Universidade Estadual Paulista “Julio de Mesquita Filho” 

Programa de Pós-graduação em Ciência e Tecnologia de Materiais 

 

 

 

 

Evandro Martin Lanzoni 

 

 

 

Electric Force Microscopy techniques on GaAs mesoscopic 

structures 

 

 

 

 

 

 

Sorocaba 

2018 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  



 

 

 

 

 

 

 

 

 

   

 

 
 

 

 

Lanzoni, Evandro Martin. 

   Electric force microscopy techniques on GaAs 

mesoscopic structures / Evandro Martin Lanzoni, 2018 

70 f. 

   Orientador: Elidiane Cipriano Rangel 

   Dissertação (Mestrado)–Universidade Estadual 

Paulista. Instituto de ciência e Tecnologia, Sorocaba, 

2018. 

1. Semicondutor. 2. KPFM. 3. AFM. I. Universidade 

Estadual Paulista. Instituto de ciência e Tecnologia. 

II. Título.    

 

  

 

 

 

 

  



 

  



 

Evandro Martin Lanzoni 

 

 

 

 

 

Electric Force Microscopy techniques on GaAs mesoscopic 

structures 

 

 

 

Dissertação apresentada como requisito à obtenção 

do título de Mestre à Universidade Estadual Paulista 

“Júlio de Mesquita Filho” - Programa de Pós-

Graduação em Ciência e Tecnologia de Materiais, 

área de concentração Ciência e Engenharia de 

Interfaces, sob a orientação da Profa. Dra. Elidiane 

Cipriano Rangel e co-orientação do Prof. Dr. 

Christoph Friedrich Deneke 

 

 

 

Sorocaba 

2018 



 

i 

Agradecimentos: 

Primeiramente agradeço a minha orientadora Profa. Dra. Elidiane Rangel a 

qual tornou possível este desafio de realizar o mestrado no programa de pós-

graduação em ciência e tecnologia de materiais.   

Agradeço meu ao co-orientador e líder Prof. Dr. Christoph Deneke que foi 

um grande guia nesta jornada, auxiliando em todo o desenvolvimento do projeto, e 

também em meu desenvolvimento pessoal crítico, que é fundamental para a formação 

de um pesquisador. 

Agradeço ao Dr. Carlos Costa que foi um grande mestre e colega na 

microscopia de varredura por sonda. A constante troca de conhecimento que tivemos 

no desenrolar do projeto permitiu que os resultados fossem obtidos com total 

confiabilidade.  

Agradeço ao Dr. Saimon Covre pelas primeiras amostras preparadas e pelas 

muitas discussões ao longo do projeto. 

Agradeço a Ailton Garcia e Matthjin Floris pelas várias amostras cedidas. 

Agradeço ao Dr. Carlos Ospina pelas imagens de HR-TEM. 

Agradeço ao Dr. Suwit Kiravittaya pelo código para o cálculo numérico de 

uma partícula em um poço de potencial. 

E por fim agradeço imensamente a toda minha família por sempre terem 

apoiado em minhas decisões, em especial à Mónica Arenas por ter dado apoio 

incondicional durante todo o decorrer do mestrado.  

  



 

ii 

Dedicatória: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dedico este trabalho ao esforço de toda minha família por acreditarem que a 

educação pode transformar vidas. 

Para meu pai Edilson Fernando Lanzoni (In Memoriam) 

  



 

iii 

Sumário 

Resumo ...................................................................................................................... v 

Abstract ................................................................................................................... vi 

1. Introduction ................................................................................................... 1 

2. Materials and methods .................................................................................. 3 

2.1. Molecular Beam Epitaxy ........................................................................... 5 

2.2. Mesoscopic GaAs Structures .................................................................... 8 

2.3. Samples Investigated ............................................................................... 11 

2.4. Electron Microscopy Based techniques for pre-

characterization ................................................................................................................. 13 

2.5. Scanning probe microscopy (SPM) ........................................................ 13 

2.6. Kelvin probe force microscopy (KPFM)................................................ 17 

2.7. Calculation and simulation ..................................................................... 22 

3. Development and results discussion ........................................................... 28 

3.1. Sample processing .................................................................................... 28 

3.2. Kelvin probe force microscopy measurements ..................................... 31 

3.3. Changing GaAs thickness ....................................................................... 35 

3.4. Changing aluminum concentration into AlGaAs barrier .................... 36 

3.5. Modelling half-finite quantum well ........................................................ 37 

3.5.1. Numerical solution ............................................................................... 37 

3.5.2. KPFM of the proposed sample ............................................................ 38 

3.6. Simulations of band diagram and electron wave function ................... 40 

3.7. Electrostatic force microscopy (EFM) without AC voltage ................. 41 

4. Conclusions .................................................................................................. 44 

5. Bibliographical References ......................................................................... 46 



 

iv 

APPENDIX – I – Used code for particle in the box ........................................ 49 

APPENDIX – II – Used code for band gap and electron wavefunction ....... 57 

 

  



 

v 

Resumo 

As técnicas de microscopia de sonda Kelvin (KPFM) e de microscopia de 

força eletrostática (EFM) são amplamente utilizadas para analisar a distribuição do 

potencial de superfície, porém com pouca aplicação em nanoestruturas 

semicondutoras auto-organizadas embutidas em um substrato. Neste trabalho, 

investigamos diretamente o acúmulo de carga dentro de estruturas mesoscópicas de 

GaAs (MGS) [1]. As estruturas são fabricadas através do crescimento sobreposto de 

um modelo de nano orifícios usando epitaxia de feixe molecular. Para tal, uma 

combinação de desoxidação assistida por Ga e ataque químico por gotículas 

localizadas foram utilizadas para criar orifícios iniciais com uma profundidade de ca. 

10 a 15nm, que são posteriormente cobertos com 15nm de barreira AlxGax-1As e 

GaAs com 1nm, 2nm, 5nm, 10nm de espessura. Microscopia de força atômica e 

microscopia eletrônica de transmissão mostraram que a forma do orifício é 

preservada durante o crescimento de AlGaAs. Em seguida, esses orifícios são 

preenchidos com GaAs formando uma estrutura alongada sobre o buraco [1]. 

Investigamos o potencial de superfície local e a distribuição das cargas nestas 

estruturas com a técnica KPFM de passagem única. Portanto, uma voltagem AC de 

5 V é aplicada a uma ponta metalizada e varremos a amostra no modo de contato 

intermitente. Observamos uma clara diferença de potencial na região central da 

estrutura, onde esperamos o furo preenchido. Então, um estudo sistemático com a 

técnica de KPFM mostrou a influência no acúmulo de carga quando a espessura de 

GaAs é alterada, bem como, quando modificamos a concentração de Al na barreira 

de AlGaAs. O cálculo simulando um poço de potencial com barreiras semi-finitas e 

finitas mostrou que não ocorre acúmulo de carga quando a espessura do GaAs é 

menor que 1,5 nm, corroborando com nossos resultados. Simulações do diagrama de 

banda e da densidade de elétron da estrutura permitem atribuir o acumulo de carga 

observado, aos diferentes níveis de energia da estrutura mesoscópica de GaAs em 

comparação com as camadas de GaAs circundantes.  

  



 

vi 

Abstract  

Kelvin probe force microscopy and electric force microscopy techniques are widely 

used to analyze the distribution of the surface potential with little application to self-

assembled semiconductor nanostructures embedded into a substrate. In this work, we 

directly investigate the charge accumulation inside mesoscopic GaAs structures  [1]. The 

structures are fabricated by overgrowth of a nanohole template using molecular beam 

epitaxy. Therefore, a combination of Ga assisted deoxidation and local droplet etching is 

used to create initial holes with a depth of ca. 10 to 15nm, which are covered subsequently 

with 15nm of AlxGax-1As barrier and GaAs caps with 1nm, 2nm, 5nm, 10nm thicknesses. 

Atomic force microscopy and transmission electron microscopy results showed that the hole 

shape is preserved during the AlGaAs overgrowth. Then filled with GaAs forming an 

elongated  mount over the hole [1]. We investigate the local potential and the charge 

distribution in these structures with a single pass Kelvin probe force microscopy technique. 

Therefore, an AC voltage of 5 V is applied to a metalized tip and scanned in tapping mode 

over the sample. We observed a clear potential difference in Kelvin probe force microscopy 

measurements in the middle of the structure, where we expect a filled hole. We 

systematically study by Kelvin probe force microscopy the influence on the charge 

accumulation when the GaAs thickness is changed, as well as the Al concentration in the 

AlGaAs barrier. Calculation of the particle in the box for semi-finite and finite barriers were 

done and show that no charge accumulation is observed for GaAs thickness lower than 

1.5nm in the semi-finite barrier, corroborating with our results. Simulations of band gap and 

electron wavefunction of the structure allow us to ascribe the charge accumulation observed, 

to the different confinement of carriers inside of the unstrained mesoscopic GaAs structure 

compared to the surrounding GaAs layers.  

  



1 

 

1. Introduction 

 

During the last decades, development in semiconductor materials has shown 

an important role in our daily lives, as well as pushing many frontiers of science such 

as in communication, transport, power generation, medicine. The silicon based 

semiconductor technology has a fundamental importance in this process. Advances 

in smaller and more efficient transistors, diodes and all kind of semiconductor 

heterostructures enable to store and process data efficiently and faster in a modern 

cellphone than a 10 years old computer. Without the silicon based semiconductors, 

vacuum tubes  [2] would still be the most advanced available technology and, 

probably, anything approaching the simplest computers available today would need 

an entire room. 

In the same way, III – V semiconductor have gain a lot of attention due to 

their electronic and optical properties [1,3–5]. A particular nanostructures class of III 

– V semiconductor self-assemble quantum dots (QD), has been widely studied 

because of the potential application in lasers device and quantum computing [6,7]. 

In this case, the quantum dot structure combines the advantages of bulk 

semiconductors with those in single atom. An example is the three-dimension 

confinement of the structures turning the density of states in a delta-function with 

discrete energy emission [7,8]. 

In the last two decades, quantum dots fabricated using the Stranski–Krastanov 

growth mode gained great attention due to their potential application in opto-

electronic devices [9,10]. In this process, the materials with large lattice mismatch 

start to grow layer by layer, but often in a critical thickness, the formation of the 3D 

structure is favored, producing some of the strain in the structure. Despite the 

advantages of the self-assemble QD, an intrinsic strain and an intermixing material 

during island formation is present [11,12].  

To overcome the inherent strain, a new class of QDs has been investigated as 

an alternative to the strained self-assemble QDs. Here, materials with almost perfect 

match of lattice parameter, like AlGaAs and GaAs, are used to produce the strain-



2 

 

free QDs. The AlGaAs/GaAs heterojunction are interesting due the well-studied optical 

properties when doing planar heterostructures (Quantum well and superlattice) as well as the 

limited intermixing at the interface make them potentially easier to model [13]. 

AlGaAs/GaAs QDs, cannot be obtained by Stranski-Krastanov growth mode and a different 

technique has to be used. One commonly used way is produce a nanohole pattern for 

subsequent deposition of the target material. Several approaches to prepare the hole 

templates, such as electron beam lithography, multistep hierarchical self-assembly, and local 

droplet etching, have been applied during the last years [1,4,13–15]. Da Silva et al. [1] 

combines the process of Ga-assisted deoxidation and local droplet etching to fabricate 

unstrained mesoscopic GaAs/AlGaAs structures (MGS). Optical characterization of this 

kind of structures has shown interesting properties with sharp line emission measured by 

micro-photoluminesce spectroscopy{Citation}. 

In the recent literature, optical characterization of strain-free QDs have been well 

explored, works in single photon sources [16,17], tuning optical properties by external 

strain  [18–20] and optical control of the nuclear spin for potential application in quantum 

computing  [3,21,22] are examples found in the literature. Until now, there is no electrical 

characterization of strain-free QDs at the nanoscale.  

In this work, we used Kelvin probe force microscopy (KPFM) to directly probe the 

electrostatic properties of the mesoscopic GaAs Structure (MGS) proposed by da Silva et al. 

KPFM technique provides quantitative information of the sample surface potential with 

lateral resolution around 25nm. The main advantage of this technique is measuring 

quantitatively the difference surface potential with 15 mV resolution without current 

between tip and sample. 

Finally, we carried out the KPFM measurement, which shows a difference surface 

potential in the middle of the MGS samples, where we would expect the initial hole. 

Increasing the thickness of the GaAs cap and changing the AlGaAs content, we observed 

changes in the difference surface potential. A numeral simulation of the half-infinite 

quantum well -band diagram and the electron wave function, assist us a better understanding 

of that, for GaAs thickness lower than 1.2nm there is no confinement, in accordance to our 

experimental results.  



3 

 

2. Materials and methods 

 

Figure 2.1 shows a schematic example of size confinement and its 

consequence for the energy level. In a bulk structure, the states of energy are closely 

spaced that the energy appear as a continuously increasing function. Reducing the 

material in one axis (2D – Quantum well, 1D – Quantum wire) the energy is 

quantized. When the material is reduced in the three dimensions (0D – Quantum dot), 

the carriers or electrons inside the material start to confine, and the structure assumes 

properties of a single atom, like the discrete quantized energies. 

 

 
Figure 2.1: Schematic of size reduction for a quantum confinement. The 3D material (bulk) 

have continuous density of states in function of the energy. When reducing the size in the x 

(quantum well), y (quantum wire) and z (quantum dot) directions the energy starts to be 

quantized.  

 

One way for size confinement is preparing two different semiconductors in 

contact to each other, doing a heterojunction. The combination of multiple 

heterojunction give rise to the heterostructures, which in the year 2000, Herbert 



4 

 

Kroemer and Zhores I. Alferov were awarded with the Nobel Prize in Physics for 

“developing semiconductor heterostructures in high-speed and opto-electronic devices”.  

devices”.  

In order to obtain good optical and electrical properties, it is essential that interface 

between the semiconductors is free of dislocations or defects. With the advances of 

molecular beam epitaxy (MBE) and the vacuum chemical deposition (CVD) high quality 

heterojunction or heterostructures start to improve the performance of semiconductor 

devices [23–26].  

The Figure 2.2 (a) illustrates a typical example of a heterostructure combining two 

different semiconductors that allow continuous transition of carriers through the interfaces 

of the two materials. The band gap difference between the semiconductors produce the 

carrier confinement effect. The Figure 2.2 (b) shows the simulation by NextNano of a 

quantum well using GaAs sandwiched between two Al0,3Ga0,7As barriers. The following 

figure show the conduction band edges and the lowest confined eigenstates for 6nm of 

AlGaAs/GaAs quantum well. 

 

Figure 2.2: (a) General semiconductor interface for carrier confinement. The different 

bandgap between the semiconductor A and B create a potential well. (b) NextNano simulation 

for a quantum well using 6nm of GaAs sandwiched with AlGaAs. 

 

 

 

 



5 

 

2.1. Molecular Beam Epitaxy 

 

Molecular Beam Epitaxy (MBE) is widely used in the manufacturing of 

semiconductor devices, and is considered one of the fundamental tools for the 

development of the nanotechnology. In the last decades, MBE has made some 

impressive contributions to fundamental studies of the growth of low dimensional 

structures and to basic concepts of thin film growth, as well as to surface studies in 

general [6,27]. 

MBE is a versatile ultrahigh vacuum technique for growing multiple epitaxial 

layers of semiconductor crystals with high precision. The extreme control of the 

MBE technique over composition variation, interface sharpness, impurity doping 

profiles, and epitaxial layer thickness to the atomic level makes it possible to 

demonstrate a wide variety of novel semiconductor structures [14,28]. Since its 

invention, nearly 40 years ago, the MBE technique has evolved from a laboratory 

apparatus for exploring new materials and novel devices to a favored tool for the 

mass production of III–V high-speed devices.  

In the MBE process, beams of atoms are incident upon a heated crystal 

atomically clean in an ultra-high vacuum environment. The arriving atoms form a 

crystalline layer following the lattice parameter of the substrate (Figure 2.3). Thus, it 

is possible to produce unique structures precisely controlled in composition. 

Examples of structures grown by MBE, include quantum well devices, superlattices 

and lasers.  



6 

 

 

Figure 2.3: Schematic of molecular beam epitaxy deposition. Inside the UVH chamber the 

interest material is evaporate in a controlled tax, and following the lattice parameter of the 

substrate. Is possible change the material during the growth and obtain a heterostructure like 

showed in TEM image (right side).  

 

In Figure 2.3 a schematic model for molecular beam epitaxy (MBE) is shown. The 

substrate is mounted on a heater plate inside the main UHV chamber. Different materials 

can be evaporated together or individually while a reflection high energy electron diffraction 

(RHEED) system provides a diffraction pattern on a phosphor-coated window allowing in-

situ monitor the ordering of the substrate surface. Therefore, layer by layer of the interest 

material grow following the lattice parameter of the substrate, as shown in Figure 2.3 (I, II 

and III). In this schematic model four effusion cells are placed at the main UHV chamber. 

In the currently MBE system that we used to prepare the sample the four effusion cells were 

Arsenic, Aluminum, Gallium, Indium (Figure 2.4)  



7 

 

 

Figure 2.4: Molecular Beam epitaxy system used in this project. Introduction, preparation and 

main chambers compose the vacuum system. The sample is grown in the main chamber, where 

four effusion cells (As, Ga, Al, In) and two doping cells (Si, Be) is used for material 

deposition, and monitored by reflection high energy electron diffraction (RHEED). 

 

Figure 2.4 shows the molecular beam epitaxy system used in this project. The 

machine is localized at Brazilian Nanotechnology National Laboratory – LNNano 

and is able to growth epitaxial thin film with the III – V elements of the periodic 

table. The current setup has four effusion cells (Ga, As, In, Al) and two doping cells 

(Si and Be). In Figure 2.4 is possible to identify some of the main components of the 

MBE among which, the main chamber with effusion cells, preparation chamber, 

introduction chamber, mechanical and ionic pump, RHEED screen monitor, cooling 

system and mass spectrometer. 

 

 

 

 



8 

 

2.2.  Mesoscopic GaAs Structures  

 

Recently, there has been a development of unstrained quantum dots [29]. In this 

process, materials with similar lattice parameter but different band gap are used to produce 

the quantum dots.  

To prepare the mesoscopic structures, normally first a hole template is created and 

then is deposited the material of interest. There are two different ways to prepare the pattern 

of holes, the ex-situ and the in-situ procedures (Figure 2.5). In the ex-situ process, an external 

technique, such as electron beam lithography, is used to locally etch the sample producing 

the holes. Precise and uniform holes are prepared by using this process, but the disadvantage 

of the ex-situ procedure, is that the sample preparation is not completely carry out inside the 

UHV chamber of the MBE. On the other hand, in the in-situ procedure, the entire process is 

carry out inside the UHV chamber of the MBE avoiding the disadvantage of the ex-situ 

patterning.  

 

Figure 2.5: Examples from the literature of ex-situ and in-situ etching process for the holes 

template. In the left side, the ex-situ process used by Kiravittaya et.al. which used the electron 

beam lithography to produce the patterns. In the right side, local droplet etching was employed 



9 

 

by Wang et.al. and Heyn et.al. in an in-situ procedure. In both procedures were observed 

photoluminescence intensity due to the unstrained QD.  

 

In this project, we used the in-situ procedure combining two methods for the 

etching process: the gallium assisted deoxidation and a nanodrill technique called 

local droplet etching (LDE). During the gallium assisted deoxidation, the native 

oxide is removed using a higher pressure of gallium to reduce the stable gallium 

oxide, which occurs by the reaction Ga2O3+4Ga→3Ga2O  [30–33]. This leads to a 

flat, and oxide-free surface that has been successfully used to patterned substrates 

with features several microns wide [15,30].  After removing the native oxide from 

the sample surface the local droplet etching is used to prepare the hole template. The 

fabrication of such nanoholes was first demonstrated by Wang et. al. [15] on GaAs 

surfaces using gallium as etchant. The LDE technique is related to the droplet epitaxy 

where Ga droplets are formed on crystalline surfaces in a Volmer–Weber–type 

growth mode, which in a subsequent step were crystallized under As 

pressure  [1,4,14,15,29]. During this process, the higher concentration of Gallium 

starts to react with GaAs substrate generating holes with controllable 3 dimension 

size. Creation of a variety of semiconductor nanostructures such as quantum dots, 

quantum rings and double rings has already been demonstrated [14,29,34,35]. The 

Figure 2.6 ((I) a, b and c) show a schematic procedure of Gallium assisted 

deoxidation and the Figure 2.6 ((I) d, e and f) show the LDE technique.  

Using the hole template, the material deposition starts for the unstrained 

quantum dots. At first step, 15nm of AlGaAs is grown over the GaAs substrate. Then 

different thickness of GaAs is used to filled the structure. AlGaAs and GaAs have 

similar lattice parameter (~ 5.653 Å) and grow epitaxially without strain. However, 

the main advantage of this system is because changing the Al concentration of the 

AlGaAs barrier, the band gap can be tuned from 1.42 eV (GaAs) to 2.16 eV 

(AlAs) [36]. Thus, samples with different Al concentration was produced. The MGS 

structures have one more barrier of AlGaAs and GaAs cap layer to protect the system. 

However, to access the internal region of the sample these two final steps were not 

carry out. 



10 

 

 
Figure 2.6: Gallium assisted deoxidation, Local droplet etching and material deposition 

procedures. High concentration of Ga removed the native oxide layer and then with As 

atmosphere starts to nanodrill the GaAs substrate producing a hole template. AlGaAs and 

GaAs is grown to produce the unstrained quantum dots. Adapted figure from Saimon C. da 

Silva thesis.   

 

Part of the structures analyzed in this work were previously studied by da Silva 

et al. [1] where is demonstrated a good controllability over size and morphology by changing 

growth parameters, as well as interesting optical properties. Figure 2.7a shows μ-

photoluminescence (μ-PL) spectra maps obtained at 10K. From this map is possible to 

clearly identify three different contributions. At 1.88eV a strong signal comes from the 

quantum well formed by the material surrounding the MGS structure. Structures with 

thickness smaller than 15nm is emitting in the energy range between 1.58 and 1.72eV, and 

the deep holes (15 to 19nm thickness) is emitting at 1.54eV. In the Figure 2.7b, μ-PL is 

carried out on a single structure. A sharp line emission is identified with linewidth of the 

peak ca. 370μeV. 



11 

 

 

Figure 2.7: (a) μ-Photoluminescence intensity map with a large spot size (5 μm). (b) 

μ-Photoluminescence spectrum of a single MGS showing a sharp line emission with high 

potentiality for optical applications. 

 

2.3. Samples Investigated 

 

Using the described methods, we prepare the set of sample shown in Table 

1. We measure the entire set of sample from this table. The sample P323 is only the 

substrate with the hole templates made by LDE. Sample P325 is the hole template 

covered is the AlGaAs barrier. Then, two sets of sample was prepared. In the first set 

(P144, P149, P152 and P155) aluminum barrier was grown with 33% Aluminum 

content and filled with different thickness of GaAs. In the second set of samples 

(P325, P326, P327, P328), we change de Aluminum content of the AlGaAs barrier. 

Finally, the process of cooling down the sample was changed in the way that the 

GaAs layer follow the shape of the hole template. With this procedure 1nm (P359) 

and 2nm (P369) thickness of GaAs was grown.  

 

 

 



12 

 

Table 1. Table with the measured samples names and showing the changed parameters for 

wach experiment. Five different set of sample were prepared: Only hole template, holes filled 

with AlGaAs barrier, holes with AlGaAs barrier and changing the thickness of the GaAs, 

changing the concentration of the AlGaAs barrier and AlGaAs/GaAs following the hole shape 

of the structures. 

 

 

 

 

 

 

 

 



13 

 

2.4. Electron Microscopy Based techniques for pre-characterization 

 

For pre-characterization of our samples, electron microscopy based technique 

was employed. The High-resolution transmission electron microscopy (HR-TEM) is 

a powerful technique that provides an image with the atomic structure of the sample. 

To check dislocations and defects, that could be prejudicial to the optical and 

electronic quality of the sample, we used the double correct transmission electron 

microscope TITAN Themis Cubed – FEI. The sample preparation for HR-TEM was 

done in the Helios NanoLab DualBeam – FEI. This equipment combines the 

scanning electron microscopy (SEM) with the focus ion beam (FIB) for 

nanomanipulation of the sample. With FIB, a cross-section cut was done right on the 

middle of the structure to observe the AlGaAs/GaAs heterojunction. Both equipment 

are located at Brazilian Nanotechnology National Laboratory – LNNano/CNPEM. 

 

2.5.  Scanning probe microscopy (SPM) 

 

Scanning probe microscopy (SPM) is a group of versatile techniques to 

imaging sample surface in three dimensions. The beginning of SPM comes from 

1981 when Binnig et.al. observed for the first time tunneling trough a controllable 

vacuum gap given rise to scanning tunneling microscopy (STM) [37]. The main idea 

of this experiment was use a tungsten (W) sharp tip connected to a piezoelectric 

material and measure the spectroscopy tunneling current when approaching to a Pt 

substrate. In 1982 the same authors combined the tunneling effect with the precise 

movement of piezoelectric material. STM with atomic spatial resolution was now 

tried out on surfaces of CaIrSn4 and Au samples in vacuum atmosphere [38]. This 

experiment awarded Binnig and Rohrer with the Nobel Prize in physics in 1986.  

The STM was the first powerful scanning probe microscopy technique 

invented, but this technique is limited to conductive material and need some special 

sample preparation for ultra-high vacuum. In 1985, using the principle of the STM 



14 

 

to control forces around 10−18𝑁 in air, Binnig and Quate propose the atomic force 

microscopy (AFM) to measure insulators with atomic resolution [39]. This experiment open 

experiment open a range of new sophisticated techniques based on scanning probe 

microscopy. Nowadays, SPM has more than 30 different techniques commercially available 

with especial emphasis to topographical modes (AFM, STM, phase contrast, etc.), electrical 

modes (EFM, KPFM, SCM, etc.) [40–43] and magnetic mode (MFM)  [44,45].  

The basic principle of the SPM is measure the different kind of interactions (e.g. Van 

der Waals force, Coulomb force, magnetic force, tunneling) between tip and sample (Figure 

2.8). A high precise system to analyze and process the interaction signal is used to build a 

3D image.  

 

Figure 2.8: Basic principle of scanning probe microscopy (SPM). A sensitive system is used 

to detect different kind of interaction between tip and sample. The result signal is interpreted 

to build a 3D map. 

 

The most known SPM technique is the atomic force microscopy (AFM). Here, a 

sharp tip is located in the edge of a silicon cantilever and placed close or sometimes in 

contact with the sample to scan the surface. The basic principle of the AFM is detect the 

Interaction 
between 

tip/sample

Signal 
detection

Electronics 
interpret the 

signal

Piezoelectric 
movement



15 

 

involved forces (Pauli interaction and van der Waals) in the Lennard-Jones potential 

between tip and sample (Figure 2.9). The AFM is based on a laser movement in a 

photodetector when different gradient forces act to the tip. Therefore, the laser is 

focused on the top side of the cantilever and the reflection light is monitored using a 

four-quadrant photodetector. There are three different modes to operate the AFM that 

are dependent on the distance d between tip and sample: contact (red square in Figure 

2.9), intermittent contact and non-contact mode (blue square in Figure 2.9). In the 

first one, the tip is located a few nanometer to the sample (in contact) where the Pauli 

interaction is the main responsible to exert forces between tip and sample. The 

generated force will deflect the tip changing the laser signal in the photodetector, and 

then the piezoelectric material correct the position of the tip to build a 3D 

topographical image. In intermittent contact mode, the tip oscillate at the resonance 

frequency of the cantilever. To build an image, the microscope try to keep a constant 

oscillation amplitude of the tip. When the tip pass over a sample step the amplitude 

change and immediately the electronic system of the microscope correct the height 

with the piezoelectric material. The non-contact mode works similar to tapping 

mode, the difference is the oscillation amplitude. As the range of van der Waals 

interaction is not higher than 10nm, in non-contact mode the oscillation amplitude 

cannot be higher than 10nm. In this way, non-contact mode use smaller values for 

oscillation amplitude compare to intermittent contact mode, and is recommended 

only for flat surface. Regardless the used mode, the AFM technique provide the 

topographical information with high resolution in 3D axis. 



16 

 

 

𝑉(𝑑) = 4𝜀 ((
1

𝑑12
) − (

1

𝑑6
)) 

Repulsive Attractive 

 

 

Figure 2.9: Lennard-Jones potential for the AFM techniques. Repulsive interaction or Pauli 

interaction, proportional to 1/d12, used in contact mode. Attractive interaction or van der Waals 

force, proportional to 1/d6, used in non-contact mode. 

 

Because of the powerful and high precise tool to acquire topographical information, 

the SPM is an excellent technique to probe long-range forces such as magnetic forces and 

electrostatic forces. As showed in Figure 2.9, van der Waals and Pauli interaction between 

tip and sample are used to acquire the topographical information. But, as the long range 

forces are much more stronger at long distances (longer than 10nm), the SPM can be applied 

to measure long-range forces keeping a constant distance to the sample surface. Therefore, 

the van der Waals interaction is insignificant compared to the long range force as 

demonstrated in Figure 2.10 by Saint Jean et al [46]. Assuming the tip geometry as a sphere-

cone model and sample surface as a plane, Saint Jean demonstrated that in long distance the 

predominant force through the tip is due the long-range forces. In this way, the SPM can be 

applied to simultaneously acquire topographic image when tip is close to the sample surface, 

and in a second pass, the electrostatic information keeping a constant distance between tip 

and sample. This technique is called electrostatic force microscopy (EFM).  



17 

 

 

Figure 2.10: Adapted figure from Saint Jean et al. [46] simulating a comparison between van 

der Waals interaction and long-range forces. Saint Jean et al. described the tip as a sphere-

cone model whereas the sample surface is supposed to be plane (left side). The long-range 

curve start to be stronger than van der Waals when distance between tip and sample is higher 

than ~8nm.  

 

Over the years, enhanced EFM techniques have been explored such as Kelvin 

probe force microscopy (KPFM). In this work, we are interested to quantitatively 

direct probe electrostatic forces over the sample surface of strain-free QDs. 

Therefore, KPFM will be employed. 

 

2.6.  Kelvin probe force microscopy (KPFM) 

 

After development of the atomic force microscopy (AFM) that essentially 

measure the involved forces in the Lennard-Jones potential, researchers start to 

explore the different forces acting between tip and sample. In 1988 Stern et.al.  [40] 

detected charges on polymethylmethacrylate and single crystal of sapphire. This was 

the first time that electrostatic charges were detected using SPM, later called 

electrostatic force microscopy (EFM). 



18 

 

EFM use a silicon tip covered with Pt/Ir to measure in double pass topography and 

electrostatic forces of the sample surface. In the first pass, the microscope monitor the 

amplitude vibration of the tip in intermittent contact mode to acquire the topography of the 

sample. In the second pass, the tip is lifted tens of nm and follow the topographical profile 

keeping constant height between tip and sample. At this distance, the long-range forces are 

predominant over the van der Waals interaction as showed in the Chapter 2.5. Therefore, 

each time the tip pass over electrostatic charge an electric field act between tip and sample 

generating an acting force. This force deflect the cantilever, changing amplitude and phase 

signals of the tip oscillation. Using a lock-in amplifier is possible to qualitatively acquire the 

electrostatic forces acting to the tip. Thereby, EFM is a powerful and well-established 

technique to qualitative detect electrostatic forces with high lateral resolution.  

In 1991 Nonnenmacher et.al. used the principle of the parallel plate capacitor 

proposed in the late 19th century by Lord Kelvin and combined it with SPM to quantitative 

measure the electrostatic force over the sample surface with nanometer lateral scale [42]. In 

the Lord Kelvin method, two conducting materials are arranged as a parallel plate capacitor 

with a small spacing in between. In a simple model, the contact potential between the two 

materials is 𝑉𝐶𝑃𝐷 =
1

𝑒
(𝜙1 − 𝜙2), where 𝜙1and 𝜙2 are the work function of the conductors 

including changes due to absorption layer on the surface. A periodic vibration of the distance 

between two plates at  frequency results in a current 𝑖(𝑡) given by 

𝑖(𝑡) =  𝑉𝐶𝑃𝐷𝜔𝐶 cos 𝜔𝑡,  (1) 

where 𝐶 is the change in capacitance. For the actual measure of 𝑉𝐶𝑃𝐷, an additional 

bucking voltage is applied between the two plates until the space in between is filed free and 

the current 𝑖(𝑡) goes to zero.  

In the KPFM technique, the acting force between tip and sample will be measured 

instead the current. Allied to this, an additional DC voltage is applied between tip and sample 

until the gradient force in between minimizes. Therefore, KPFM technique uses the fact of 

tip and sample build a parallel plate capacitor when close to each other, and the electrostatic 

force 𝐹 can be written as: 



19 

 

𝐹 =  −
1

2

𝑑𝐶

𝑑𝑧
𝑉2 

(2) 

where 𝐶 is the tip to sample capacitance. 

We assume that the voltage 𝑉 is composed of the contact potential 𝑉𝐶𝑃𝐷 plus 

applied DC and AC voltages (𝑉𝐷𝐶 and 𝑉𝐴𝐶 respectively). Then, the electric force can 

be written as the following equations: 

𝐹 =  −
1

2

𝑑𝐶

𝑑𝑧
[(𝑉𝐷𝐶 + 𝑉𝐶𝑃𝐷 + 𝑉𝐴𝐶)2] 

(3) 

The main advantage of KPFM comes from the fact that, topographical and 

quantitative electrostatic surface charge maps are obtained simultaneously in a single 

pass. The Figure 2.11 shows a schematic diagram of the KPFM method. A silicon tip 

covered with Pt/Ir is oscillated at its mechanical resonance frequency (𝜔𝑚). The lock-

in amplifier I will analyses the deflection signal of the tip at 𝜔𝑚 frequency in order 

to the Z Servo build the topography image. Simultaneously an AC voltage with 

frequency 𝜔𝑒 is applied between tip and sample. When the local electric field is 

changing, the amplitude and phase of the AC signal will change too. At this point, 

lock-in amplifier II set at 𝜔𝑒 frequency read phase and amplitude oscillation. 

Subsequently the KPFM Servo applied a DC voltage in order to minimize the 

mechanical oscillation at 𝜔𝑒 frequency. When lock-in amplifier II identifies no 

mechanical oscillation, the DC tip voltage will be the same as the unknown surface 

potential. Then the DC signal is recorded by the software, and used to build the 

electrostatic surface potential maps. 



20 

 

 

Figure 2.11: Schematic diagram of the KPFM method. In single pass measurements, 

Topography and surface potential is acquired simultaneously. The “lock-in amplifier I” 

analyses the cantilever deflection at resonance frequency of the tip and try to keep constant 

oscillation amplitude to acquire topography image. At the same time, the lock-in amplifier II 

analyses the AC frequency and use a servo to apply DC bias nullifying tip oscillation at AC 

frequency [47] (Adapted image). 

 

The KPFM is a well establish technique and has been used as an important tool at the 

semiconductor world [48]. In the literature is possible to find a great variety of works using 

the KPFM  [48–51], but this kind of characterization has not yet applied to directly probe 

the charge accumulation inside unstrained quantum dots.  

Kelvin probe force microscopy (KPFM) technique was used to measure the surface 

potential of the samples. The KPFM acquires in a single pass the topography and the surface 

potential simultaneously. In intermittent contact mode, the mechanical oscillation of the tip 

is tracked by the photodetector and analyzed by two feedback loops. The first loop is used 

in the conventional way to control the distance between tip and sample surface, while 



21 

 

scanning the sample at constant oscillation amplitude. The second loop is used to 

minimize the electric field between tip and sample by adjusting the DC tip bias 

voltage to minimize electrostatic forces between tip and sample. In this way, an AC 

signal is applied to the tip at 13 kHz or 17 kHz (depending the tip used). A second 

lock-in amplifier is monitoring the AC frequency, so that for each point, a DC bias 

is add to the tip to nullify any oscillation caused by electrostatic forces.  

In this project we used the following scanning probe microscopy: NX-10 – 

Park Systems, FlexAFM – Nanosurf and NanoIR2-s – Anasys Instruments operated 

in EFM-based techniques using PPP-EFM – Nanosensors tips. The entire images 

were treated using the Gwyddion software correlating topography and electrostatic 

potential data at the same position. Almost all the samples presented an average 

surface potential around 0.6 V that we normalized for all the measurement, in order 

to get a comparable scale between 0 and 100 mV. This procedure do not interfere to 

the project conclusion since we are not interest in the quantitative value for the 

surface potential. We are looking only for the difference potential between the 

structures and the surrounding surface. 

 

 

 

 

 

 

 

 

 



22 

 

2.7.  Calculation and simulation 

   

For the mathematical understanding of our sample, we use the fact of the different 

band gap in the GaAs/AlGaAs heterojunction generated a well-known problem of a particle 

in the box with half-infinite potential barrier Figure 2.12.  

 

Figure 2.12: Potential energy for a particle in a half-infinite box in one dimension.  

 

In this case, the potential energy is given by 

𝑉(𝑥) = {
∞, 𝑥 < 0
0, 0 ≤ 𝑥 ≤ 𝐿
𝑉0, 𝑥 > 𝐿

} (4) 

Solving the Schrodinger equation for the region I, II and III: 

Region III solution  

The Schrödinger equation in region III is 



23 

 

{−
ℏ2

2𝑚

𝑑2

𝑑𝑥2
+ ∞} 𝜓𝐼𝐼𝐼(𝑥) = −𝐸𝜓𝐼𝐼𝐼(𝑥) (5) 

Since 𝐸 − ∞ is equal to −∞, the equation can be simplified, 

{−
ℏ2

2𝑚

𝑑2

𝑑𝑥2
} 𝜓𝐼𝐼𝐼(𝑥) = −∞𝜓𝐼𝐼𝐼(𝑥) (6) 

Both sides can be divided by the constant ∞, and since 
1

∞
 is zero, the solution 

for wavefunction in the region III is. 

𝜓𝐼𝐼𝐼(𝑥) = 0 (7) 

Region I solution 

The Schrödinger equation in region I is 

{−
ℏ2

2𝑚

𝑑2

𝑑𝑥2
+ 0} 𝜓𝐼(𝑥) = −𝐸𝜓𝐼(𝑥) (8) 

It is most convenient write the wavefunction in the trigonometric function 

form.  

𝜓𝐼(𝑥) = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥 (9) 

With A, B and k arbitrary constants. Substituting in the Schrödinger equation, 

−
ℏ2

2𝑚

𝑑2

𝑑𝑥2
{𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥} = −𝐸{𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥} (10) 

Taking the second derivative on the left side and simplifying yield 



24 

 

 
ℏ2𝑘2

2𝑚
{𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥} = 𝐸{𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥} (11) 

And then, 

ℏ2𝑘2

2𝑚
= 𝐸 → 𝑘 =

√2𝑚𝐸

ℏ
 (12) 

Region II solution 

The Schrödinger equation for the region II is 

{−
ℏ2

2𝑚

𝑑2

𝑑𝑥2
+ 𝑉0} 𝜓𝐼𝐼(𝑥) = −𝐸𝜓𝐼𝐼(𝑥) (13) 

The general solution in the region II can be written  

𝜓𝐼𝐼(𝑥) = 𝐶𝑒𝜆𝑥 + 𝐷𝑒−𝜆𝑥 (14) 

Where C, D are arbitrary constant and 𝜆 is a real positive constant. Substituting 𝜓𝐼𝐼(𝑥) 

in the Schrödinger equation.  

{−
ℏ2

2𝑚

𝑑2

𝑑𝑥2
+ 𝑉0} (𝐶𝑒𝜆𝑥 + 𝐷𝑒−𝜆𝑥) = −𝐸(𝐶𝑒𝜆𝑥 + 𝐷𝑒−𝜆𝑥) (15) 

Moving the constant terms that is multiplying the wavefunction to the right side, 

taking the second derivative and simplifying yields 

−
ℏ2𝜆2

2𝑚
(𝐶𝑒𝜆𝑥 + 𝐷𝑒−𝜆𝑥) = (𝐸 − 𝑉0)(𝐶𝑒𝜆𝑥 + 𝐷𝑒−𝜆𝑥) (16) 

Then, 



25 

 

−
ℏ2𝜆2

2𝑚
= (𝐸 + 𝑉0) ⇒ 𝜆 =

√2𝑚(𝑉0 + 𝐸)

ℏ
 (17) 

Because the wavefunction must be finite, it must be testes to ensure that it 

does not blow up as 𝑥 → ±∞. In the region II, the first term 𝐶𝑒𝜆𝑥 blows up when 𝑥

→ ∞ and makes the wavefunction in this region blow up, reducing the equation to 

𝜓𝐼𝐼(𝑥) = 𝐷𝑒−𝜆𝑥 (18) 

Boundary conditions 

Since the potential is infinite in region III, only the wavefunction must be 

continuous across the boundary between region I and III (𝑥 = 0).  

𝜓𝐼𝐼𝐼(0) = 𝜓𝐼(0) (18) 

Substituting in the equation for the wavefunction in regions I and III gives 

𝐵 = 0. Therefore, the continuity requirement leads to: 

𝜓𝐼(𝑥) = 𝐴 sin 𝑘𝑥 (19) 

At 𝑥 = 𝐿 the wavefunction in the region I and II, and its first derivative must 

be continuous across the boundary. 

𝜓𝐼(𝐿) = 𝜓𝐼𝐼(𝐿)

𝜓𝐼
′(𝐿) = 𝜓𝐼𝐼

′ (𝐿)
 (20) 

Dividing 𝜓𝐼(𝐿) by 𝜓𝐼
′(𝐿) and 𝜓𝐼𝐼(𝐿) by 𝜓𝐼𝐼

′ (𝐿) 

𝐴 sin 𝑘𝐿

𝐴𝑘 cos 𝑘𝐿
=

𝐷𝑒−𝜆𝑥

−𝐷𝜆𝑒−𝜆𝑥
 (21) 

Cancelling the constants A and D, the equation becomes 



26 

 

𝜆 = −𝑘𝑐𝑜t 𝑘𝐿 (22) 

Or 

𝑐𝑜t {
√2𝑚𝐺𝑎𝐴𝑠𝑚𝑒𝐸

ℏ
𝐿} = −√

𝑚𝐺𝑎𝐴𝑠

𝑚𝐴𝑙𝐺𝑎𝐴𝑠

𝐸

(𝑉0 + 𝐸)
 (23) 

Where, 𝑚𝐺𝑎𝐴𝑠 and 𝑚𝐴𝑙𝐺𝑎𝐴𝑠 are coefficient of the effective mass for the GaAs and 

AlGaAs, 𝑚𝑒 is the electron mass, E is the energy of the confined state, 𝑉0 the potential 

difference, 𝐿 is the thickness of GaAs and ℏ is the Planck constant. Using 0.0632 and 0.0909 

values [36] as 𝑚𝐺𝑎𝐴𝑠 and 𝑚𝐴𝑙𝐺𝑎𝐴𝑠 respectively and solving numerically the equation 23 is 

possible to find the energy of the confined state below the conduction band energy of the 

AlGaAs, in function of the GaAs thickness. If the found energy value is negative, the electron 

is confined. The used code was adapted to IgorPro from the Suwit Kiravitaya Matlab codes. 

See APPENDIX – I – Used code for particle in the box. 

More sophisticated calculation was done with Nextnano3 software. Nextnano is a 

commercial package for the simulation of electronic and optoelectronic devices. The 

software solves the 𝑘 ∙ 𝑝 perturbation theory and the Poisson equation to calculate the band 

gap diagram and the electron density of the structure. The entering parameter was the shape 

of the structure, doping concentration and the Schottky barrier. Figure 2.13 shows the used 

structure model for the Nextnano simulation. Cross-section TEM image (Figure 3.1) was 

used as template to create this simple trapezoid model. The orange bottom side represents 

the GaAs hole template, yellow part represents the 20nm AlGaAs barrier, and the orange top 

side represents the GaAs thickness used to fulfilled the holes. We assume the concentration 

of 3.0x1014 of carbon doping layer and a Schottky barrier pinned at 0.7eV. The used code is 

attached in APPENDIX – II – Used code for band diagram and electron density calculations. 



27 

 

 

Figure 2.13: Structure model used for Nextnano simulation. The structure shape was 

simplified according to the TEM image, and representing the GaAs hole template (orange 

bottom side) covered with AlGaAs barrier (yellow) and fulfilled with GaAs (orange top side).  

  



28 

 

3. Development and results discussion  

 

In this work, we used the Kelvin probe force microscopy (KPFM) to explore the 

nano-electrostatic properties of strain-free mesoscopic GaAs structures (MGS). The 

structures were fabricated by overgrowing a nanohole template using molecular beam 

epitaxy  [1]. Therefore, the combination of Ga assisted deoxidation and local hole etching is 

used to create initial holes with a depth of ca. 10 to 15nm, which were covered subsequently 

with AlGaAs changing the concentration of Al and filled with GaAs of different thicknesses. 

The local potential and charge distribution in these structures was investigated using single 

pass KPFM. The NextNano3 software package was used to calculate band bending and 

electron wave function of the MGS.  

 

3.1. Sample processing 

 

The entire sample used in this work were grown by molecular beam epitaxy (MBE) 

at Brazilian Nanotechnology National Laboratory (LNNano). In this equipment is possible 

to growth epitaxially thin films of the III – V elements of periodic table. The current setup 

has four effusion cells (Ga, As, In, Al) and two doping cells (Si and Be) to grow high quality 

semiconductor heterostructures. 

For the sample preparation, two different techniques were combined in order to 

produce the templates for unstrained quantum dots. In the first moment, Gallium assisted 

deoxidation technique removed the native gallium oxide (Ga2O3) from the top of GaAs (001) 

substrate. In this procedure, a Gallium atmosphere is used to react with the Ga2O3 generating 

Ga2O (not stable). After the complete removal of native gallium oxide, the local droplet 

etching is used to produce the hole templates on top of the substrate. In this process, the 

exceed Gallium of deoxidation process, start to produce droplets on top of the substrate. The 

droplets in presence of low pressure of As will react with the surface of GaAs producing 

holes. The result is a homogeneous template of holes with ~ 20nm depth that was used to 

produce the unstrained QDs or mesoscopic GaAs structure (MGS). 



29 

 

With the hole template in hand, the first step to grow the MGS structure is 

filled the holes with AlGaAs. A few layer of AlGaAs will work as a potential barrier 

for carrier confinement of the QDs. In this study, we systematically used different 

thickness of AlGaAs barrier (15nm and 20nm) and changed the Al concentration of 

33%, 50%, 70% and 90%. Then different thickness of GaAs was grown on top of the 

barrier. For the Kelvin probe measurements we stop the growth at this point because 

is necessary to access the center of the structure. The final MGS structure have one 

more layer of AlGaAs barrier and a GaAs cap to protect the structures as showed in 

the cross-section transmission electron microscopy (TEM) of the structure (Figure 

3.1). For this measurement, the focus ion beam (FIB) technique was employed to 

cross-sectioned right on the middle of the structure.  

 

Figure 3.1: Cross-section transmission electron microscopy (TEM) of the strain-free quantum 

dots. The bright color shows the GaAs material and the dark color shows the AlGaAs material. 

The confinement potential gave by the dot formation is marked in the middle of the structure.  

 

In Figure 3.2 we cannot see any dislocation or defects of the crystal in the 

AlGaAs/GaAs/AlGaAs junction of the sample. The dark region shows the AlGaAs 

barrier and the bright region is the GaAs layer. This is an important information for 

optical application, where is necessary high quality crystal for optical emission. In 

the same way, if we have dislocation or defects at the junction between the two 

semiconductors the charge carrier transport will change, modifying the quantum 

confinement. 



30 

 

 

Figure 3.2: High-resolution transmission electron microscopy HR-TEM of the AlGaAs/GaAs/ 

AlGaAs heterojunction cross-section. The dark and bright regions show the AlGaAs barrier 

and the GaAs layer respectively . No dislocation of defects is observed in this region of the 

sample, where the atoms made a perfect crystal.  

 

In this work, we prepare a set of samples to comprehend the charge accumulation, 

following the steps above and outlined in Figure 2.1: 

-  (I) Only holes made by LDE 

-  (II) Holes filled with AlGaAs barrier 

- (III) Holes filled with AlGaAs barrier and changing the amount of GaAs 

- (IV) Holes filled with AlGaAs barrier and GaAs but changing the Al 

concentration 



31 

 

 

Figure 3.3: Schematic showing set of the analyzed sample. (I) only substrate with GaAs holes; 

(II) holes with AlGaAs barrier; (III) holed with AlGaAs barrier and changing GaAs thickness; 

(IV) holes with different concentration of Aluminum in the AlGaAs barrier and filled with 

2nm GaAs.  

 

3.2. Kelvin probe force microscopy measurements 

 

In the Figure 3.4 the measurement of initial holes template is showed. We use 

this image to confirm that the observed difference potential in the final structure is 

not coming from topography artifacts.   

 

Figure 3.4: Topography (a) and KPFM (b) images of the template holes. Depth of the holes is 

around 15nm. The electrostatic potential is homogenous under the surface.  

 



32 

 

Figure 3.4 (a) shows the topography of GaAs substrate with the hole templates. The 

gallium droplets etching make holes with 20nm depth approximately. As showed in section 

2.2 an expected GaAs accumulation with 5nm height is observed beside the holes and 

following the substrate orientation. The KPFM results is shown in Figure 3.4 (b). The 

electrostatic surface potential is uniform over the sample since we have only GaAs over the 

surface. This result proofs that the irregular topography of the structure is not generating 

artifacts to the potential data.  

In the second set of sample, we grown 20nm of Al0.33Ga0.67As barrier was deposit 

over the hole template and measured in the same way (Figure 3.5). 

  

 

Figure 3.5: Topography (a, c) and KPFM (b, d) of the hole template after 15nm AlGaAs barrier 

deposition. The topography looks similar to the initial hole template indicating AlGaAs follow 



33 

 

evenly the shape of the holes. The potential data shows difference electrostatic potential over 

the structures, with higher contrast in the mounts beside the holes. 

 

Figure 3.5 shows the topography (a, c) and KPFM (b,d) results of these 

structures. As observed on Figure 3.4, the topography does not change so much from 

the initial hole template. This behavior indicates an epitaxial and uniform growth of 

the AlGaAs barrier, by following the shape of the holes. From the KPFM result, we 

observe a huge difference in the electrostatic surface potential. Regions with the 

structures relief (mounts and hole) shows lower electrostatic potential, but the mounts 

region have higher contrast if compared with the holes region. A suggestion for the 

observed effect is because an “inverted” quantum dot formed over the mounts around 

the hole.  

The Figure 3.6 shows a set of images from MGS structures. We used the holes 

template showed in Figure 3.4 and deposit 20nm of Al0,33Ga0,67As and 2nm of GaAs.  

 

Figure 3.6: Difference electrostatic potential observed on MGS structures. (a) Topography 

showing the shape of the unstrained quantum dots. (b) KPFM measurement showing a contrast 



34 

 

right on the middle of the structures (holes template). (c) Overlay topography and KPFM data 

to identify the exactly position where we observe the contrast in electrostatic potential.  

 

Topography images shows an elongation of the structure in the direction [1-10] due 

10] due the preferential material diffusion (Figure 3.6 (a)). The GaAs completely filled the 

initial holes generating a region with higher mounts (height distribution of 5.05nm Figure 

3.7) and dimensions around 800nm x 200nm. KPFM measurements shows a contrast in the 

potential data (Figure 3.6 (b)). The black spots region means that surface potential at the 

MGS region is lower than surrounding. An overlay (Figure 3.6 (c)) confirm that the 

difference at the potential data is situated right at the middle of the structures, where we 

expected the holes are. This results indicates a charge confinement inside the MGS structures 

but only at the hole position. Afterwards, we prepared samples with different amount of 

GaAs. 

 
Figure 3.7: Height histogram of the sample with 2nm thickness of GaAs. The red color shows 

the number of counts in each height of the structure. Black curve is the Gaussian plot of the 

histogram with maximum value 5.05nm. 

 



35 

 

3.3.  Changing GaAs thickness 

After observed the charge accumulation inside the MGS structure, we prepare 

of samples to check the confinement when changing the GaAs layer thickness. Figure 

shows topography and potential data for 1nm, 2nm, 5nm and 10nm thickness of 

 

Figure 3.8: Topography (a-d) and KPFM (e-h) images acquired for different thickness of 

GaAs. The structures elongate when the thickness was increased and the difference 

electrostatic potential disappear with 10nm thickness of GaAs.  

 

 The images (Figure 3.8 (a-d)) shows the topography image for 

different thickness of GaAs. We observed a higher elongation of the structures and a 

decrease in height when the thickness increases. With increasing GaAs layer 

thickness covering the hole-mound structures, the height distribution for this 

sequence of samples was 6.6nm 5.05nm, 2.2nm and 2.7nm respectively for 1nm, 

2nm, 5nm and 10nm thickness of GaAs. These topographical images is illustrating 

that the [1-10] direction grows faster than the [110] direction. This effect was already 

observed by Silva et.al.  [1], where the asymmetric Ga surface diffusion during the 

hole formation, result in an increased growth rate in the [1–10] direction of the 



36 

 

crystal. Hence, the structures does not symmetrically flatten out as naively expected, but 

flattens by elongation in one direction and by continuously decreasing the mound height 

over the surrounding flat GaAs surface. 

On the other hand, the Figure 3.8 (e-f) show the KPFM results when changing the 

thickness of GaAs. For the sample with 1nm and 2nm thickness we observed a surface 

potential contrast around 100mV between the structure and the surrounding. The contrast is 

reduced when the GaAs thickness increase to 5nm and is close to zero when increase to 

10nm. 

  

3.4. Changing aluminum concentration into AlGaAs barrier  

 

Samples with different concentration of Aluminum in the AlGaAs barrier were 

measured. Changing the concentration of Al into the barrier the band gap will change and 

consequently the carrier confinement will change. The Figure 3.9 shows the topography (a-

d) and KPFM (e-h) measurements of the samples with different concentration of Al. In the 

topographical images of the 0.3, 0.5 and 0.7 Aluminum content, is possible to identify the 

MGS structures as before, with the AlGaAs barrier following the hole shape. However, for 

0.9 Aluminum content, we observed a huge change in the topographical images. The change 

in topographical image is because the Aluminum easily oxidizes under ambient condition, 

and as higher is the Al content, easily oxidation is observed for these samples without the 

GaAs layer.  

The Figure 3.9 (e-h) shows the potential maps for these structures. For 0.3, 0.5 and 

0.7 contents, a clear contrast between the MGS structures and the surrounding was observed 

in the potential data, with the 0.5 Al content showing the higher contrast. Lower contrast is 

observed for the 0.7 in comparison to 0.5 probably due the direct-to-indirect transition of the 

band gap. In the 0.9 content, the sample surface oxidized as observed in the topographical 

images, for this reason the surface potential is homogenous over the surface. In other words, 

no confinement is archived with the complete oxidation of the surface.  



37 

 

 

Figure 3.9: Topography (a-d) and KPFM (e-h) measurement of the holes with AlxGax-1As 

barrier in different concentration of Aluminum. The images with x = 0.3, 0.5, 0.7 shows black 

spots in potential data. The concentration 0.9 looks like oxidized surface and does not show 

any contrast in potential data. 

  

3.5. Modelling half-finite quantum well 

3.5.1. Numerical solution 

 

Assuming that the observed signal measured in our samples behave as a 

particle in a half-finite quantum well, we solve the classical equation for a potential 

deep to calculate the energy difference between the valence and conduction band in 

the confined state. We compare the energy solution for the finite and half-finite 

quantum well. In Figure 3.10a and Figure 3.10b is outlined the finite both sides 

quantum well and the half-finite quantum well, respectively, formed by the 

AlGaAs/GaAs heterostructure. Figure 3.10.d shows the energy plot for both 

conditions. The energy is plotted relatively to the conduction band of the AlGaAs 

and let the electron drop into the potential deep. Therefore the calculated energy gives 



38 

 

the energy of the confined state below the conduction band energy of AlGaAs - if it is 

negative, the electron is confined. Thus, from this plot, if the GaAs thickness is lower than 

1.5nm no confinement is observed. With this theoretical results we change the sample 

processing in the way that the GaAs follow the shape of the holes as showed in the schematic 

of Figure 3.10c.  

 

Figure 3.10: Modelling the finite and the half-finite quantum-well. (a) General schematic for 

a quantum well (a) finite barrier of both sides and (b) half-finite. (c) Schematic of the grown 

sample. (d) Simulation of the energy in function of the well width.     

 

3.5.2. KPFM of the proposed sample 

Figure 3.11 shows the samples grown with 1nm and 2nm thickness of GaAs and 

following the schematic of Figure 3.10c. To the GaAs follow the hole shape is necessary to 

abrupt cooling down the temperature of the sample after grow. Doing this the GaAs will 

follow the shape of the hole template. The topography image of these samples is showed in 

Figure 3.11(a-b). From the topography, we see the hole shape is still preserved, indicating 

that GaAs layer is following the hole template. In the potential data, the sample with 1nm 

(Figure 3.11c) thickness of GaAs doesn’t show any contrast, while, the sample 2nm 

thickness of GaAs show a clear contrast between the holes and the surrounding. These results 

are in agreement with our calculations where thickness lower than 1.5nm does not confine 

any charge.   



39 

 

 

Figure 3.11: Structures grown following the hole template. Topography image for (a) 1nm 

and (b) 2nm GaAs thickness. The potential image for (c) 1nm doesn’t show any contrast and 

for (d) 2nm shows a clear contrast.   

 

 

 

 

 

 



40 

 

3.6.  Simulations of band diagram and electron wave function 

The Nextnano3 software package was used to simulate the band diagram (Figure 

(Figure 3.12) and the electron wave function (Figure 3.13) of the MGS structures.  

 

Figure 3.12: Band diagram of the unstrained quantum dot. The conduction and valence band 

was plotted (a) inside and (b) beside the dot. In both graph a small band bending was observed 

due the fermi level pinning.  

 

The conducting and valence band inside and beside the dots were plotted in Figure 

3.12. In both cases we observe a band bending of the conduction and valence band, indicating 

the Fermi level pinning phenomena caused by surface potential in the interface. This band 

bending makes possible a charge accumulation inside the dots, with the electron moving for 

states with lower energies close to the surface. For this kind of system, we should naturally 

see a charge accumulation in the AlGaAs barrier, but when we applied the AC voltage, used 

during KPFM measurements, this behavior was intensified.   

The Figure 3.13 shows the electron wavefunction calculation changing the thickness 

of GaAs. When using 2nm GaAs thickness the green spot is not dispersed, in other words, 

electron intensity is more localized indicating a higher probability of charge accumulation. 



41 

 

This simulation follow according to the experimental results when we observe a 

higher charge accumulation decreasing the GaAs thickness.   

  

 

Figure 3.13: Electron probability simulation of the unstrained quantum dot. Increasing the 

thickness of the GaAs the probability of the electron dissipate over the sample increase. 

   

 

3.7. Electrostatic force microscopy (EFM) without AC voltage 

 

Electrostatic force microscopy (EFM) was done for the sample with 50% 

Aluminum content and 2nm GaAs thickness. The Figure 3.14a-b show the 

topography image of the sample and the Figure 3.14c-d show the EFM phase signal 

at the same region using 20nm and 100nm lift. Using 100nm lift, we see only a small 

bright dot in the EFM image that comes from surface contamination, also observed 

in the topography image. When the lift is reduced to 20nm, we start to see some 

contrast in the EFM image, which is not a real EFM contrast. As described in the 

Chapter 2, the EFM technique works in lift mode and without AC voltage applied 

between tip and sample. In this way, the tip is always keeping a constant distance to 



42 

 

the sample surface, and, when far enough, the tip is probing only the electrostatic forces and 

not the Van der Waals interaction. Therefore, if we look carefully for the EFM image with 

20nm lift, we see that always in the left side of the structure the contrast is dark, and the left 

side in bright. These facts are an indicative that the EFM signal is contaminated with the 

Van der Waals/topography information, where the signal move to one side when the tip 

moves up and to the other side when tip moves down.    

From the previous KPFM results, we expect to see charge accumulation right on the 

middle of the MGS structures. However, for the EFM measurement (Figure 3.14c-d) no clear 

difference potential signal was observed for this sample. The explanation for this behavior 

is because, at this condition, we are not applying to the tip, the AC voltage used in the KPFM 

technique. We suggest, that the AC voltage are inducing charges on the sample surface, 

which, due to the different band gap of the material, immediately confine inside the MGS 

structures.    

 

Figure 3.14: Topography images (a-b) of the sample with 0.5 Aluminum content and 2nm 

GaAs thickness. EFM phase signal (c-d) with100nm and 20nm lift respectively. No clear 



43 

 

contrast is observed for the 100nm lift (c). The shallow contrast for 20nm lift (d) is due the tip 

fell the Van der Waals interaction, and should not be consider as a real signal.  

  



44 

 

4. Conclusions 

The analyzed samples allowed to access the internal region of the mesoscopic GaAs 

structures and measure their electrostatic properties. Kelvin probe measurement 

demonstrates a huge capability to direct probe difference in the electrostatic surface potential 

with nanometric resolution.  

The observed charge accumulation was right on the middle of the structures where 

we expect the holes are. This is due to the difference band gap of the materials that produce 

a carrier confinement similar to a particle in the box. In the first set of sample, when changing 

the GaAs thickness, we observed that the difference potential between the MGS structures 

and the surrounding is higher for 2nm GaAs thickness. Increasing the GaAs thickness, we 

observe the difference potential reducing, because lower confinement is archived as thicker 

is the GaAs. 

Changing the Aluminum content, second set of samples, we observed a complete 

oxidation of the surface for Al concentration in 90%. The oxidation interfere in the KPFM 

measurement where no difference potential at the holes were observed. The sample with Al 

concentration around 50% present higher difference band gap compared to the 30%. For this 

reason we see higher difference potential in this sample. The same reasoning cannot be 

followed for the sample with 70% Al concentration because the sample probably start the 

oxidation process.   

Forcing the GaAs layer to follow the shape of the hole template we observe no charge 

accumulation inside the structures for the 1nm GaAs thickness. This result corroborate with 

the simulation of the half-finite quantum well where no confinement is observed for 1nm 

thickness. 

The simulations using NextNano package software held us to better describe the 

charge accumulation. The band diagram shows a band bending with lower values close to 

the surface, making possible the charge accumulation. The electron density plot, move in the 

same direction of the experimental data, with better confinement for the 2nm GaAs 

structures.  



45 

 

Electrostatic force microscopy (EFM) in lift mode shows that the huge signal 

observed in the KPFM measurement is intensified by the AC voltage applied to the 

tip. This voltage is inducing charges on the sample surface, which gets confined due 

the band gap difference of the materials. 

  



46 

 

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49 

 

APPENDIX – I – Used code for particle in the box 

 

 

#pragma rtGlobals=3  // Use modern global access method and 

strict wave access. 

 

//Define a grenarl Constant 

 

Constant PlanckConstantReduced=1.054571800e-34,electronmass=9.10938356e-

31,electroncharge=1.6021766208e-19 

Constant eVtoJoul=1.602176565e-19 

 

//Functions to calculate QW energy levels 

 

//Material properties functions 

 

Function AlGaAs_prop(Al_content,properties,[q]) 

 

//Calculates the band gap and electron masses for unstrained GaAs 

// 

// Based on a Mathematica sheet by Suwit 

// 

//Version 1.0 (27.01.2017) 

// 

// Materials parameters after E. H. Li, Physica E 5, 215 (200) and C. Van 

de Walle, PRB 39, 1871 (1989) 

// Li works with band offset theory and is not so clear, what is the 

bandoffset ratio (should be 50:50 according to  

// S. Achachi, GaAs and related materials (World Scientific) 

// Suwit uses model-solid theory, which will provied a band position and 

a spin-orbit coupling offset. 

// 

//Values to be written in properties wave 

//properties[1]: Bandgap 

//properties[2]: Energy Valence band 

//properties[3]: Energy Conduction band 

//properties[4]: Mass electrons 

//properties[5]: Mass heavy holes 

 

Wave/D properties 

Variable Al_content,q   

 

//Define soem internal variables 

 

Variable evav,delta 

 

evav=-6.92*(1-Al_content)-7.49*Al_content  //model-solid valencband 

energy at delta point after PRB  

delta=0.341-0.066*Al_content // Spin-orbit splitting after Li for the 

valence band (our zero) 

 

properties[0]=1.5194+(1.36*Al_content)+(0.22*Al_content^2) //Bandgap at 

low temp after Li 

properties[1]=evav+delta/3 //Position of the valence band for given 

concentration - Valence band has an average position and a splting due to 

spin orbit 

properties[2]=properties[0]+properties[1] // The conduction band energy 

is valenceband energy plus band gap 

properties[3]=0.0665+(0.1006 * Al_content)+(0.0137*Al_content^2) 



50 

 

properties[4]=0.34+0.16*Al_content 

 

//Print results if q=0 

 

if( ParamIsDefault(q)) 

printf "Bandgap: %g\r", properties[0] 

printf "Valenceband energy: %g\r", properties[1] 

printf "Conduction band energy: %g\r", properties[2] 

printf "Electron mass: %g\r", properties[3] 

printf "Heavy hole mass: %g\r", properties[4] 

endif 

 

End 

 

Function Symteric_finiteQW(coeff,x) 

 

// Condition for symetric solution of a QW after Suwit. Zero is the 

barrier and the carries are confined by a negative energy inside the well 

dip. 

//  

// To find theEnergylevel, this function has to return zero 

// 

// coeff[0]: Potential difference 

// coeff[1]: Mass carrier in Well 

// coeff[2]: Mass carrier in Barrier 

// coeff[3]: thickness of the QW 

 

Wave/D coeff 

Variable x 

 

Variable left,right,k,kappa 

 

//k=sqrt((2*coeff[1]*electronmass*x*eVtoJoul)/PlanckConstantReduced^2) 

//kappa=sqrt((2*coeff[2]*electronmass*(coeff[0]-

x)*eVtoJoul)/PlanckConstantReduced^2) 

 

//left=tan(0.5*k*coeff[3]*1e-9) 

//right=kappa/k 

 

left=tan(0.5*coeff[3]*1e-

9*(sqrt(2*coeff[1]*electronmass*(coeff[0]+x)*eVtoJoul)/PlanckConstantRedu

ced)) 

right=sqrt(coeff[1]/coeff[2]*(-x/(coeff[0]+x))) 

 

return left-right 

 

End 

 

 

//-----------------------------------------------------------------------

-------------------------------------------------------------------------

------------------------------------------------------------------------- 

 

Function semifinite_QW(coeff,x) 

 

// Condition for symetric solution of a QW after Suwit and modifit for a 

hakf infinit barrier. Zero is the barrier and the carries are confined by 

a negative energy inside the well dip. 

// Difference is only to take away a factor of two before and we have to 

use the cotan instead of tan. 



51 

 

//  

// To find theEnergylevel, this function has to return zero 

// 

// coeff[0]: Potential difference 

// coeff[1]: Mass carrier in Well 

// coeff[2]: Mass carrier in Barrier 

// coeff[3]: thickness of the QW 

 

Wave/D coeff 

Variable x 

 

Variable left,right 

 

left=cot(coeff[3]*1e-

9*(sqrt(2*coeff[1]*electronmass*(coeff[0]+x)*eVtoJoul)/PlanckConstantRedu

ced)) 

right=-sqrt(coeff[1]/coeff[2]*(-x/(coeff[0]+x))) 

 

return left-right 

//return left 

 

End 

 

//-----------------------------------------------------------------------

-------------------------------------------------------------------------

------------------------------------------------------------------------- 

 

Function AlGaAs_semifiniteQW_transition(al_content,thickness,[q]) 

 

//Calcualtes the transition energy in an unstrained AlGaAs/GaAs Qw 

assuming semifinite barrieres of infinite thickness and a QW thickness 

// One side has infinite potential, the other side has an AlGaAs barrier 

// 

//Fuinction follows Suwits Mathematica sheet 

// 

//Version 1.0 27.01.2017 

// 

//returns transition energie and needs Al content of Barrier and GaAs 

thickness as input 

// 

// Eq. for QW can be found inS. Adachi, GaAs and Related Materials, World 

Scientific 

// The given solution is wrong after Suwit and Wikipedia 

// 

 

Variable al_content,thickness,q  

 

Variable V0, E1c,E1v, lower 

 

Make/D /N=10 propertiesQW,propertiesbarrier,coeff 

 

AlGaAs_prop(0,propertiesQW,q=1) 

AlGaAs_prop(al_content,propertiesbarrier,q=1) 

 

//calculate energy state of the vallence band 

 

coeff[0]=0.57*(propertiesbarrier[0]-propertiesQW[0]) //Band discontinuety 

or confinment potential of conduction band using band gap nand band 

offset ratios (ca. 0.5) from literature 



52 

 

//coeff[0]= propertiesbarrier[2]-propertiesQW[2] //Band discontinuety or 

confinment potential of conduction band using model-solid theory 

coeff[1]=propertiesQW[3] //electron mass in conduction band of QW 

coeff[2]=propertiesbarrier[3] //electron mass in conduction band of 

barrier 

coeff[3]=thickness 

 

if( ParamIsDefault(q)) 

printf "Potential: %g\r",coeff[0] 

printf "Carrier mass QW: %g\r", coeff[1] 

printf "Carrier mass barrier: %g\r", coeff[2] 

endif 

 

//FindRoots /L=0 /H=(0.10) /Q Symteric_finiteQW, coeff 

 

lower=(pi*PlanckConstantReduced)^2/((coeff[3]*1e-

9)^2*2*coeff[1]*electronmass*eVtoJoul)-coeff[0] // Looking for the extrma 

in the tan fkt. We have to restrict search to the lower part before the 

first extrema 

if (lower>0)  

lower=-0.000001 

endif 

 

FindRoots /L=(-coeff[0]) /H=(lower) /Q semifinite_QW, coeff //Finding the 

zeos - care has to be taking tofind the first state and not the excited 

states. 

 

if (V_flag !=0)   

printf "Problems with E1c at thickness: %g\r", coeff[3]  

printf "Error: %g\r", V_flag 

endif 

 

E1c=V_root // Energy state in conduction band 

 

//Redo the same for the valence bands and the holes inside 

 

coeff[0]=0.43*(propertiesbarrier[0]-propertiesQW[0]) //Band discontinuety 

or confinment potential of conduction band 

//coeff[0]=propertiesQW[1]-propertiesbarrier[1] 

coeff[1]=propertiesQW[4] //hole mass in valenceband of QW 

coeff[2]=propertiesbarrier[4] //hole mass in valenceband of barrier 

 

if( ParamIsDefault(q)) 

printf "Potential: %g\r",coeff[0] 

printf "Carrier mass QW: %g\r", coeff[1] 

printf "Carrier mass barrier: %g\r", coeff[2] 

endif 

 

//FindRoots  /L=0 /H=(coeff[0]/coeff[3]) /Q  /T=1e-20 Symteric_finiteQW, 

coeff 

 

lower=(pi*PlanckConstantReduced)^2/((coeff[3]*1e-

9)^2*2*coeff[1]*electronmass*eVtoJoul)-coeff[0] 

 

if (lower>0)  

lower=-0.000001 

endif 

 

FindRoots /L=(-coeff[0]) /H=(lower) /Q semifinite_QW, coeff 

 



53 

 

if (V_flag !=0)   

printf "Problems with E1v at thickness: %g\r", coeff[3]  

printf "Error: %g\r", V_flag 

endif 

 

E1v=V_root 

 

// Define the transition energy as barrier band gap plus the two 

negatibve energies found for valence and conduction band. 

 

V0=propertiesbarrier[0]+E1c+E1v 

 

if( ParamIsDefault(q)) 

printf "Energy Conductionband: %g\r", E1c 

printf "Energy Valenzband: %g\r",  E1v 

printf "Transitionenergy: %g\r", V0 

endif 

 

killwaves propertiesQW,propertiesbarrier,coeff 

 

if (q==1)  

   return V0 

elseif (q==2) 

   return E1c 

elseif (q==3)  

   return E1v 

else  

   return V0 

endif 

 

End 

 
Function AlGaAs_QW_transition(al_content,thickness,[q]) 

 

//Calcualtes the transition energy in an unstrained AlGaAs/GaAs Qw 

assuming finite barrieres of infinite thickness and a QW thickness 

// 

//Fuinction follows Suwits Mathematica sheet 

// 

//Version 1.0 27.01.2017 

// 

//returns transition energie and needs Al content of Barrier and GaAs 

thickness as input 

// 

// Eq. for QW can be found inS. Adachi, GaAs and Related Materials, World 

Scientific 

// The given solution is wrong after Suwit and Wikipedia 

// 

 

Variable al_content,thickness,q  

 

Variable V0, E1c,E1v, lower 

 

Make/D /N=10 propertiesQW,propertiesbarrier,coeff 

 

AlGaAs_prop(0,propertiesQW,q=1) 

AlGaAs_prop(al_content,propertiesbarrier,q=1) 

 

//calculate energy state of the vallence band 



54 

 

 

coeff[0]=0.57*(propertiesbarrier[0]-propertiesQW[0]) //Band discontinuety 

or confinment potential of conduction band using band gap nand band 

offset ratios (ca. 0.5) from literature 

//coeff[0]= propertiesbarrier[2]-propertiesQW[2] //Band discontinuety or 

confinment potential of conduction band using model-solid theory 

coeff[1]=propertiesQW[3] //electron mass in conduction band of QW 

coeff[2]=propertiesbarrier[3] //electron mass in conduction band of 

barrier 

coeff[3]=thickness 

 

if( ParamIsDefault(q)) 

printf "Potential: %g\r",coeff[0] 

printf "Carrier mass QW: %g\r", coeff[1] 

printf "Carrier mass barrier: %g\r", coeff[2] 

endif 

 

//FindRoots /L=0 /H=(0.10) /Q Symteric_finiteQW, coeff 

 

lower=((pi*PlanckConstantReduced)/(coeff[3]*1e-

9*sqrt(2*coeff[1]*electronmass*eVtoJoul)))^2-coeff[0] // Locing for the 

extrma in the tan fkt. We have to restrict search to the lower part 

before the first extrema 

if (lower>-0.1)  

lower=0 

endif 

 

FindRoots /L=(-coeff[0]) /H=(lower) /Q Symteric_finiteQW, coeff //Finding 

the zeos - care has to be taking tofind the first state and not the 

excited states. 

 

if (V_flag !=0)   

printf "Problems with E1c at thickness: %g\r", coeff[3]  

printf "Error: %g\r", V_flag 

endif 

 

E1c=V_root // Energy state in conduction band 

 

//Redo the same for the valence bands and the holes inside 

 

coeff[0]=0.43*(propertiesbarrier[0]-propertiesQW[0]) //Band discontinuety 

or confinment potential of conduction band 

//coeff[0]=propertiesQW[1]-propertiesbarrier[1] 

coeff[1]=propertiesQW[4] //hole mass in valenceband of QW 

coeff[2]=propertiesbarrier[4] //hole mass in valenceband of barrier 

 

if( ParamIsDefault(q)) 

printf "Potential: %g\r",coeff[0] 

printf "Carrier mass QW: %g\r", coeff[1] 

printf "Carrier mass barrier: %g\r", coeff[2] 

endif 

 

//FindRoots  /L=0 /H=(coeff[0]/coeff[3]) /Q  /T=1e-20 Symteric_finiteQW, 

coeff 

 

lower=((pi*PlanckConstantReduced)/(coeff[3]*1e-

9*sqrt(2*coeff[1]*electronmass*eVtoJoul)))^2-coeff[0] 

 

if (lower>-0.1)  

lower=0 



55 

 

endif 

 

FindRoots /L=(-coeff[0]) /H=(lower) /Q Symteric_finiteQW, coeff 

 

if (V_flag !=0)   

printf "Problems with E1v at thickness: %g\r", coeff[3]  

printf "Error: %g\r", V_flag 

endif 

 

E1v=V_root 

 

// Define the transition energy as barrier band gap plus the two 

negatibve energies found for valence and conduction band. 

 

V0=propertiesbarrier[0]+E1c+E1v 

 

if( ParamIsDefault(q)) 

printf "Energy Conductionband: %g\r", E1c 

printf "Energy Valenzband: %g\r",  E1v 

printf "Transitionenergy: %g\r", V0 

endif 

 

killwaves propertiesQW,propertiesbarrier,coeff 

 

 

if (q==1)  

   return V0 

elseif (q==2) 

   return E1c 

elseif (q==3)  

   return E1v 

else  

   return V0 

endif 

 

End 

 

//-----------------------------------------------------------------------

-------------------------------------------------------------------------

-------------------------------------------------- 

 

Function Plot_value(Al_content,In_content,Property) 

 

// Returns a material property value for a given set of parameters. 

Practical, if you need 

// to know some specific parameter or want to plot it. Also usefull, if 

you need to calculate strain  

// for any substrate 

 

Variable Al_content,In_content,Property 

 

Wave properties 

Variable value 

 

Make /N=10 properties 

 

InAlGaAs_prop(In_content,Al_content,1,properties,q=1) 

 

value=properties[Property] 

 



56 

 

killwaves properties 

 

return value 

 

End 

  



57 

 

APPENDIX – II – Used code for band gap and electron wavefunction 

 

# 1D sample for solving quantum-poisson 

 

#Begin NextnanoWizard parameter section 

$BIAS =0         # Gate voltage(V) 

$CAP = 2        # Thickness cap 

$ALLOY = 0.3       # Al content of AlGaAs layer 

$SIZE = 20         # size of AlGaAs layer (nm) 

$DOT = 20          # Quantum dot deepth 

$Dot_top = 290    # Top width 

$Dot_bottom = 120 # Bottom width 

$DOPECONC = 3.0e14 # donor concentration of 10nm doping layer (cm^-3) 

$DOPEPOS = 65      # distance between AlGaAs-GaAs interface and delta 

doping position (nm) 

$MINGRID = 0.1     # minimum grid spacing (nm) 

$NUMEV = 20        # number of eigenvalues to be calculated 

$BARRIER = 0.7     # height of left Schottky barrier (eV) 

#End NextnanoWizard parameter section 

 

 

global{ 

   simulate2D{} 

    

   temperature = 300 

 

   substrate{ name = "GaAs" } 

 

   crystal_zb{  

      x_hkl = [0, 0, 1]  

      y_hkl = [1, 1, 0]  

   } 

}<> 

 

 

grid{ 

   xgrid{ 

      line{ pos = 0                   spacing = 0.1 } #cap 

      line{ pos = $CAP                spacing = 0.1 } #begin AlGaAs 

      line{ pos = $CAP+$DOT           spacing = 0.1 } 

      line{ pos = $CAP+$SIZE          spacing = 0.1 } #end AlGaAs and 

2DEG 

      line{ pos = $CAP+$DOT+$SIZE     spacing = 0.1 } 

      line{ pos = 400                 spacing = 5 } #after baggate 

   } 

   ygrid{ 

      line{ pos = 0       spacing = 5 } 

      line{ pos = 500-$SIZE-($Dot_top)   spacing = 5} 

      line{ pos = 500-$SIZE-($Dot_top/2) spacing = 1} #left end of 

barrier 

      line{ pos = 500-$SIZE              spacing = 1} #Dot left 

      line{ pos = 500+$SIZE              spacing = 1} #Dot right 

      line{ pos = 500+$SIZE+($Dot_top/2) spacing = 1} #Right beginning of 

barrier 

      line{ pos = 500+$SIZE+($Dot_top)   spacing = 5} 

      line{ pos = 1000                   spacing = 5.0 } 

   } 

 

   periodic{ 



58 

 

      x = no 

      y = no 

   } 

}<> 

 

 

impurities{ 

   acceptor{ name = "C" energy = 0.027 degeneracy = 4 } # fully ionized 

}<> 

 

 

# Defines a pinned GaAs cap to 0.7 eV 

contacts{ 

   schottky{  

      name = "gate" 

      bias = 0 

      barrier = $BARRIER 

   } 

 

   fermi{  

      name = "backgate" 

      bias = -$BIAS 

   } 

}<> 

 

 

structure{ 

   output_material_index{} 

   output_contact_index{} 

   output_impurities{} 

 

   # AlGaAs layer 

   region{ 

      everywhere{} # default material 

      ternary_constant{ name = "Al(x)Ga(1-x)As" alloy_x = $ALLOY } 

   } 

 

   # GaAs cap layer 

   region{ 

      rectangle{ x = [0, $CAP] y=[0,1000]} 

      binary{ name = "GaAs" } 

   } 

 

   # doping layer 

     region{ 

        rectangle{ x= [1,395] y=[0,1000] } 

        doping{ constant{ name = "C" conc = $DOPECONC} } 

     } 

 

   # GaAs substrate 

   region{ 

      rectangle{ x = [$CAP+$SIZE+$DOT, 400] y=[0,1000]} 

      binary{ name = "GaAs" } 

   } 

 

   #GaAs dot 

 

 #  region{ 

 #     rectangle{ x = [$CAP, $CAP+$DOT] y=[200,300]} 

 #     binary{ name = "GaAs" } 



59 

 

 #  } 

 

    region{    

      trapezoid{                          # 2D object e.g. a simple 

trapezoid along the x axis 

         base_x     = [ $CAP, $CAP]       # base line extends in x 

direction from 5 to 15 nm 

         base_y     = [500-($Dot_top/2), 500+($Dot_top/2)]       # base 

line has a constant y coordinate y = 25 nm 

         top_x      = [ $CAP+$DOT, $CAP+$DOT]       # top line extends in 

x direction from 8 to 12 nm 

         top_y      = [500-($Dot_bottom/2),500+($Dot_bottom/2)]       # 

top line has a constant y coordinate y = 30 nm 

         } 

       binary{ name = "GaAs" } 

      } 

 

#   region{ 

#      rectangle{ x = [$CAP+$DOT, $CAP+$DOT+$SIZE] y=[0,200-$SIZE]} 

#      binary{ name = "GaAs" } 

#  } 

 

    region{    

      trapezoid{                          # 2D object e.g. a simple 

trapezoid along the x axis 

         base_x     = [ $CAP+$DOT, $CAP+$DOT]       # base line extends 

in x direction from 5 to 15 nm 

         base_y     = [0,500-($Dot_top/2)-$SIZE]       # base line has a 

constant y coordinate y = 25 nm 

         top_x      = [ $CAP+$DOT+$SIZE, $CAP+$DOT+$SIZE]       # top 

line extends in x direction from 8 to 12 nm 

         top_y      = [0,500-($Dot_bottom/2)-$SIZE]       # top line has 

a constant y coordinate y = 30 nm 

         } 

       binary{ name = "GaAs" } 

      } 

 

 #  region{ 

 #     rectangle{ x = [$CAP+$DOT, $CAP+$DOT+$SIZE] y=[300+$SIZE, 500]} 

 #     binary{ name = "GaAs" } 

 #  } 

 

    region{    

      trapezoid{                          # 2D object e.g. a simple 

trapezoid along the x axis 

         base_x     = [ $CAP+$DOT, $CAP+$DOT]       # base line extends 

in x direction from 5 to 15 nm 

         base_y     = [500+($Dot_top/2)+$SIZE,1000]       # base line has 

a constant y coordinate y = 25 nm 

         top_x      = [ $CAP+$DOT+$SIZE, $CAP+$DOT+$SIZE]       # top 

line extends in x direction from 8 to 12 nm 

         top_y      = [500+($Dot_bottom/2)+$SIZE,1000]       # top line 

has a constant y coordinate y = 30 nm 

         } 

       binary{ name = "GaAs" } 

      } 

 

   #  Top gate (Schottky contact) 

    region{  

      rectangle{ x = [0, 1] y=[0,1000]} 



60 

 

      contact{ name = "gate" } 

   } 

 

   # Back gate (Ohmic contact) 

   region{  

      rectangle{ x = [390, 400] y=[0,1000]} 

      contact{ name = "backgate" } 

   } 

}<> 

 

 

classical{ 

   Gamma{} 

   L{} 

   X{} 

   HH{} 

   LH{} 

   SO{} 

 

   output_bandedges{averaged = no} 

   output_carrier_densities{} 

   output_ionized_dopant_densities{} 

   output_intrinsic_density{} 

   output_bandgap{} 

}<> 

 

 

quantum { 

   region{ 

      name = "DOT" 

      x = [0, 100] y=[500-$Dot_top,500+$Dot_top] 

      Gamma{ num_ev = $NUMEV } 

      HH { num_ev = $NUMEV } 

      LH { num_ev = $NUMEV } 

      output_wavefunctions{ probabilities = no  max_num = $NUMEV } 

   } 

}<> 

 

 

poisson{ 

   newton_solver{ residual = 1e-9 } 

   output_potential{} 

   output_electric_field{} 

}<> 

 

 

currents{ 

   insulator_bandgap = 1 

   output_fermi_levels{} 

}<> 

 

 

output{ 

 

  # format2D  = Origin 

   section1D{                 # output a 1D section of the simulation 

area (1D slice) 

      name = "x1"              # name of section enters file name 

 

      y = 500-($Dot_top/2)    # 1D slice at y = 100 nm  



61 

 

   } 

 

   section1D{                 # output a 1D section of the simulation 

area (1D slice) 

      name = "x2"              # name of section enters file name 

 

      y = 500.0               # 1D slice at y = 100 nm  

 

   } 

   section{   name    = "zoom" 

              range_x = [0,100]          # x = [0, 100] y=[500-

$Dot_top,500+$Dot_top] 

              range_y = [500-$Dot_top,500+$Dot_top] }  

}<> 

 

run{ 

   solve_poisson{} 

   solve_quantum{} 

   outer_iteration{} # solve coupled Schrödinger-Poisson equations 

}<>