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Raman spectroscopy of InAs/GaAs quantum dots patterned by nano-indentation

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Raman spectroscopy of InAs/GaAs quantum dots patterned by nano-indentation
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Hussey, Lindsay K
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ABSTRACT: Patterns of InAs/GaAs quantum dots (QDs) grown by the combination of nanoindentation technique and molecular beam epitaxy were studied. The resulting QDs tend to preferentially nucleate on indented areas rather than other regions. We studied the strain on the indentations, regions surrounding the indents, and non-indented areas. The QD LO mode for the patterned areas shifts by 7 cm-1 when compared to the non-patterned area. The biaxial strain in the indented areas producing this shift is four times larger than that in non-indented areas, explaining the QD preferential formation within these areas. This larger strain suggests that QDs on the indentations can be formed by depositing a smaller InAs amount than that required to form QDs on non-indented areas, thus obtaining QDs only on the pattern.
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Thesis (M.S.)--University of South Florida, 2007.
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Raman Spectroscopy of InAs/GaAs Quantum Dots Patterned by Nano-Indentation by Lindsay K. Hussey A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Physics College of Arts and Sciences University of South Florida Major Professor: Martin Munoz, Ph.D. Xiaomei Jiang, Ph.D. George S. Nolas, Ph.D. Date of Approval: May 11, 2007 Keywords: scattering, optics, nanostructure, semiconductor, physics Copyright 2007, Lindsay K. Hussey

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Dedication This work is dedicated to Amalia, Ariana and Alexandra Fragoso.

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Acknowledgements I would like to thank Dr Martin Munoz, Dr. Cu rtis Taylor, Dr. Durig Lewis and Laura Akesson. Without their help this work would not have been possible. I would also like to thank my parents, my family and my wonderful friends.

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Table of Contents List of Figures................................................................................................................ ......i List of Tables................................................................................................................. .....ii Abstract....................................................................................................................... .......iii Chapter 1. Fundamentals of Raman Spectroscopy.............................................................1 1.1 Introduction........................................................................................................1 1.2 Classical Treatment of Raman Shift..................................................................2 1.3 The Raman Tensor and Selection Rules............................................................4 1.4 Experimental Setup............................................................................................8 Chapter 2. Raman Spectroscopy of Nano-Indented Quantum Dots.................................13 2.1 Introduction......................................................................................................13 2.2 Sample Description..........................................................................................14 2.3 AFM Results....................................................................................................15 2.4 Raman Results.............................................................................................................1 6 2.5 Analysis: Identification of Raman Modes...................................................................18 2.6 Analysis: Strain Calculation............................................................................22 Chapter 3. Conclusions.....................................................................................................26 References..................................................................................................................... .....27 Appendices..................................................................................................................... ....28 Appendix A Mathematica Calculation of Quantum Dot Interface Modes........................29

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List of Figures Figure 1: Comparison of Raman and PL signal intensities..................................................1 Figure 2: Illustration of x, y and z directions within sample geometry...............................7 Figure 3: Internal components of HR 800...........................................................................9 Figure 4: Horiba Jobin Yvon HR 800 micro-Raman System............................................10 Figure 5: Path of light in Raman System...........................................................................11 Figure 6: Microscope Objectiv e and Spectrograph Grating..............................................12 Figure 7: InAs coverage versus 3D Island Density...........................................................14 Figure 8: Schematic of InAs dot on GaAs buffer layer.....................................................15 Figure 9: AFM Images of InAs QDs gr own by nano-indentation (increasing tip load).......................................................................................................................... .........15 Figure 10: Scattering Intensity versus wavenumber of 200N nano-indented samples........................................................................................................................ .......17 Figure 11: Scattering Intensity versus wavenumber of 500nm nano-indented samples........................................................................................................................ .......17 Figure 12: QD orientation diagram and hydr ostatic and biaxial strain distribution along the center of a pyramidal QD (Z (001) direction)....................................................22 Figure 13: Wavenumb er vs. QD height.............................................................................25 i

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List of Tables Table 1: Selection Rules for zincblende materials3............................................................8 Table 2: Samples with fixed load......................................................................................16 Table 3: Samples with fi xed indentation spacing.............................................................16 ii

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Raman Spectroscopy of InAs/GaAs Quantum Dots Patterned by Nano-Indentation Lindsay K. Hussey ABSTRACT Patterns of InAs/GaAs quantum dots (QDs) grown by the combination of nanoindentation technique and molecular beam epitaxy were studied. The resulting QDs tend to preferentially nucleate on indented areas rather than other regions. We studied the strain on the indentations, regions surrounding the indents, and non-indented areas. The QD LO mode for the patterned areas shifts by 7 cm-1 when compared to the nonpatterned area. The biaxial stra in in the indented areas produc ing this shift is four times larger than that in non-indented areas, expl aining the QD preferential formation within these areas. This larger strain suggests that QDs on the indentations can be formed by depositing a smaller InAs amount than that required to form QDs on non-indented areas, thus obtaining QDs only on the pattern. iii

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Chapter 1. Fundamentals of Raman Spectroscopy 1.1 Introduction When light is incident on a material se veral processes can occur; light can be reflected, transmitted, absorbed or scattered. Raman Scattering is a second order process consisting of the inelastic scattering of inci dent photons by phonons (lattice vibrations). As illustrated by Figure 11, Raman Scattering (RS) is extremely weak when compared to other optical processes such as Photolumines cence. However, with the development of lasers and ultra-sensitive detectors, Rama n Spectroscopy (RS) has become an extremely useful tool for investigating properties of semiconductors such as phonon modes, doping levels, crystalline quality, impur ity concentration and strain. Fig.1 Comparison of Raman and PL signal intensities of Cd0.72Mn0.28 Te film on a GaAs substrate1 1

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1.2 Classical Treatment of Raman Shift The interactions of the in cident photon with the phonon modes within a material lead to a shift in photon frequency, referre d to as the Raman shift. Some outgoing photons will be shifted to higher energy (a phonon is absorbed) or lower energy (a phonon is emitted). These shifts are called anti -Stokes (higher energy) and Stokes (lower energy), respectively. The quantum mechani cal treatment of RS is very complex1 and will not be provided here but can be found in Ref. 2. In this thesis RS will be treated in a classical perspective for the reader to achieve a basic unde rstanding of the process. The nonlinear Raman interaction depends on the polarizablity of the material on which light is incident, we note that the dielect ric function will vary slightly with lattice spacing and the electric susceptibility cha nges due to the excita tion of the material1. We begin our discussion by introducing a monochromtic electromagnetic plane wave of the form, ,,cosiiiitt FrFkkr (1) into some medium. This electric field will induce a polarization ,,cosiiiitt PrPkkr (2) The polarization will have the amplitude: ,,,iiiiii PkkFk, (3) where is the electric susceptibility. The fre quency and wavevector of the polarization is the same as those of the incident radiation. The electric susceptibility will fluctuate due 2

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to atomic vibrations3 The normal modes of these atom ic vibrations are called phonons. We can express the displacement of atoms associated with a phonon as plane waves: t tph ph r q q Q r Q cos , (4) where ph and q are the phonon frequency and wa vevector, respectively. The electron frequencies which contribute to the electric susceptib ility are much higher than ph therefore we can take to be a function of Q and expand it as a Taylor series in Q ( r ,t): 0,,,,...iiiit 0kQkQr Q (5) The first term of equation (5) represents the electric susceptibility with no atomic fluctuations and the second te rm represents the fluctuations induced by the phonon wave Q Using this expression for the electric susceptibility we can rewrite the polarization as: ,,,,,indttt 0PrQPrPrQ (6) where, t ti i i i i i i r k k F k r P0cos , ,0 (7) and, t t ti i i i i ph ph ind r k k F r q q Q Q Q r P cos cos , ,. (8) Equation (7) is a polarizati on vibrating in phase with the incident electromagnetic radiation and Eq. (8) is a polar ization induced by the phonon. We determine the frequency and wavevect or of the polarization induced by the 3

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phonon by rewriting Eq. (8) as t t tph i i ph i i i i i ph ind r q k r q k k F q Q Q Q r P cos cos , 2 1 ,0 (9) From Eq. 9 we can see that th is polarization will give origin to electromagnetic waves of frequency i ph, ) ( ) ( 12 2 0 2 2 2t t t t c r P r E (10) This form of the phonon-induced polarization re veals two sinusoidal waves, the Stokes and Anti-Stokes shifted waves di scussed previously. The Stok es and Anti-Stokes shifted waves have the wavevectors and frequencies ,SiSiph kkq (11) and A SiASiph kkq (12) respectively. 1.3 The Raman Tensor and Selection Rules Now that we understand the origin of the Raman shift we will use this information to determine the intensity of the scattered ra diation. We will also discuss the importance of scattering geometries and the determination of Raman-active phonons through the Raman selection rules. The intensity of the scattered radiation can be expressed as3: Stokes N Stokes Anti N LV Ip p s i s s12 4e e (13) 4

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where s is the scattered frequency, L is the scattering length, V is the scattering volume L times the excitation beam area), Np is the phonon density which is expressed as 1 exp1p ph BN kT (14) and most importantly we have the Raman tensor 0ˆ 0Q Q (15) The Raman tensor is derived from the polar ization induced by the phonon, Eq. (8). carries geometric information and will vary for different materials depending on their symmetry and magnetic properties. The Ra man tensor is symmetric only when the frequency difference between the incide nt and scattered ra diation is ignored2. In practice this symmetry is assumed because the frequency of the laser is much larger than that of the phonon, i ph i ph i AS or S (16) Other non-symmetric component s can appear in the tensor when a magnetic field is present. Symmetry requirements lead to the non-existence of scattered radiation for certain scattering geometries, in other wo rds for a given geometry some phonons are allowed and some phonons are forbidden. To better acquaint ourselves with this t opic, we will discuss the Raman selection rules for zincblende materials, such as GaAs and InAs which are the base materials of our structures. For zinc-blende mate rials, vibrations polarized in the x = [100], y = [010] and 5

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z = [001] directions have the following Raman tensors3 0000000 00,000 and00 0000000xyzdd dd dd (17) where d is the one linearly independent component. Now that we have the Raman tensors in Eq. (17) we can determine the selection rules for RS in GaAs for different scattering geometries. Backscattering geometry is the geometry in which the directions of the inci dent and scattered photons are antiparallel to each other. Since this geomet ry is the one used in our experiments we will use it in our example. Let’s say we want to determine if the LO phonon is allowed for backscattering geometry on the (100) surface of GaAs. We begin with the Raman tensor 000 00 00xd d (18) We use this tensor because the LO phonon is de fined as the vibration in the x-direction. The selection rules will be illu strated in the backscattering ge ometry over a (100) surface. Figure 2 illustrates the x, y and z directions with respect to the sample geometry. 6

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z = [001] y = [010] x = [100] light LO phonon ezey z = [001] y = [010] x = [100] light LO phonon ezey Figure 2 Illustration of x, y, and z directions. Next we determine where the mode is allowed by imposing the incident and scattering polarizations: 00000 0010 000xyd dd e (19) and we have =0 and ,yxyzxyLOd eeee 0 = and 0 0xzyzzLOzxzdd eeeee (20) which shows that the LO phonon is allowed in this geometry4. A similar procedure can be followed for other scattering directions Table 1 gives selec tion rules for other backscattering geometries on the (100) surface of zinc-blende materials. In this table, we are using Porto’s notation given by: 7

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s s i ik e e k ) ( (21) where ki and ks are the directions of the incident and scattered photons, respectively, and ei and es are the polarizations of th e incident and scattered phot ons, respectively. In table 1 y’ and z’ denote the [011] and [0 11] directions, respectively. Selection Rule for (100) surface Scattering Geometry TO phonon LO phonon x(y,y) x ; x(z,z) x 0 0 x(y,z) x ; x(z,y) x 0 2 LOd x(y’,z’) x ; x(z’,y’) x 0 0 x(y’,y') x ; x(z’,z’) x 0 2 LOd y'(x,x) y 0 0 y'(z’,x) y 2 TOd 0 y'(z’,z’) y 2 TOd 0 Table 1 Selection Rules for zincblende materials on (100) surfaces3. 1.4 Experimental Setup A general micro-Raman system setup illustrating the paths of the excitation laser and scattered radiation is shown in Figure 3. The incoming laser radiation is reflected by the notch filter and then focused by the microscope objective on the sample. The reflected laser and scattered light are collected by the objective. The notch filter only allows the transmission of the scattered light and avoids the laser transmission to the spectrometer. 8

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NOTCH FILTER NOTCH FILTER Fig. 3 Internal components in the HR 800 system1 The experiments in this thesis were pe rformed using a Horiba Jobin Yvon HR 800 micro-Raman System where the 800 refers to the 800mm focal length of the spectrograph, an illustration of the e quipment is provided in Figure 4. 9

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Fig.4 Horiba Jobin Yvon HR 800 micro-Ra man System used in our experiments4 The Raman system includes an internal HeNe 20mW laser with a wavelength of 632.8 nm and an external HeCd 50mW lase r with a wavelength of 325nm. Scattered radiation is detected by a ch arge coupled device (CCD) dete ctor cooled to 77K by liquid nitrogen cooling. The computer on the setup uses LabSpec software. As shown in Figure 5, the exciting laser ra diation travels through a laser filter to remove plasma lines, afterward it passes thr ough a pinhole mirrors and notch filter and is focused on the sample by the microscope objective. The HR800 system includes six different neutral density filters to reduce the laser intensity according to DI I 100 (22) 10

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where D can range from zero to four dependi ng on the filter density. These neutral density filters are often used to reduce phot oluminescence that may appear in a scan. Fig. 5 Path of light in Raman Sy stem (white line represents light)4 Since we have a backscattering setup, the Raman Scattered light is transmitted back through the microscope objective and through the notch filter where it travels to the monochromator. This notch f ilter rejects the la ser light and allows only the scattered light to arrive at the spectrograph. The spectrograph scanning range is limited by the grating chosen. The HR800 is equipped with two gratings, 600g/mm and 2400g/mm. The 600g/mm has a scanning range of 0-2600n m and the 2400g/mm has a scanning range of 0-650nm. The spectrometer scans over a selected range and the CCD detects the intensity of the scattered light. The CCD de tector was calibrated using the Raman shift of a standard silicon samples as a reference. 11 Laser Filter Notch Filter

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The microscope consists of four differ ent objectives, one in which UV light may pass and three objectives of 10X, 50X and 100X magnification to vi sible light, Figure 6 is a picture of the system’s microscope a nd spectrograph. Along with an adjustable confocal hole (0-1000nm), we are able to contro l the size of the laser spot on the sample. The size of the confocal hole will influe nce spectral and spatial resolution. Fig. 6 Microscope Objectiv e and Spectrograph Gratings4 12

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Chapter 2. Raman Spectroscopy of Nano-Indented Quantum Dots 2.1 Introduction Quantum Dots (QDs) are semiconductor nanos tructures of particular interest due to their confinement in three dimensions and sharp density of states. Possible applications of QDs include qubits for quantum computing, optical modulators5, photonic crystals to control and manipul ate the flow of light, and lase rs with ultra-low threshold current6. The pathway towards QD devices depends on the ability to control their size and morphology. Strain is the driving fo rce behind the formation of QDs7. In Molecular Beam Epitaxy (MBE) QDs are grown by depositing a semiconductor with a lattice constant larger than that of the substrate, for instan ce InAs on GaAs or CdSe on ZnSe. There is a critical amount of the larger la ttice constant material necessary to create QD structures, as shown for InAs in Figure 7. Below this criti cal amount only a thin, st rained layer of InAs is formed. The formation of QDs by MBE ha s become a standard procedure however the structures that form are disorganized and vary in size and position on the surface. If the size and position of QDs can be more contro lled, applications of QDs which demand more organized structures, such as qubits and photonic crystals, can be achieved with higher accuracy and confidence. 13

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Fig. 7 InAs coverage versus 3D Island Density8 Patterning techniques are an excellent development towards preferential QD growth9. Nano-indentation is a tech nique used to create a pred efined indentation pattern on which QDs are formed preferentially. Nanoindentation can lead to a decrease in the amount of InAs necessary to create QDs, resulti ng in the organized formation of QDs. In this thesis we have used Raman Spectroscopy to study the strain in the nano-indented QD structures. 2.2 Sample Description Samples were provided by Dr. Curtis Taylor, Euclydes Marega and Ajay Malshe of the Mechanical Engineering Department at Virginia Commonwealth University, the Physics Department of the University of Arkansas and the Mechanical Engineering Depart ment of the University of Arkansas, respectively. Samples were created by first removing th e oxide layer of the GaAs substrate at 580C under an As flux inside of a MBE chamber. A GaAs buffer layer of approximately 0.1m is then deposited. 14

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The sample is removed from the chamber and a diamond tip is pressed in to the buffer layer to create the nano-indented patt ern. Later, the sample is reintroduced into the chamber and once again the oxide layer is removed from the sample and a GaAs buffer layer of 28nm is created. Finally, two monolayers of InAs are deposited on the sample. Fig.8 Schematic of InAs dot on GaAs buffer layer8. 2.2 AFM Results It was observed by Atomic Force Microscopy (AFM) 8 that the QDs have a tendency to form in the indented areas. Figure 9 shows AFM images of InAs QDs of various pattern sizes and tip loads. Fig. 9 AFM image of InAs QDs grown by nano-indentation (increasing tip load) 8. 15

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2.4 Raman Results Raman measurements were taken with a HR 800 micro-Raman system using a spot size of 2x2 m2 and a 632.8 nm HeNe laser at room temperature. The details of this system were provided in Secti on 1.4. Data was taken on di fferent indentat ion patterns which varied in load and indent spacing. Th e RS measurements were done on two sets of patterned samples, one with fixed indentation spacing (500nm) and the other with fixed load (200N). The details of these sample se ts, which correspond with the Raman data in Figures 10 and 11, are presente d in Tables 2 and 3. Load Separation 200N 250nm 200N 500nm 200N 1m Table 2: Samples with fixed load. Load Separation 400N 500nm 200N 500nm 100N 500nm Table 3: Samples with fixe d indentation separation. For comparison purposes, measurements were also taken on the non-patterned portion of the sample. The non-patterned area includes QD struct ures that are not organized and are not influenced by the nano-indents. AFM images of Figure 9 show examples of the unorganized QD structures formed on the outer perimeter of the nanoindent regions. Raman peak s were observed from 200-320 cm-1. Figure 10 shows Raman data of the 200N load with variable pattern spac ing and Figure 11 shows data of 500nm spacing with variable tip load. 16

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200220240260280300320 1 2 3 262 cm-1GaAs LO IF QD~7 cm-1 295.4 cm-1269 cm-1221.2 cm-1off pattern 1m 500nm 250nm Wavenumber (cm-1)Intensity (x103counts/s)200N Fig. 10 Scattering Intensity versus wave number of 200N nano-indented samples. 200220240260280300320 1 2 3 262 cm-1~7 cm-1 295.4 cm-1GaAs LO 269 cm-1QD221.2 cm-1IF 200N 400N 100N off pattern500nm SpacingIntensity (x103counts/s)Wavenumber (cm-1) Fig. 11 Scattering Intensity versus wave number of 500nm nano-indented samples. 17

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2.5 Analysis: Identification of Raman Modes From the Raman data in Fig. 10 and 11 one observes three distinct peaks. The most easily identifiable peak is the bulk GaAs LO mode at 295 cm-1. In order to determine the origin of the other peaks we performed calculations to determine the presence interface modes and compare our data to reported values for capped InAs/GaAs QDs. Interface modes were determined by following the method of Knipp and Reinecke10 in which a dielectric continuum appro ach is used. We begin by defining an interface mode as the propagation of electro magnetic radiation at the interface between two dielectric media11. From Maxwell’s equation ()i DE (23) and with the assumption that there is no net charge within either material, we have in both the InAs and GaAs materials 0 ) ( E i. (24) In order to fulfill this condition we have two possibilities 20 or 0 E (25) and 0 ) ( i. (26) The first possibility corresponds to the case of transverse (TO) modes and the second to longitudinal (LO) modes. 18

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In the simplest case of two slab layers the condition to have an IF mode of frequency ( IF ) is11 12()()IFIF (27) The general expression of the dielectric func tion of a media in terms of non-damped LO and TO modes is given by3 2 1 ) (2 2 2 2 , ii TO i LO i i (28) where the subscript i is 1 for the dot and 2 for the media in which the dot is embedded. By substituting (28) into (27) and solving for the frequencies of the interface modes are determined. To determine the IF modes we will derive the electric field from a scalar potential ( ) satisfying Laplace’s equation (20 ). The scalar potential is determined by imposing traditional electrostatic boundary cond itions at the interface of the two media. The tangential components of the electric field E and the normal components of the displacement field D must be continuous. There are few shapes for which the IF modes can be calculated analytically. These shapes are slab layers, the sphere, the spheroid and the ellipsoid. We will illustrate the general procedure for the case of a sphere of radius R, we have that the electrostatic potential in spheri cal coordinates is R r r R R r R r P e rl l m l im, / / ) (cos ) , (1 (29) where Pm l are Legendre polynomials of th e first kind and the integers l ( 0) and 19

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m ( m l ) are interface mode quantum numbers6. From the continuity condition of the electric displacement acr oss the boundary (Eqs. 26 and 27), it follows that the eigenfrequency condition llm lm1 12 1 (30) Substituting Eq. 28 into Eq. 30 will make it po ssible to find the IF mode frequencies allowed for spherical QDs. Equation 30 was solved with Mathematica for several values of l and m but these values did not closely corr espond to the observed IF modes. This result is reasonable since the QDs look more like ellipsoids than spheres as evidenced by the AFM images in Fig. 9. For our case we will assume the QDs have an ellipsoidal form. Following the same procedure described earlier for spheri cal QDs, the following eigenfrequencies can be determined for ellipsoidal dots6: 2 2/ 2 1' ln lnE P PR R R x x m l m l lm lmx P x x Q x (31) where Qm l are Legendre functions of the second kind, RP is the polar radius of the dot and RE is the equatorial radius of the dot. The interface modes are determined by substituting known values of the GaAs and the InAs LO mode, TO mode, and high frequency dielectric constant into Eq. 28, inserting this result in to Equation 31, and solving for Eq. 31 was solved with Math ematica for several values of l and m see Appendix A. This resulted in an IF mode value of 220.5 cm-1 ( l = 1, m = 0), corresponding to a mode found in experiment in both the indented and non-indented areas. 20

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A QD mode was identified at 262 cm-1 off of the pattern. This is in good agreement with the experimentally determ ined capped InAs/GaAs QD mode identified between 255-258 cm-1 in Ref. 7 and with the calculated mode of 258.9 cm-1 in Ref. 12. On the patterned regions our data reveals another mode at 269 cm-1. The intensity of this mode increased as the pattern spacing decrea sed (Fig. 10 and Table 2) and as the indent load increased (Fig. 11 and Table 3). As evident by the AFM images of Figure 9, the density of QDs increases as spacing decrea ses and as load increases. Therefore the increase in intensity of this mode with decreased spacing and increased indent load suggests that it is the QD mode in the nano-in dentations. Therefor e the QD mode for the patterned areas is shif ted approximately 7 cm-1 from the non-patterned QD mode. 2.6 Analysis: Strain Calculation At the time of this work there are no reports about the strain distribution in spherical or uncapped QDs, therefore we us ed the strain distri bution for InAs/GaAs pyramidal capped QDs, shown in Figure 1212. The hydrostatic and biaxial strain are defined as yy xx zz B zz yy xx H 2. (32) Where xx yy ,and zz ,are components of the strain tensor. 21

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-10-5051015 -15 -10 -5 0 5 10 15 QD Base Z (nm)% Strain Hydrostatic Biaxial Fig. 12 QD orientation diagram13 and hydrostatic and biaxial st rain distribution along the center of a pyramidal QD (Z (001) direction). Figure 12 shows the hydrostatic and biax ial strain distribut ion along the center axis of the pyramid. The strain distribution in Fig. 12 shows that the hydrostatic strain distribution for a capped QD does not cha nge considerably within the dot. In order to calculate the strain in the nano-indented QD structures, we begin with the Hamiltonian for diamond-type se miconductor materials with strain taken into consideration14, 0 00 00 0 0 0 00 0 00 00(2)/6() // ()/6(2) (2)/6() // ()/6(2) (2)/6() // ()/6(2)xxyyzz xyxz yyzzxz xxyyzz xy yz xxzzyy xxyyzz xzyz xxyyzzXYZ pq rr pq pq rr pq pq rr pq (33) where p, q and r are deformation potentials that de scribe changes in restoring force constants and 0 is the unperturbed bulk frequency. 22

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Equation 33 can be diagonalized for a uniaxial strain along the [001] or [111] directions. This Hamiltonian has three solu tions; a singlet freque ncy with eigenvectors parallel to the strain axis, denoted with an LO subscript, and a doublet frequency with eigenvectors in the plane perpendicular to the strain axis, denoted with a TO subscript. The expressions for the phonon frequency change s due to biaxial strain that follow from this Hamiltonian in zincblende materials are9 s H TO TO 3 2 (34) s H TO TO 3 1 (35) s H LO LO 3 1 (36) where H and S represent the frequency changes due to hydrostatic and shear strain, respectively. Equations 34, 35 and 36 are ex pressed in terms of the hydrostatic and biaxial strain components as H Hq p 06 2 (37) and B Sq p 04 (38) To incorporate the frequency shift observed in our experimental results we rewrite the equations (34)-(36) for the TO and LO frequencies as, 23

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2 0 2 0 2 01 6 1 12 1 3TO H B TO TO H B TO LO H B LOpq p q pq (39) where 2 0(2) 6 p q (40) is known as the Gruneisen parameter and LO ( TO) is the unperturbed LO (TO) frequency. The coefficients of the hydros tatic and biaxial stra in components in the expressions for the change in frequency are known values for different materials. For InAs9, 220.85,0.8191.012LOTOpqpq (41) Substituting the values given in Eq. 41 into Eq. 39 we have 0.850.1691 0.850.084 0.850.068TO H B TO TO H B TO LO H B LO (42) Using Eq. 42 and the strain distribution in Fig. 12, we are able to determine the TO and LO modes along the QD, as shown in Figure 13. The dotted line in Fig. 13 represents the mode found in Ref. 12 in capped QDs and corresponds well with the QD LO mode calculated with this method. 24

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The LO mode in our experiment, with uncapped QDs was slightly higher (262 cm-1). -10-5051015210 220 230 240 250 260 270 255-258 cm-1 Exp. Wavenumber (cm-1)Z (nm) LO TO1 T2 Fig. 13 Wavenumber vs. Capped QD height We use the experimentally observed shift in the LO phonon frequency (7 cm-1) of our uncapped QDs the hydrostatic strain valu e for capped QDs and Eq. 41 to obtain an expression for the biaxial strain within a patterned area, ) (30 4 57 0capped B base QD B (43) This value for the biaxial strain within the un capped QDs is four times higher than that of capped QDs. This larger strain explains the preferential growth of QDs within the indented areas. 25

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Chapter 3. Conclusions Nano-indentation is a promising technique for forming predefined patterns of QDs. We have determined the strain that is the origin of the preferential formation of QDs. We found that this stra in is approximately four times larger than in non-indented regions and therefore results in the prefer ential formation on indented regions. The additional strain promises the creation of QDs with smaller amounts of InAs than the critical quantity of 1.6 MLs. By depositing less than 1.6 MLs of InAs on nano-indented samples, the QD formation will occur only in the indented areas thus allowing for more organized QD structure which may lead to ap plications such as photonic crystals and quantum qubits. 26

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References 1. S. Perkowitz. Optical Characterization of Semiconductors: Infrared, Raman, and Photoluminescence Spectroscopy Academic Press(1993). 2. Hayes and Loudon. Scattering of Light by Crystals Dover Publications. (1978). 3. P. Yu and M. Cardona. Fundamentals of Semiconductors: Physics and Materials Properties. Springer. (2005). 4. LABRAM HR Pre-Installation Manual. 5. T. H. Wood, C. A. Burrus, D.A.B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, “Optic al modulation in semiconductor multiple quantum wells”, IEEE J. Quantum Electron 21 117, (1985). 6. D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures John Wiley, England 1999. 7. Yu. A. Pusep, G. Zanelatto, et, al. Phys. Rev B 58. R1770 (1998). 8. C. R. Taylor, et al. “Directed Se lf-Assembly of Quantum Structures by Nanomechanical Stamping Using Probe Tips,” submitted for publication July 2007. 9. H.T. Johnson, R. Bose. Journal of the Mechanics and Physics of Solids.51 2085 (2003). 10. P.A. Knipp and T.L. Reinecke. Phys. Rev B 46. 10 310 (1992). 11. Pochi Yeh. Optical Waves in Layered Media. John Wiley & Sons. (1988). 12. M. Grundmann, O. Steir, and D. Bimberg. Phys. Rev B 52. 11 969 (1995). 13. M. A. Cusack, P.R. Briddon, and M. Jaros. Phys. Rev B 54. No.4. (1996). 14. Fred H. Pollak. Strained-Layer Superlat tices: Physics. Vol. 1. Academic Press. (1990). 27

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Appendices 28

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Appendix A: Mathematica Code for th e Calculation of Interface Modes Spherical Approximation SphericalApproximation InAsQuantumdotsonGaAsClear,a,b,c,l; Print;ParametersforInAsquantumDotLO1243;TO1218;112.25; Print"Quantumdotparameters:TO",TO1,"cm1LO", LO1,"cm1",1 ParametersforsurroundingGaAsmaterialLO2295;TO2270;210.89; Print"Barrierparameters:TO",TO2,"cm1LO",LO2, "cm1",2l_: 2 111 l; al_:l1; bl_:LO12TO22lTO12LO22; cl_:lTO12LO22LO12TO22; Forl1,l4,l, PrintSqrtblSqrtblbl4alcl2al;; Print; Forl1,l4,l, PrintSqrtblSqrtblbl4alcl2al;; 29

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Appendix A: (Continued) Ellipsoidal/Spheroidal Approximation InAsQuantumdotsonGaAsCleara,b,c,d,r,,l,m; Print;ParametersforInAsquantumDotLO1243;TO1218;112.25; Print"Quantumdotparameters:TO",TO1,"cm1LO", LO1,"cm1",1 Parametersforsurr oundingGaAsmaterialLO2295;TO2270;210.89; Print"Barrierparameters:TO",TO2,"cm1LO", LO2,"cm1",2 HeightRadiusR718; Print"HeightRadiusratio:",Rrl_,m_,x_:EvaluateDLogLegendreQl,m,3,x,xDLogLegendrePl,m,3,x,x;l_,m_:rl,m,1Sqrt1R21rl,m,1Sqrt1R21t1Tablem,1,m,m,0,1N; t2Tablem,2,m,m,0,2N; t3Tablem,3,m,m,0,3N; t4Tablem,4,m,m,0,4N; t5Tablem,5,m,m,0,4N; 30

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Appendix A: (Continued) PrintStyleForm"\nPositiveRoot", FontWeight"Bold"; l1; Form10,m11,m1, s21rl,m1,1Sqrt1R2N; a1s; bsLO22TO12LO12TO22; cLO12TO22sTO12LO22; Print"l1,m",m1,"Energycm1:", SqrtbSqrtb24ac2a l2;Print; Form10,m1l,m1, s21rl,m1,1Sqrt1R2N; a1s; bsLO22TO12LO12TO22; cLO12TO22sTO12LO22; Print"l2,m",m1,"Energycm1:", SqrtbSqrtb24ac2a l3;Print; Form10,m1l,m1, s21rl,m1,1Sqrt1R2N; a1s; bsLO22TO12LO12TO22; cLO12TO22sTO12LO22; Print"l3,m",m1,"Energycm1:", SqrtbSqrtb24ac2a 31

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Appendix A: (Continued) PrintStyleForm"\nNegativeRo ot",FontWeight"Bold"; l1; Form10,m1l,m1, s21rl,m1,1Sqrt1R2N; a1s; bsLO22TO12LO12TO22; cLO12TO22sTO12LO22; Print"l1,m",m1,"Energycm1:", SqrtbSqrtb24ac2a l2;Print; Form10,m1l,m1, s21rl,m1,1Sqrt1R2N; a1s; bsLO22TO12LO12TO22; cLO12TO22sTO12LO22; Print"l2,m",m1,"Energycm1:", SqrtbSqrtb24ac2a l3;Print; Form10,m1l,m1, s21rl,m1,1Sqrt1R2N; a1s; bsLO22TO12LO12TO22; cLO12TO22sTO12LO22; Print"l3,m",m1,"Energycm1:", SqrtbSqrtb24ac2a 32


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ABSTRACT: Patterns of InAs/GaAs quantum dots (QDs) grown by the combination of nanoindentation technique and molecular beam epitaxy were studied. The resulting QDs tend to preferentially nucleate on indented areas rather than other regions. We studied the strain on the indentations, regions surrounding the indents, and non-indented areas. The QD LO mode for the patterned areas shifts by 7 cm-1 when compared to the non-patterned area. The biaxial strain in the indented areas producing this shift is four times larger than that in non-indented areas, explaining the QD preferential formation within these areas. This larger strain suggests that QDs on the indentations can be formed by depositing a smaller InAs amount than that required to form QDs on non-indented areas, thus obtaining QDs only on the pattern.
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