Considering the actual positions of the atoms in both unit cells, a misfit of approximately 0.16 is obtained.
2.1.2
In
xGa
1−xN phase diagram
There are many reports of inhomogeneities of In concentration x in InxGa1−xN, and differ-ent origins of these fluctuations are discussed. One explanation is based on the predicted miscibility gap of InN in GaN [15–17]. Consider the Gibbs free energy G of the system InxGa1−xN as a function of the concentration x. Depending on the actual system, G can have more than one minimum, as is shown schematically in figure 2.2. In a composition re-gion given by the common tangent method [18] (approximately given by two neighbouring minima), G can be reduced by decomposition. This can either take place by nucleation or spinodal decomposition. Both processes are determined by diffusion when they comprise changes of concentration.
Ho and Stringfellow [15, 16] calculated the equilibrium binodal and chemical spinodal curves of the ternary sphalerite InxGa1−xN system. Stating that the sphalerite and wurtzite structure are different only concerning the stacking sequence, they claim that their results are applicable also for the wurtzite material. The binodal curve was approximated to be given by the minima of the Gibbs free energy G in dependence of the In concentration x (dG/dx = 0). The region of spinodal decomposition was calculated by d2G
/dx2 ≤ 0, i. e. if
G(x) is concave, a small decomposition will decrease G. Their result for the binodal and spinodal curves is given in figure 2.3 a.
Nevertheless, MOVPE and MBE are no equilibrium processes. In addition, nano sized InxGa1−xN structures such as quantum wells (QWs) or quantum dots (QDs) are usually intended to grow completely strained on the GaN substrate. The resulting additional strain energy influences the phase diagram significantly, as calculated by Karpov [17]. The phase diagram obtained for a completely strained InxGa1−xN layer on (0001) GaN is shown in figure 2.3 b. As can be seen the miscibility gap for this case is comparably small.
a)
b)
c)
Figure 2.2: Schematic of the Gibbs free energy G for two temperatures a T1 and b T2 < T1 in dependence of the concentration x. A spontaneous reaction can take place if G is decreased. If G has more than one minimum, G can be reduced by decomposition in a composition range given by the common tangent rule, as is marked with arrows in a and b. This concentration range is approximately given by the minima of G(x), i. e. dG(x)/dx= 0. The minima positions in dependence of the temperature can be used to define the binodal curve of the phase diagram (solid curve in c). If G is concave, i. e. d2G(x)
/dx2 ≤ 0, G is
reduced by any small concentration fluctuation (spinodal decomposition). For an initial alloy of concentration x0 decomposition can take place, resulting in concentrations x1, x2, and a reduction of Gibbs free energy of ∆G as shown in a. The inflexion pointsd2G(x)/dx2 = 0
in dependence of the temperature, inbetween which the spinodal decomposition can be observed, define the spinodal curve of the phase diagram (dashed curve in c).
a) b)
Figure 2.3: Binodal (solid lines) and spinodal (dashed lines) curves for InxGa1−xN. In a the phase diagram for relaxed InxGa1−xN [15] is given. b shows the phase diagram for completely strained InxGa1−xN grown on (0001) GaN [17].
2.1.3
Doping
For application of InxGa1−xN/GaN heterostructures in electronic and optoelectronic de-vices, p- and n-type doping is necessary. Undoped GaN is unintentionally n-conductive, which is attributed to either nitrogen vacancies (VN) or to the incorporation of oxygen (with
oxygen located at nitrogen sites, ON) or silicon (SiGa) impurities acting as donors [19–21].
For intentional n-type doping usually Si is used, and sufficiently high doping levels can be achieved. Liliental-Weber et al. [22] report a decrease of surface roughness and threading dislocation (TD) density for increasing Si doping concentration up to 1 · 1019 cm−3. Nev-ertheless, Nakamura et al. [23] observed a decrease of photo luminescence (PL) intensity due to formation of V-shaped defects for Si concentration exceeding 4.2 · 1019 cm−3. These defects are surrounded by {1011} facets. Recently, Schmidt et al. [24, 25] found that Si segregates to the surface and that the Si concentration at the facets of observed cracks is increased in comparison to the flat surface. The angle between the facet and the (0001) plane is evaluated to be 63.2◦ ± 3.1◦ and 69◦± 7◦. As Si is discussed to stabilise certain facets, it is interesting to compare the facets of the V-shaped defects with the facets of the cracks at which an Si enhancement is observed. The angle measured by Schmidt et al. [25] does not match with the {1011} facets of the reported V-shaped defects which have an angle of 54.7◦ to the (0001) plane, but is in reasonable good agreement with the angle between (0001) and {1121} of 67.8◦.
p-type doping for GaN has proven more difficult than n-type doping. Different dopants such as Li, Be, Na, Ca etc. were discussed (see e. g. [26]). The most suitable p-dopant which
is commonly used is magnesium (MgGa, [19]). Also Mg segregates to the surface [25,27–30].
Furthermore, its effectiveness for contribution to the free hole concentration is hampered at high Mg concentrations of > 1019 cm−3 by formation of pyramidal defects (PDs) [31]. These will be discussed in more detail in chapter 4.
2.1.4
Optical properties
GaN and InN are direct semiconductors. The bandgap energies Eg(T < 10 K) of the binary materials vary between 3.508 eV for GaN [32] and about 0.7 eV for InN [33–39]. The bandgap energy of the ternary alloy InxGa1−xN cannot be linearly interpolated as is the case for the lattice parameters, but is described by the introduction of an additional term containing the bowing parameter b:
Eg(x) = Eg(0)(1 − x) + Eg(1)x − bx(1 − x) , (2.2)
which amounts to b = 1.43 eV [33, 40, 41]. The dependence of the bandgap energy on the a lattice parameter and the In concentration x for InxGa1−xN is shown in figure 2.4. The bandgap energy of sapphire at 300 K amounts to (8.1 . . . 8.6) eV [14] and is marked at a as given by the epitaxial relationship to GaN. The background in the image denotes the colour of the emitted light for a given Eg. It can be seen that the whole visible spectrum is covered by variation of the In concentration. If e. g. blue luminescence is desired, an In concentration of approximately 0.2 would be favourable.
The dependence of the bandgap energy on the temperature can be described by the empirical Varshni formula Eg(T ) = Eg(0 K) − αT
2
T +β [42]. α and β are material constants. α amounts to 0.909 meV/K for GaN and 0.245 meV/K for InN, and β is 830 K and 624 K, respectively [41], i. e. the bandgap energy decreases with increasing temperature.
For heterostructures the band alignment between the used materials constituting the structure is important. If e. g. a material with lower bandgap energy is embedded in a barrier material with larger bandgap energy, different types of band alignment are distin-guishable. A type II band alignment is present if the bandgap of the material with the lower bandgap energy is not entirely located within in the bandgap of the barrier material. A schematic of one case of a type II band alignment which can give rise to bound states of holes in the valence band is given in figure 2.5 a. If the bandgap of the embedded material is completely contained in the bandgap of the barrier material, a type I band alignment is obtained as sketchend in figure 2.5 b. The energy levels of the ground states of a bound electron and hole and the corresponding wave functions are indicated. InxGa1−xN embed-ded in GaN is a type I heterostructure. For h0001i grown GaN/InxGa1−xN it has to be considered that h0001i is a polar axis in wurtzite GaN or InN. Applying pressure in e. g. the (0001) plane leads to a piezoelectric field Epiezo. The piezoelectric field causes a defor-mation of the band structure in embedded quantum structures, as is shown schematically
Figure 2.4: Bandgap energy Eg versus a lattice parameter and In concentration x, varying between GaN and InN. The background symbolises the colour of the emitted light for the corresponding Eg.
a) b) c)
Figure 2.5: a Type II band alignment. The lower bandgap of the embedded material is not completely located in the larger bandgap of the barrier material. b Type I band alignment. The bandgap of the embedded material lies completely in the bandgap of the barrier material. The ground state energy levels together with electron and hole wave functions are sketched. c In presence of a piezoelectric field Epiezo the band structure is deformed, resulting in a spatial separation of electron and hole wave functions and a red shift of the emission.
in figure 2.5 c [43]. The deformation leads to a red shift of the emission and causes a separation of the hole and electron wave functions as is indicated. This charge separation induces a screening field in the opposite direction of Epiezo, i. e. the piezoelectric field depends on the charge carrier density. In addition, the separation of the electron and hole wave functions leads to a decrease of the luminescence intensity as the overlap integral is reduced.
For a bulk material, the density of states D(E) is given in figure 2.6 a for a quadratic dispersion relation E(k) ∝ k2. If charge carriers are confined along one dimension to an extension of about the de-Broglie wavelength λ of the charge carrier (quantum confine-ment), with λ = hp = √ h
3m∗k
BT [44] (h Planck’s constant, p momentum, m
∗ effective mass, kB Boltzmann constant, and T temperature; see table A.1 on p. 137 for used values of physical constants), the density of states becomes a step function (figure 2.6 b, quantum well (QW)). For size restriction in all three dimensions a quantum dot (QD) is obtained. The density of states is a series of δ functions (figure 2.6 c) and the energy levels are discrete as is the case for atoms. The reduction of dimensionality results in a blue shift of the emission. The discrete energy levels in QDs lead to sharp emission lines with a full width at half maximum (FWHM) in the range of 10−4 eV.
Due to the small size of a QD, excitons will be formed if an electron and a hole are present in one QD. The Coulomb interaction between the electron and hole can be taken into account using perturbation theory.
Optoelectronic devices containing QDs in their active region can show an improved performance due to the properties of the QDs. For example QD lasers are predicted to show a low and approximately temperature independent threshold current (as long as the charge carriers are not excited into the barrier).
a) b) c)
Figure 2.6: Density of states D(E) in dependence of the energy E for a bulk material, b a QW, and c a QD.
2.2
Epitaxial growth
GaN based samples for this work were grown by MOVPE and MBE in the semiconductor epitaxy group (Prof. Dr. D. Hommel) at the University of Bremen. These two growth methods will be described in this section.
2.2.1
Metalorganic vapour phase epitaxy
The MOVPE samples were grown in a so called “vertical closed coupled showerhead re-actor” (Thomas Swan Scientific Equipment Limited). MOVPE is based on the chemical reaction between different precursors. For growth of GaN based heterostructures, the pre-cursors are given in table 2.2. A schematic of a showerhead reactor is shown in figure 2.7.
element precursor chemical symbol
Ga trimethylgallium (TMG) (CH3)3Ga
In trimethylindium (TMI) (CH3)3In
Al trimethylaluminium (TMA) (CH3)3Al
N ammonia NH3
Mg magnesocene / bis-cyclopentadienyl magnesium (Cp2Mg) (CH3C5H4)2Mg
Si silane SiH4
Table 2.2: Precursors for MOVPE growth of GaN based structures. For each element of the final film the precursor and the chemical symbol is given.
The liquid metalorganic precursors are mixed with a carrier gas, i. e. H2 or N2 for In containing films, and the desired gas flow is adjusted prior to introducing the gases into the reactor. Due to the showerhead design the group-III precursors and the ammonia molecules are evenly distributed separately from each other over a quite large area, before they are intermixed just about a centimeter above the susceptor, as can be seen in figure 2.7. The wafer on which the film has to be deposited is located on the susceptor, which can be resistively heated up to a temperature of 1373 K. Due to the heat, TMG, TMI, and TMA dissociate. Above the susceptor the following reaction takes place (X = Ga, In, Al) [45]:
X(g) + NH3 −→ XN(s) +3/2H2 . (2.3)
The carrier gas and the remaining carbon and hydrogen compounds are conducted out of the reactor chamber. The compostion of e. g. the InxGa1−xN films can be varied by changing the growth temperature, i. e. the temperature of the substrate, the gas flow, or the ratio between the group-III and group-V precursors. The pressure in the MOVPE chamber is usually in the range of 0.01 to 0.1 MPa. Therefore most in situ methods to
Figure 2.7: Schematic of a showerhead MOVPE reactor. Above the susceptor the molecules dissociate due to the resistive heating. The group-III precursors are transported via a carrier gas (not depicted). symbolises the ligands of the precursors.
control the film growth such as reflection high energy electron diffraction (RHEED) are not applicable, as the mean free path of the electrons is too short. Hence only optical methods such as reflectometry can be employed [45].
2.2.2
Molecular beam epitaxy
GaN/InGaN structures were also grown by MBE in an EPI 930 growth chamber. High purity elements such as metallic Ga or In or gaseous N2 are used. The solid elements are evaporated in effusion cells at temperature Te. The gaseous N2 is dissociated to atomic nitrogen, in the case of this work by a radio frequency plasma at 13.56 MHz. The molecular beams are directed to the rotatable substrate (figure 2.8). A shutter in front of the effusion cells can be used to block the molecular beam of individual cells. The substrate can be heated up to 1273 K and is either glued on a silicon wafer using In or clamped in a holder.
The growth temperature Tg of the substrate is measured with a pyrometer. As Tg can differ from Te, MBE growth does not take place in thermal equilibrium. To minimise the incorporation of impurities, growth requires ultra high vacuum (UHV) (appendix A.2). This also ensures that the mean free path of the atoms is larger than the distance from the effusion cells to the substrate, i. e. the molecular beams reach the substrate. Additionally, this allows the use of in situ investigation techniques using electrons as probes such as RHEED.
For in situ RHEED analysis during MBE, electrons accelerated by a high voltage of (5 . . . 30) keV are directed at the surface at grazing incidence (< 5◦). For a two dimensional surface, a streaky RHEED pattern is observed [46]. When a change to three dimensional growth of i. e. islands occurs, the RHEED pattern shows spots. This transition can be used to monitor the onset of three dimensional island growth.
Figure 2.8: Schematic of a MBE chamber. In the effusion cells the high purity materials are evaporated. Due to the ultra high vacuum in the chamber, RHEED is possible.
2.3
Growth modes
For epitaxial growth, three different growth modes are reported, which mainly depend on the surface energy [47]. Considering epitaxial growth of a film material F on a smooth substrate S under vacuum V, the surface energies for the different interfaces are γFS, γFV, and γSV.
For γSV > γFS+ γFV, a layer of material F grows two dimensionally on the substrate. This growth mode is called Frank-van der Merwe growth mode [47, 48] (figure 2.9 a). The above relation is only fulfilled for small lattice misfits f = (aF−aS)/aS between S and F.
If the film (with f 6= 0) exceeds a critical thickness, the strained layer stores significant amounts of elastic strain energy and generation of misfit dislocations can become energet-ically favourable [49].
For larger lattice misfits, the material F can first be grown as two dimensional film (wet-ting layer). For a certain film thickness a transition to three dimensional island growth can be observed (2d-3d transition). In this Stranski-Krastanow growth mode [50] (figure 2.9 b), the islands grow self assembled and may give rise to defect free islands. Furthermore, the size and composition distribution of luminescence centres in InGaN, expecially for stacked layers, is possibly more homogeneous than one would expect from other growth approaches (see section 5.1).
If γSV < γFS+ γFV, the material F grows three dimensionally and islands are formed directly on the substrate (Volmer-Weber growth mode [51], figure 2.9 c). On further growth, the size of the islands increases until they coalesce.
a) b) c)
Figure 2.9: Growth modes of epitaxial films. a Frank-van der Merwe growth mode of two dimensional films, b Stranski-Krastanow growth mode, and c Volmer-Weber growth mode.
TEM theory and evaluation
In a transmission electron microscope, an approximately plane electron wave is directed on a specimen. The interaction of the electron wave with the specimen can be described by the kinematic or dynamic theory of electron diffraction. The kinematic theory and the Bloch wave and multislice approach of the dynamic theory are described in the first section of this chapter (section 3.1). The result of the calculation is the object exit wave function. The object exit wave then propagates through the lens system of the transmission electron microscope, and the final image appears in the image plane. This image formation is the topic of section 3.2. In section 3.3 first the strain state analysis is described, i. e. the derivation of the In concentration of InxGa1−xN structures embedded in GaN from measurement of the local c lattice parameter or d0002 lattice spacing using elasticity theory. Then different sources of error in the derivation of the In concentration are discussed. The optimised imaging conditions resulting from a minimisation of artefacts in the image are explained. As they impose the necessity to distinguish the reflections of the (0002) and (0002) lattice planes, a procedure for measurement of the polarity in GaN during a TEM session is described in section 3.4.
3.1
Electron diffraction
In a transmission electron microscope, electrons travel at more than half of the speed of light c. Hence relativistic corrections will be needed. These are given by
mr = me 1 + eU mec2 =: meγ (3.1) Ur = U 1 + eU 2mec2 =: U + ǫU2 (3.2) λr = h r 2meeU 1 + 2meU ec2 = h √ 2meeUr (3.3)
where mr, Ur, and λrare the relativistically corrected electron mass, acceleration potential, and wave length, me is the electron rest mass, e is the magnitude of the electron charge, and U is the uncorrected acceleration potential. The wave length λ of an electron wave is connected to its wave vector k and its momentum p via the Planck’s constant h: p = h k and | k | = 1/λ.
3.1.1
Kinematic theory
For kinematic diffraction theory multiple scattering is not taken into account. Furthermore it is assumed in the following, that an incident plane electron wave is scattered elastically.
Electron scattering at a single atom
The incident plane electron wave with wave vector k0 can be described by
Ψ0( r ) = e−2πik0·r , (3.4)
where the amplitude of the wave is set equal to one. As the scattering is elastic, the wave vector k = k0+ ∆ k (figure 3.1) of the scattered wave is of the same magnitude as the wave vector of the incident wave. The electron wave is scattered at the Coulomb potential φa( r ) of the atom. φa( r ) is connected with the charge density of the atom by the Poisson equation
∆rφa( r ) = − e ǫ0
Figure 3.1: Scattering of an incident plane electron wave with wave vector k0 at a single atom with nucleus positioned at r0.
with ∆r the Laplace operator acting on the components of r , ǫ0the permittivity of vacuum and ρn,e( r ) the charge density of the nucleus and the electrons of the atom. ρn( r ) can be approximated by Zδ( r ) for an atom with atomic number Z located at the origin ( r0 = 0). The stationary, relativistically corrected Schr¨odinger equation [52] is given by
∆rΨ( r ) + 8π2m ee h2 UrΨ( r ) = − 8π2m ee h2 γφa( r )Ψ( r ) . (3.6)
This inhomogeneous differential equation can be transformed into an integral equation [53] using the Green’s function resulting in
Ψ( r ) = Ψ0( r ) + σ Z r′ e−2πik| r − r′| | r − r′| φa( r ′)Ψ( r′)d r′ , (3.7)
with the norm of the relativistically corrected wave vector in vacuum k := | k | =q2meeUr
h2
(equation (3.3)) and the interaction constant σ := 2πmre
h2 . Ψ0( r ) is the solution of the homogeneous differential equation (3.6) for φa( r ) = 0 , i. e. identical to the incident plane wave (equation (3.4)). For r := | r | ≫ r′ := | r′|, | r − r′| can be approximated by r − r′k· k, and by use of the first Born approximation, equation (3.7) can be written as
Ψ( r ) = Ψ0( r ) + e−2πikr r σ Z r′ e2πi∆ k· r′φa( r′)d r′ = Ψ0( r ) + ΨS( r ) . (3.8)
This means that scattering of a plane electron wave at a Coulomb potential results in a spherical scattered wave ΨS( r ) of amplitude
feB(∆ k ) := σ Z r′ e2πi∆ k · r′φa( r′)d r′ (B.1) = σFr→∆ kφa( r ) (3.9) = σ Z r′ e2πi k· r′φa( r′)e−2πi k0· r ′ d r′
= σ he−2πi k · r|φa( r )|e−2πi k0· ri
| {z } (3.10)
Fr →∆ kφa( r ) := fe(∆ k ) . (3.11)
feB(∆ k ) is a former definition of the atomic electron scattering amplitude and is derived by Born approximation. It is proportional to the nowadays commonly used atomic electron scattering amplitude fe(∆ k ), which is defined as the propability that the incident plane wave Ψ0( r ) = e−2πi k0· r is scattered by the potential φa( r ), resulting in a wave propagating in direction of k . This definition of fe(∆ k ) is equivalent with the definition as the Fourier transform of the Coulomb potential φa( r ) of the atom. The x-ray scattering amplitude is defined analogously by
fx(∆ k ) := he−2πi k · r|ρe( r )|e−2πi k0· ri = Fr →∆ kρe( r ) . (3.12)
With aid of the Poisson equation (3.5), the following relation between the electron scat-tering amplitude fe(∆ k ) and the x-ray scattering amplitude fx(∆ k ) is obtained
fe(∆ k ) = e
4π2ǫ
0∆k2(Z − f
x(∆ k )) , (3.13)
with ∆k := |∆ k |. Introducing this result for fe(∆ k ) into equation (3.10) results in feB(∆ k ) = mre
2
2πǫ0h2∆k2(Z − f x
(∆ k )) . (3.14)
The atomic scattering amplitude feB was calculated for many elements by e. g. Doyle and Turner [54]. They give a parameterised form feB( s ) without relativistic correction (i. e. me instead of mr in equation (3.14)) and define the variable s = ∆ k/2. In figure 3.2, the atomic electron scattering factors feB and fe in dependence of ∆k and s := | s | are given for Ga, In, N, and Al. The values of the reciprocal lattice spacings for the {1010}, {0002}, and {1120} planes of wurtzite GaN are indicated.
0 5 10 15 20 ∆k [nm-1] 0 0.1 0.2 0.3 0.4 0.5 f e [Vnm 3 ] 0.0 2.5 5.0 7.5 10.0 s [nm-1] 0 0.5 1.0 1.5 f eB (200 kV) [nm] N In Ga Al 10 1 0 0002 11 2 0
Figure 3.2: Atomic electron scattering amplitudes for N, Ga, In, and Al [54]. fe according to equation (3.13) is shown on the left hand side and feB according to equation (3.14) for electrons accelerated by 200 kV is shown on the right hand side. The dashed vertical lines mark the reciprocal lattice spacings for the indicated lattice planes of wurtzite GaN.
The derivation of the atomic electron scattering amplitude fe(∆ k ) as given above assumes the atoms to be at rest at position r0. Strictly, this is never the case. The effect due to a time dependent displacement ∆ r0(t) from r0 of the atom on the atomic scattering amplitude can be described by regarding the fact that the velocity of the atom due to lattice vibrations is orders of magnitude smaller than the velocity of the electrons in the transmission electron microscope. For a time integral corresponding to several periods of the lattice vibration, the influence of the vibration can therefore be approximated by an atomic potential which is spread over a larger area φa( r0 + ∆ r0). Using the Taylor expansion of the time averaged φa up to the second derivative of φa, the corresponding time averaged atomic scattering amplitude can be expressed as fe(∆ k )e−∆ k · B ·∆ k. The Debye-Waller factor B depends on the time averaged square of the vibrational amplitude h∆ r2
0it and increases with increasing temperature. In the general case, B is a tensor, as the atomic vibrations depend on the crystal orientation.
Electron scattering at a periodic lattice
In a crystal lattice consisting of atoms in unit cells which are periodically arranged at lattice vectors R = u a1 + v a2+ w a3 (u, v, w ∈ Z, basis vectors ai), the total scattered
wave ΨtS( r ) is determined by the total potential of all atoms. This can be derived by first summing up the contribution of the potentials of all atoms in a unit cell, which gives the
structure amplitude Fs(∆ k ), and a subsequent summation over the lattice vectors of the unit cells, which gives the lattice amplitude G(∆ k ).
With the nomenclature r′ = R + r
0i+ r′′ as given in figure 3.3 and N atoms per unit cell, the total scattered wave at r is given by
ΨtS( r ) = e−2πikr r σ X u,v,w N X i=1 Z r′ e2πi∆ k· r′φai( r′ − r0i− R )d r′ (3.15) r ′= R + r 0i+ r′′ = e−2πikr r σ X u,v,w e2πi∆ k · R N X i=1 e2πi∆ k· r0i Z r′′ e2πi∆ k· r′′φai( r′′)d r′′ (3.16) (3.11) = e −2πikr r σ X u,v,w e2πi∆ k · R | {z } N X i=1 e2πi∆ k · r0ife i(∆ k ) | {z } . (3.17) := G(∆ k ) := Fs(∆ k ) (3.18)
With the reciprocal lattice vectors G = h a∗
1+k a∗2+l a∗3 with h, k, l ∈Zand the reciprocal
Figure 3.3: Geometry and nomenclature of scattering at a lattice consisting of unit cells with basis vectors ai.
basis vectors a∗ i defined by a∗1 := a2× a3 a1· ( a2× a3) a∗2 := a3× a1 a1· ( a2× a3) a∗3 := a1× a2 a1· ( a2× a3) , (3.19)
it can be seen that for an infinite crystal, the lattice amplitude G(∆ k ) gives only contribu-tions at ∆ k · R = n, i. e. for ∆ k = G . The intensity of a diffracted beam is thus propor-tional to |Fs( G )|2. For the wurtzite unit cell with position vectors r0i (p. 5) of the group III- and N-atoms approximated by r0,Ga = [1/3,2/3, 0] and [2/3,1/3,1/2], r0,N = [1/3,2/3,3/8] and [2/3,1/3,7/8] one obtains
|Fs( G )|2 = fGae ( G ) e2πi(h3+2k3) + e2πi(2h3 + k 3+ l 2) +fNe( G )e2πi(h3+2k3+3l8) + e2πi(2h3 + k 3+7l8) 2 = 2(fGae ( G )2+ fNe( G )2){1 + cos 2π h 3 − k 3 + l 2 }+ +2fGae ( G )fNe( G )· ·{2 cos 2π 3l 8 + cos 2π h 3 − k 3 + l 8 + cos 2π h 3 − k 3 + 7l 8 } . (3.20) From this formula one can obtain the kinematic systematic absences in dependence of h, k, and l for the wurtzite structure (Miller indexing). For example, the 001 reflection is kinematically forbidden. Nevertheless, it can be seen in diffraction in the transmission electron microscope. This is due to the fact that the kinematic approximation is only valid if multiple scattering does not occur, i. e. for sample thicknesses well below 10 nm. If the sample thickness gives rise to multiple scattering events, dynamic diffraction theory has to be applied.
3.1.2
Dynamic theory
In this subchapter, two different methods for computation of the object exit wave func-tion using dynamic theory will be presented. The starting point for both methods is the governing equation, which will be derived first. This is followed by a description of both methods, which are the Bloch wave approach and the multislice approach. The Bloch wave approach needs comparable long computation times [55] and is well suited for calculation of the object exit wave function in case of a perfect crystal. It was applied in this work for calculating structure amplitudes and intensities of diffracted beams and convergent beam electron diffraction (CBED) patterns including higher order Laue zone (HOLZ) lines from perfect crystals. The multislice approach can be applied to perfect crystals and to crys-tals containing defects. It was used in this work for the simulation of HRTEM images of inversion domain boundaries (IDBs).
The governing equation
In a crystal the electrons experience not only the acceleration voltage U, but an additional acceleration due to the crystal potential φc( r ), hence the relativistic total acceleration voltage according to equation (3.2) is given by Utr ≈ Ur+ γφc( r ), where it has been used that φc( r )≪ U. This is expressed by transformation of equation (3.6)
∆rΨ( r ) + 8π2m ee h2 (Ur+ γφc( r )) Ψ( r ) = 0 . (3.21)
Absorption can be described phenomenologically by adding an imaginary term to the crystal potential [56]. φc( r ) can be expressed as a Fourier series
φc( r ) = φk = 0 + X k6= 0
φke2πi k· r , (3.22)
where φ0 is the mean inner potential of the crystal. The mean wave vector of the electron beam inside the crystal can then be described by k2
c =
2mee(Ur+γφ0)
h2 , and with Uc( r ) = 2mee
h2 γ(φc( r ) − φ0) equation (3.21) can be written as
∆rΨ( r ) + 4π2kc2Ψ( r ) = −4π2Uc( r )Ψ( r ) , (3.23)
which is called the governing equation. In the following, the object exit wave function is obtained from this equation by the Bloch wave approach and the multislice approach.
Bloch wave approach
In the periodic lattice of a crystal the wave functions which solve equation (3.23) can be described with aid of the Bloch wave ansatz Ψk( r ) = uk( r )e−2πi k · r, where the coefficients uk( r ) have the periodicity of the crystal lattice. These coefficients can be expressed as Fourier series uk( r ) =
P
g Cgke−2πi g · r, where the Cgk are called Bloch wave coefficients.
Ψk( r ) is then given by Ψk( r ) = Pg Cgke−2πi( g + k )· r. Introducing this and the Fourier expansion of Uc( r ) into the governing equation (3.23), one obtains the secular equation
k2c − ( k + g )2Cgk + X h 6= g
Ug − hChk = 0 . (3.24)
k can be decomposed into its components parallel and perpendicular to the specimen surface k = kk + k⊥, with kk = kck = constant due to the continuity of the wave function at the specimen surface. This means that k and kc differ by a small amount ∆ k⊥ = k − kc perpendicular to the specimen surface. Using the high energy approxima-tion, where backscattering is neglected, one can approximate k2
c − ( k + g )2 ≈ 2kc⊥(kc⊥− k⊥) − ( kck+ g )2. Substitution into the secular equation (3.24) results in
− kck+ g 2
Cgk + X h6= g
Ug − hChk = −2kc⊥(kc⊥− k⊥)Chk . (3.25)
Writing this as matrix and considering N beams gn, one obtains the eigenvalue equation −( kck+ g0)2 Ug0− g1 · · · Ug0− gN Ug1− g0 −( kck+ g1) 2 ... ... . .. ... UgN− g0 . . . −( kck+ gN) 2 · Ck g0 .. . .. . Ck gN = −2kc⊥(kc⊥− k⊥) Ck g0 .. . .. . Ck gN . (3.26)
From the N eigenvalues of this equation km = kck+ km⊥ is calculated, and the N vectors C km can be derived. The total wave function is then given by the superposition of the
Bloch waves Ψ( r ) = N X m=0 αkm N X n=0 Ckm gn e −2πi( gn+ km⊥+ kck)· r , (3.27)
where αkm are the excitation amplitudes of the corresponding Bloch waves and are derived
from the constraint PNm=0αkmC
km
g0 = 1. Interpreting αkm as vector α and (C
km
gn ) as
matrix C with the inverted matrix D := C −1 containing components Dkm
gn, it can be
seen that αkm = D
k0
gm. For specimen thickness t and with the unit vector ˆe⊥perpendicular
to the specimen surface, r = r⊥+ rk = t ˆe⊥+ rk is splitted into its components parallel and perpendicular to the specimen surface analogously to k , and one obtains the object exit wave function
Ψ( r ) = N X n=0 N X m=0 Dk0 gmC km gn e −2πikm⊥t | {z } e−2πi( gn+ kck)· rk , (3.28) := F ( gn) (3.29)
with the amplitude of a scattered beam F ( gn) depending on the specimen thickness t. The normalised intensities |F ( g )|2
/|F ( 0 , t=0)|2 for GaN with incident electron wave along h2110i
and h1100i are shown in figure 3.4 for the transmitted, 0002, and 0002 beams. For the calculation 183 beams were considered, corresponding to a square of 0.5 nm−1 around the chosen zone axis (ZA). Comparison for calculation with 500 beams shows for example a maximum deviation of the normalised intensity of the transmitted beam in the thick-ness range up to 40 nm of 0.0029. Therefore the use of only 183 beams for Bloch wave calculations in ZA orientation is sufficiently precise.
a) 0 10 20 30 40 specimen thickness t [nm] 0 1 |F g | 2 / | F 0000 (t =0)| 2 0000 0002 0002 b) 0 10 20 30 40 specimen thickness t [nm] 0 1 |F g | 2 / | F 0000 (t =0)| 2 0000 0002 0002
Figure 3.4: Normalised intensities in dependence of the specimen thickness t of the 0000, 0002, and 0002 beams calculated for a h2110i and b h1100i ZA orientation of wurtzite GaN. The Bloch wave calculations were performed for ∼200 beams using the EMS program package [57].
Multislice approach
A description of the object exit wave as a modulated plane wave Ψ( r ) = ψ( r )e−2πi kc· r
and insertion into the governing equation (3.23) results in
∆rψ( r ) − 4πi kc · ∇rψ( r ) = −4π2Uc( r )ψ( r ) . (3.30) With the approximation that kc k ˆez, i. e. kc· ∇rψ = kc∂ψ∂z, the terms which depend on z can be separated: ∂2ψ ∂z2 + ∆x,yψ − 4πikc ∂ψ ∂z = −4π 2U cψ . (3.31)
Using again the high energy approximation, the second derivative can be neglected and equation (3.31) results in ∂ψ ∂z = − i 4πkc ∆x,yψ − iπUc kc ψ = ∆ψ + V ψ . (3.32)
The crystal potential only influences the second term on the right side of equation (3.32) (through Uc in V , see p. 26). For ∆ = 0, equation (3.32) is called phase grating equation, and its solution is given by ψ = eR0zV dz =: e−iλσφp(z), with kc = 1/λ and the interaction
constant σ as defined on p. 21. This means that the initial plane wave is changed by the phase factor ψ which depends on the projected potential φp(z).
If V = 0, equation (3.32) is called propagator equation, with the solution ψ(x, y, z) = ez∆ψ(x, y, 0) describing the changes occurring when the wave propagates the distance z. This equation can be transformed by using the Fourier and inverse Fourier transform and the representation of the exponential function as power series. With ψ( R , z) := ψ(x, y, z), ψ( R ) := ψ(x, y, 0), and q a two dimensional reciprocal space vector one obtains
ψ( R , z) = F−1q→ R{ eψ( q )eiπλz q
2
} = ψ( R ) ⊗ e−iπR 2λz = ψ( R ) ⊗ P
z( R ) , (3.33) where kc =1/λwas inserted. Pz( R ) is the Fresnel propagator describing the free propaga-tion of the wave ψ( R ) over the distance z.
For the multislice approach, the specimen of thickness t is subdivided into n thin slices of thickness ǫ. The potential of each slice is projected onto a plane in the slice. After the wave ψ[n−1] incident on slice n interacted with the projected potential φp [n], the resulting wave ψ[n] propagates to the next plane containing the following projected potential φp [n+1]. If the potential of the slices is projected on the lower surface of the slice, ψ( R , t) at the object exit plane of the specimen is given by
3.2
Image formation
After passing the specimen, the object exit wave function Ψ present at the lower surface of the sample propagates through the lens system of the transmission electron microscope (figure 3.5). In the back focal plane of a lens, the wave function is described by the Fourier transform of the wave function in the object plane. The lenses not only lead to a magnification of the final image, but also distort the wave function due to aberrations. Defocus and apertures introduce an additional change. The objective lens is the first lens after the specimen and limits the resolution in a transmission electron microscope. The spherical aberration is the most important aberration of the objective lens and cannot be
Figure 3.5: Path of an electron wave through specimen and objective lens. An incident wave Ψ0 interacts with an object. At the lower surface of the object, the object exit wave function Ψ is present. Only the transmitted and one diffracted beam is sketched. The wave propagates through the objective lens. The beams for imaging are chosen with an aperture in the back focal plane (eΨBF P). Due to aberration, the images of the object formed by the diffracted and the transmitted beam are shifted against each other in the image plane (delocalisation).
corrected in the CM20 UT microscope, which was mainly used for this work. Due to the spherical aberration, a phase shift χs( k ) is introduced which depends on the distance to the optical axis and is described by
χs( k ) = π 2Csλ
3k4 , (3.35)
with the coordinate in Fourier space k and the spherical aberration constant Cs, which amounts to 0.5 mm for the CM20 UT microscope. If the object exit wave function is not in the object plane of the objective lens, this defocus ∆f introduces an additional phase shift given by
χ∆f( k ) = π∆f λ k2 . (3.36)
The total phase shift χ( k ) = χs( k ) + χ∆f( k ) is plotted in figure 3.6 for defocus values ranging from −100 nm to 50 nm. The spherical aberration constant Cs = 0.5 mm of the CM20 UT microscope operated at 200 kV was used for the calculation of χ( k ).
The wave function in the back focal plane of a lens is described by the Fourier transform of the wave function in the object plane. With the phase shifts due to defocus and spherical aberration, the object exit wave function Ψ after propagation to the back focal plane of the objective lens is given by
Fr→ kΨBF P( r ) = eΨBF P( k ) = eΨ( k )eiχ( k ) . (3.37) In presence of an objective aperture localised in the back focal plane, eΨBF P has to be multiplied with the aperture function A( k ) which is equal to 1 inside the hole of the aperture with radius | kA| (i. e. | k | ≤ | kA|) and 0 elsewhere.
The wave function in the image plane is finally given by the inverse Fourier transform of the wave function in the back focal plane:
ΨI( r ) = F−1k → r( eΨ( k )eiχ( k ))
(B.7)
= Ψ( r ) ⊗ F−1k → reiχ( k ) . (3.38) By approximating the aberration function with the first terms of its Taylor expansion χ( k ) = χ( ghkil)+∇kχ( k )|ghkil·( k − ghkil) around the reflection ghkil, the wave function
in the image plane is expressed as
ΨI( r ) = ei (χ( ghkil)−∇kχ( k )|g hkil· ghkil)Ψ( r )⊗ F−1k → rei∇kχ( k )|g hkil· k (3.39)
(B.11, B.12)
= ei (χ( ghkil)−∇kχ( k )|g hkil· ghkil)Ψ( r )⊗ δ
r − 2π1 ∇kχ( k )|ghkil
(3.40)
(B.9)
= ei (χ( ghkil)−∇kχ( k )|g hkil· ghkil)Ψ
r − 1
2π∇kχ( k )|ghkil
−10 −5 0 5 10 −100 −50 0 50 −20 0 20 40 60 80 100 120 140
reciprocal space coordinate [nm
−1]
defocus
∆ f [nm]
χ
Figure 3.6: Aberration function χ( k ) due to spherical aberration and defocus ∆f in depen-dence of the reciprocal space coordinate k for different defocus values ∆f . The parameters for the spherical aberration constant Cs = 0.5 mm and the wave length correspond to the CM20 UT microscope operated at 200 kV.
It is obvious that additional to the phase shift, the wave function in the image plane reproduces the object exit wave function shifted by 1/2π∇
kχ( k )|ghkil. If only one beam
is used for imaging, the shift does not influence the measurements. If two beam imaging conditions for HRTEM are used, both beams will show a different shift as sketched in figure 3.5. This shift is called delocalisation. For two beam imaging, the delocalisation can be adjusted to zero by putting one beam on the optical axis and choosing the defocus in such a way that ∇kχ( k )|ghkil = 0 for the used ghkil, i. e. to adjust a minimum of χ( k )
(figure 3.6). The corresponding defocus values for zero delocalisation are given in table 3.1 for two beam conditions with different hkil beams for GaN and InN.
∆f [nm] for zero delocalisation hkil GaN InN 0002 -47 -38 1100 -41 -34 1120 -124 -101
Table 3.1: Defocus values ∆f for two beam imaging with zero delocalisation for the CM20 UT microscope. The defocus values are given for different beams for wurtzite GaN and InN.
3.2.1
Effect of incoherence
Until now, only coherent imaging was described, where wave functions are summed. Intro-ducing the nomenclature eiχ( k ) =: t( k ), usually called coherent transfer function (CTF), the diffractogram eI( k ) = Fr → kI( r ) = Fr → k(|ΨI( r )|2), i. e. the Fourier transform of the image intensity, is calculated for coherent imaging to
e I( k ) (3.38)= Z k′ e Ψ( k′+ k ) eΨ∗( k′)t( k′+ k )t∗( k′)d k′ . (3.42)
Nevertheless, the electron beam in a transmission electron microscope is not completely coherent. It can be distinguished between spatial incoherence due to a not perfect plane electron wave, i. e. the electron beam is convergent with semi convergence angle αs, and temporal incoherence, whose effects can be described by a defocus spread ∆f. If the electrons are incoherent, the intensities instead of the wave functions are summed.
In case of the spatial incoherence, the incident electron wave has not only one wave vector k0, which is assumed to point along the optical axis in the following, but a distri-bution of wave vectors. This distridistri-bution called source spread function can be described by a Gaussian distribution according to
s( q ) = 1 πα2 s e− q 2 α2s , (3.43)
where q denotes the component of the k vector perpendicular to the optical axis, nor-malised to | k0|. Each wave described by Ψ( r ; q ) with tilt to the optical axis corresponding to q produces an image. The intensity of the final image is then given by summation of
all individual images Is( r ) = Z q I( r ; q )s( q )d q (3.44) (3.38) = Z q Ψ( r ; q ) ⊗ F −1t( k ) Ψ∗( r ; q ) ⊗ F−1t∗( k )s( q )d q . (3.45)
Carrying out this integration the diffractogram can be calculated to e Is( k ) = Z k′ e Ψ( k′+ k ) eΨ∗( k′)t( k′+ k )t∗( k′)E s( k′+ k , k′)d k′ , (3.46)
with the spatial incoherence envelope function Es( k′+ k , k ) given by Es( k′+ k , k ) = e−
α2s 4λ2(∇χ( k
′+ k )−∇χ( k′))2
, (3.47)
which imposes an exponential damping.
Temporal incoherence can be present due to fluctuations of the lens current, of the acceleration voltage or due to the energy spread of the electrons leaving the electron gun. The effect is a change of the defocus from ∆f to ∆f + δf . In analogy to the source spread function the defocus spread function is obtained and can again be described as Gaussian distribution: τ (δf ) = √1 π∆f e− δf 2 ∆2f . (3.48)
The defocus spread ∆f can be calculated by ∆f = Cc s σ2(V ) V2 + 4σ2(I l) I2 l + σ 2(E) E2 , (3.49)
with the chromatic aberration coefficient Cc = 1.0 mm for the used CM20 UT microscope, the standard deviations σ of the acceleration voltage V , the objective lens current Il, and the energy of the electrons E. Carrying out the integration analogous to equation (3.44), the image intensity Iτ( r ) is obtained. The diffractogram eIτ( k ) is then given by an expres-sion comparable to equation (3.46), but with the temporal incoherence envelope function
Eτ( k′+ k , k ) = e− ∆2f 2 “∂χ( k ′+ k ) ∂∆f − ∂χ( k ) ∂∆f ”2 , (3.50)
which again imposes an exponential damping.
The damping effect of the spatial and temporal incoherence functions on the CTF is shown in figure 3.7. The parameters used for the calculations correspond to the CM20 UT operated at 200 kV.
a) b)
c) d)
Figure 3.7: The spatial incoherence envelope function (a) and the temporal incoherence envelope function (b) are exponentially decreasing functions with increasing spatial fre-quency. The CTF (c, imaginary part) is damped by both envelope functions (d). The parameters for the calculations correspond to the CM20 UT microscope operated at 200 kV with defocus set to zero (Cs= 0.5 mm, ∆f = 7 nm, αs = 1.6 mrad).
3.3
Evaluation of In concentration in
In
xGa
1−xN structures
The basis for strain state analysis is the knowledge of the lattice spacings. Inside a perfect crystal this is dhkil=1/| ghkil|. Regarding wurtzite InxGa1−xN embedded in GaN with h0001i growth direction, the in-plane lattice parameter of the InxGa1−xN a(x) is compressed. Relaxation will occur in growth direction, resulting in a tetragonally distorted InxGa1−xN lattice with lattice parameters at(x) and ct(x). The lattice distances in h0001i direction will thus change from the lattice distance of GaN to that of InxGa1−xN . In the first part of this section, an expression for the lattice spacings containing this change will be derived, which actually describes the lattice distances inside the crystal. This is followed by a description how the lattice spacings are derived from HRTEM fringe images.
In the second part of this section, the tetragonal distortion of the InxGa1−xN structure which changes the c lattice parameter or equivalently the {0002} lattice plane spacing is regarded. By preparation of a thin cross section sample, the InxGa1−xN structures can relax furthermore in the direction of the intended TEM viewing direction. This elastic relaxation is taken into account using elasticity theory, and the derivation of the In concentration of such a sample is described.
Finally, the errors which have to be considered for strain state analysis are discussed in a third part.
3.3.1
Derivation of lattice distances
For evaluation of the In concentration x in InxGa1−xN/GaN heterostructures the difference in lattice spacing is analysed according to Vegard’s rule (see equation (2.1)) and using elasticity theory. Hence one main basis is the measurement of the lattice spacings. In the first part of this subsection, an expression for the lattice spacing of {hkil} planes will be derived in the framework of linear elasticity theory, which can be used if a change from a reference lattice spacing is present. For this work, mainly the lattice fringe distance caused by {0002} planes was analysed, which is half the c lattice parameter. The second part of this subsection describes the procedure how the fringe images were taken and evaluated.
Real lattice spacings
Consider the distance dhkilbetween {hkil} lattice planes in direction z of the plane normal ˆehkil. If an InxGa1−xN film (for simplicity with film normal ˆehkil) of concentration x is grown on or embedded in GaN, the lattice distance is changed. A reference lattice of spacing drefhkil in e. g. the GaN area can be defined and extrapolated as is sketched in figure 3.8. The positions of the InxGa1−xN planes can be correlated with positions of the reference lattice. The displacement of the position of one InxGa1−xN plane to the plane of
Figure 3.8: Each position of the InxGa1−xN lattice can be correlated with a position of the extrapolated GaN reference lattice of spacing drefhkil. The difference in z direction between the InxGa1−xN lattice position and its correlated reference lattice position is the projected displacement R. n denotes the position of a lattice plane and can be expressed as a function of z.
the reference lattice defines the projected displacement R, which is zero inside the reference lattice. With aid of the reference lattice, the lattice spacing in dependence of z can be expressed as dhkil(z) = drefhkil(1 + ǫ), with the strain ǫ. This can be transformed to
dhkil(z) = drefhkil
1 −1 + ǫǫ −1
. (3.51)
If the first InxGa1−xN lattice plane is located at z = 0, the n-th lattice plane is located at z = n(z)drefhkil(1 + ǫ) ⇔ n(z) = z
drefhkil(1 + ǫ) . (3.52)
The displacement of the n-th plane to the corresponding plane in the reference lattice is given by
R(z) = n(z)drefhkil(1 + ǫ) − n(z)drefhkil = n(z)drefhkilǫ . (3.53) Introducing equation (3.52) into (3.53) results in
R(z) = z ǫ 1 + ǫ ⇒ dR(z)dz = ǫ 1 + ǫ . (3.54)
With the definition of the geometric phase Ghkil(z) := 2πghkilR(z) this can be expressed as dR(z)dz = 2πg1
hkil
dG(z)
dz =
ǫ
1+ǫ. Inserting this into equation (3.51) and with d ref
hkil=1/ghkil one
obtains dhkil(z) =
ghkil− 2π1 dG(z)dz −1
. In the general case for three dimensions one derives
dhkil( r ) =
1
| ghkil| −2π1 ∇rGhkil( r ) · ˆehkil
. (3.55)
This formula thus describes the lattice spacing dhkil( r ) in a crystal, when the lattice distance changes from dhkil to dhkil(1 + ǫ).
HRTEM fringe analyses and evaluation of fringe distances from HRTEM images All analysed GaN based material for this work was grown on {0001} sapphire, i. e. in case of InxGa1−xN layers the normal of a layer is ˆe0001. As the InxGa1−xN can be compressed in the {0001} plane to the a lattice parameter of GaN, the change of distances in [000l] is measured. Therefore the samples were prepared in cross section by a tripod method. After mechanically polishing the samples, they were ion milled in a PIPS (Gatan) with Ar+ ions at (5.0 . . . 5.4) kV and an angle of ±5◦. For HRTEM analyses a CM20 UT microscope equipped with a LaB6 filament and operated at 200 kV was used. The used
