THEORETICAL MODELING OF QUANTUM DOT IN A WELL INFRARED PHOTODETECTORS

By

Soumitra Roy Joy (0606001) Golam Md. Imran Hossain (0606029) Tonmoy Kumar Bhowmick (0606049)

A thesis submitted for the Degree of Bachelor of Science 2012

Department of Electrical and Electronic Engineering Bangladesh University of Engineering and Technology

To our loving parents.

Acknowledgements

We would like to acknowledge the various supports that we have received throughout my research work and in writing this dissertation. We would like to thank our supervisor, Dr. Farseem Mannan Mohammedy, for trusting us with this work and giving us the opportunity to be involved in such a sophisticated and interesting topic of research. We would like to thank him for proposing many of the ideas presented here, which we was able to study in depth. Prof. Mohammedyhas been nothing short of a great mentor to us, offering his guidance throughout the intricacies of this research. The amount of energy and devotion he allocates for his work never ceases to amaze us, and he continues to set an example in organization, professionalism, and scholarly prowess that is very hard to follow.

Abstract

Quantum dot in a well (DWELL) infrared photodetectors have emerged as a promising technology in the mid- and far-infrared (3 − 25µm) for medical and environmental sensing that have a lot of advantages over current technologies based on Mercury Cadmium Telluride (MCT) and quantum well (QW) infrared photodetectors (QWIPs). In addition to the uniform and stable surface growth of III/V semiconductors suitable for large area focal plane applications and thermal imaging, the three dimension confinement in QDs allow sensitivity to normal incidence, high responsivity, low darkcurrent and high operating temperature. The growth, processing and characterizations of these detectors are costly and take a lot of time. So, developing theoretical models based on the physical operating principals will be so useful in characterizing and optimizing the device performance. Theoretical models based on non-equilibrium Green’s functions have been developed to electrically and optically characterize different structures of DWELL s. The advantage of the model over the previous developed classical and semiclassical models is that it fairly describes quantum transport phenomenon playing a significant role in the performance of such nano-devices and considers the microscopic device structure including the shape and size of QDs, heterostructure device structure and doping density. The model calculates the density of states from which all possible energy transitions can be obtained and hence obtains the operating wavelengths for intersubband transitions. The responsivity due to intersubband transitions is estimated and the effect of having different sizes and different height-to-diameter ratio QDs can be obtained for optimization.

Theoretical modeling developed in the thesis give good description to the QDIP different characteristics that will help in getting good estimation to their physical performance and hence allow for successful device design with optimized performance and creating new devices, thus saving both time and money.

Contents Contents

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List of Figures

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Nomenclature

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1 Introduction 1.1 Quantum Dots . . . . . . . . . . . . . . . . . . 1.2 Quantum Dot Infrared Photodetectors (QDIPs) 1.3 Importance of DWELL Design . . . . . . . . . . 1.4 Multi-Band Radiation Detection . . . . . . . . . 1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . 1.6 Thesis Objective . . . . . . . . . . . . . . . . .

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2 Non-equilibrium Green’s functions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminary Concept . . . . . . . . . . . . . . . . . . . . . 2.3 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Why do we want to calculate the Greens function? 2.3.2 Self-Energy . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The spectral function . . . . . . . . . . . . . . . . . 2.4 Response to an incoming wave . . . . . . . . . . . . . . . . 2.5 Charge density matrix . . . . . . . . . . . . . . . . . . . . 2.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 2.7 The semiclassical limit . . . . . . . . . . . . . . . . . . . .

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CONTENTS 2.8 2.9 2.10 2.11

Quantum well structures: Why NEGF is essential Limits and simplifications in NEGF . . . . . . . . THz quantum cascade lasers: a classics for NEGF Comparison between NEGF and Monte -Carlo . . 2.11.1 Monte-Carlo . . . . . . . . . . . . . . . . . 2.12 Why nonequilibrium Green functions? . . . . . . 2.13 Conclusions . . . . . . . . . . . . . . . . . . . . .

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3 QDIPs density of states modeling 31 3.1 DWELL QDIP Structure . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Calculation of Self Energy: . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Computational Challenges . . . . . . . . . . . . . . . . . . . . . . 39 3.5 How to confront the challenges . . . . . . . . . . . . . . . . . . . . 41 3.5.1 Reduce the number of grid . . . . . . . . . . . . . . . . . . 41 3.5.2 Non-uniform distribution of grid point . . . . . . . . . . . 41 3.5.3 Disintegrating a large matrix into several smaller matrices 42 3.5.4 Dealing with Sparse Matrix other than conventional matrix 43 3.5.5 Using all the processors in a computer in a parallel process 43 3.6 Result- Our Findings . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Comparing Experimental results with simulated ones . . . . . . . 47 4 Dipole Moment and Absorption Coefficient 4.1 Transitional Dipole Moment . . . . . . . . . . . . . 4.2 Optical Transition using Fermi’s Golden Rule . . . 4.3 The Electron-Photon Interaction Hamiltonian . . . 4.4 Transition Rate due to Electron-Photon Interaction 4.5 Optical Absorption Coefficient . . . . . . . . . . . . 4.6 Wavefunction Calculation . . . . . . . . . . . . . . 4.7 Our Findings and Results . . . . . . . . . . . . . . 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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5 Conclusions 5.1 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS 5.2

FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 INTERMIXING AND STRAIN EFFECT . . . . . . . . . 5.2.2 NON-UNIFORMITY OF QDS SIZE, AND DISTRIBUTION 5.2.3 INTERACTION WITH THE LATTICE . . . . . . . . . . 5.2.4 TRANSIENT RESPONSE . . . . . . . . . . . . . . . . . .

References

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List of Figures 1.1 1.2 1.3 1.4 2.1 2.2

2.3

Band structure of a GaAs/AlGaAs Quantum well . . . . . . . . . Density of states, bandstructure and carrier distribution for (a) bulk, (b) quantum well, (c) quantum wire and (d) quantum dots. Band structure of . . . . . . . . . . . . . . . . . . . . . . . . . . Various transitions in the dots-in-well (DWELL) QDIP . . . . . .

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System Contact with Channel . . . . . . . . . . . . . . . . . . . . (Color online) Left: Calculated carrier density for a homogeneously doped semiconductor at zero bias, attached to leads with an equilibrium distribution. The expected homogeneous density is obtained only if the leads include the same type of scattering self energies as the device (red curve). Otherwise one obtains artificial charge accumulation (blue curve). Right: GaAs n-i-n structure at room temperature under bias with asymmetric doping profile as indicated by the grey lines.Once a current is flowing. the charge distribution within the leads must be a suitably shifted Fermi distribution that reflects global current conservation (results shown by thc red curve). Otherwise, NEGF calculations yield artificial pinch-off effects (blue lines) . . . . . . . . . . . . . . . . . . . . . Comparison of fully self-consistent NEGF and semiclassical carrier dynamics calculations for a standard GaAs resistor (50 nm n-i-n structure) at zero bias and room temperature . . . . . . . . . . .

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LIST OF FIGURES 2.4

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(Color onlinc) Carrier dynamics calculation for 50 nm n-i-n structure at room tcmperature with a 12 nm InGaAs quantum well as intrinsic zone attached to field-free GaAs leads of the same ndensity.The applicd voltage is 150 mV across thc structure.Red curve: calculalion in terms of charge-self-consistent semiclassical Boltzmann equation.Blue curve: Calculation in terms of strictly ballistic NEGF, equivalent to thc solution of Schrodinger equation of open system. Black curve: Fully self-consistent NEGF calculation Contour graph of calculated energy resolved electron density for 50 nm n-i-n structure at room temperature with a 12 nm lnGaAs quantum well as intrinsic zone for zero applied bias. The density scale is the analogous to the one in Fig. 5. but for lower doping. (a) Strictly ballistic NEGF calculation (no scattering included). (b) Fully self-consistent NEGF calculation . . . . . . . . . . . . . Contour graph of calculated energy resolved electron density for 50 nm n-i-n structure at room temperature with a 12 nm InGaAs quantum well as intrinsic zone for zero applied bias. (a) Fully self consistent NEGF calculation. (b) NEGF calculation with coupling Comparison between experimental (Ref. [27]) and calculated (Ref. ([13]) current-voltage characteristics for AIGaAs/GaAs quantum cascade structure . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-section schematic of a 10 layer InAs/InGaAs quantum dot in a well detector, . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-section TEM image of a single QD layer of DWELL [14] . . Cross-section TEM image of an InAs/InGaAs DWELL Heterostructures [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Modeling of DWELL QDIP Structure . . . . . . . . . Conduction band offsets and energy levels of QDWELL [14] . . . Cross section of a cylinder is disintegrated into a number of grids Device structure with grid division . . . . . . . . . . . . . . . . . Device cross section . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF FIGURES 3.9

3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 4.1 4.2

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Progressive red shift in the peak operating wavelength of the detector as the width of the bottom InGaAs is increased from 10 to 60 angstrom. The spectre has been vertically displaced for clarity. [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicolour response in the mid-wave, long-wave and very longwavelength regimes with the associated transitions in the inset.[11] The very long-wave infrared (VLWIR) response was observed till 80 K [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DOS profile for bottom well width=6nm . . . . . . . . . . . . . . Effect of changing well width on DOS . . . . . . . . . . . . . . . . Effect of changing well width on DOS . . . . . . . . . . . . . . . . Potential Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . LDOS at E=-270.3208mev . . . . . . . . . . . . . . . . . . . . . . LDOS at E=-61.8225mev . . . . . . . . . . . . . . . . . . . . . . . LDOS at E=-15mev . . . . . . . . . . . . . . . . . . . . . . . . . LDOS at E=77mev . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dipole moment versus wavelength for S polarized incident light in a standard dot-in-a-well infrared photodetector. QD has Height/Base=6.5nm/11nm 64 Absorption coefficient versus wavelength for S polarized incident light in a standard dot-in-a-well infrared photodetector. QD has Height/Base=6.5nm/11nm . . . . . . . . . . . . . . . . . . . . . . 65 Comparison of absorption coefficient of different hypothetical QDIPs having various Height to Base ratio. Incident Light is S polarized (in plane incidence) . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Comparison of absorption coefficient of different hypothetical QDIPs having various Height to Base ratio. Incident Light is P polarized (45 degree incidence to growth plane) . . . . . . . . . . . . . . . . 66 Relative change of absorption peak with dot dimension change. Absorption peak found for 7nm QD base is taken as unity, when incident light is S polarized. . . . . . . . . . . . . . . . . . . . . . 67

LIST OF FIGURES 4.6

Relative change of absorption peak with dot dimension change. Absorption peak found for 7nm QD base is taken as unity, when incident light is P polarized. . . . . . . . . . . . . . . . . . . . . .

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Chapter 1 Introduction High performance Infrared photodetectors in the mid- and far-infrared (3- 25 m) wavelength range have attracted much interest due to their important sensing applications . They are used in medical and environmental sensing, optical communications, thermal imaging, night vision cameras, and missile tracking and recognition. It is required to obtain a technology that gives high performance at high operating temperature and with low cost. Current technologies based on Mercury Cadmium Telluride (MCT) and quantum well (QW) infrared detectors (QWIPs) have some disadvantages that lower the overall performance of the sensing devices. The MCTs epitaxial growth problems limit the manufacturing yield of large area focal plane arrays (FPAs) applications. QWIPs do not support normal incidence detections and so need complicated optical coupling in addition to the requirement of operating at very low temperature . The advance in epitaxial growth of heterostructure semiconductors allows for the fabrication of devices at nano scale dimensions. These nano-devices have new physical operating principles and novel performance characteristics. Quantum dots (QDs) grown by the self-assembled epitaxial technique have attracted much interest in recent years for laser and photodetector applications. In addition to the low cost, stable and uniform surface epitaxial growth of the III/V semiconductors, suitable for large area FPAs application in thermal imaging, the three-dimensional confinement in QDs has many advantages such as the intrinsic sensitivity to normal-incidence light which simplifies the optical configuration for any application, reduced electronphonon scattering, long-lived excited states, low dark current and high temper-

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CHAPTER 1. INTRODUCTION

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atureoperation . These advantages make quantum dot infrared photodetectors (QDIPs) emerge as an alternative technology to replace QWIP and MCT infrared detectors. Therefore, improving the QDIP performance by optimizing the device design through accurate modeling is useful for obtaining the required characteristics. In this research work, theoretical models based on non-equilibrium Greens functions will be developed to describe the electrical and optical characteristics of QDIPs. The model results will be compared to the available experimental results and the models will be used for optimizing the device performance.

1.1

Quantum Dots

A quantum well (QW) is a thin layer which can confine electrons or holes in the dimension perpendicular to the layer surface, while the movement with in the layer is not restricted . A QW is formed when a lower band gap material is sandwiched between higher band gap materials. Figure illustrates the band structure of a GaAs/AlGaAs11 based QW . As seen in the figure, the electron is confined in the z-direction, or normal to the surface of the layer, by the QW. As a layer thickness approaches de-Broglie wavelength (i.e. about 10 nm), quantum effects can be seen. Therefore, usually the QW thickness is in the order of 1 to 15 nm. Similarly, in the quantum dot the carrier is confined in all three dimensions. The change in confinement can be better understood if we compare the density of states for bulk (0-dimension), QW (1-dimension), Quantum dash or wire (2dimensions) and QD (3-dimensions). As seen in Figure, the density of states function for bulk is continuous and proportional to the square root of energy . The density of states decreases for QW compared to the bulk and is a step function. For the quantum wire the density of states further decreases compared to the QW. For QDs, the density of states decreases compared to a quantum wire and is a delta function in energy. For real devices made of QDs, however, the density of states has a line broadening due to variations in dot size. The low density of states and small size of the dots means that fewer carriers are needed to invert the carrier population, which results in low threshold current density and high characteristic temperature when incorporated as active region in a laser. In terms of detector, the absorption of the dots can be easily saturated due to the

CHAPTER 1. INTRODUCTION

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Figure 1.1: Band structure of a GaAs/AlGaAs Quantum well

finite density of states.

1.2

Quantum Dot Infrared Photodetectors (QDIPs)

The three-dimensional confinement of QDs helps in localization of carriers reducing the thermionic emission which in turn lowers the dark current . The intersubband energy level spacing in the QDs is greater than the phonon energy and, therefore, reduces the phonon scattering, which is a dominant scattering mechanism in bulk and QWs. This is the reason for long carrier relaxation times in QDIPs, which in turn increases the photoconductive gain. The responsivity and detectivity are also increased due to the increase in gain and photocurrent . In addition, QDIPs are sensitive to normal incidence radiation, which is not possible in QWIPs, due to polarization selection rules, and requires specialized gratings to direct the radiation into the detector. The QDs are normally doped to about than 1-2 electrons per dot in order to prevent carriers from occupying the excited state which will increase the dark current. The thickness of the barriers surrounding the quantum dots, the doping concentration, and the number

CHAPTER 1. INTRODUCTION

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Figure 1.2: Density of states, bandstructure and carrier distribution for (a) bulk, (b) quantum well, (c) quantum wire and (d) quantum dots.

CHAPTER 1. INTRODUCTION

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of quantum dot layers are important parameters to consider while designing a QDIP. QDIPs suffer from low QE due to low absorption cross section resulting from low density of QDs and finite spacing between the dots . This thesis discusses different ways to improve the quantum efficiency and other performance parameters like responsivity, detectivity, operating temperature, of QDIPs based on dots-in-well (DWELL) design.

1.3

Importance of DWELL Design

In the DWELL design, the active region consists of 2.4 MLs of InAs QDs placed in an 11nm In0.15Ga0.85As QW sandwiched between 50 nm thick GaAs barriers, which in turn is placed in a GaAs matrix. The DWELL design is shown in Figure 1.6. As seen in the figure, there is a large conduction band offset (i.e. 250 meV) between the ground electronic state of the InAs QD and the conduction band edge of the GaAs barrier, which reduces the thermionic emission and therefore low dark current . Due to low dark current, QDIPs are expected to have higher operating temperatures than QWIPs. The total band offset is calculated from photoluminescence (PL) spectrum and using the 60-40split. The excited state is obtained from spectral response and theoretical modeling . The spacing between the ground electronic state and excited state is found to be about 50 -60 meV. The PL spectrum of the 10-period InAs/ In0.15Ga0.85As dots-in-well (DWELL) QDIP is shown in Figure . Infrared detectors based on DWELL design primarily work on bound-tobound transitions from the ground electronic state of the InAs QD to the In0.15Ga0.85As QW and bound-to-continuum transition from the ground electronic state of the InAs QD to a state in the GaAs barrier as illustrated in Figure . Depending on the bias, one of the transitions is observed in the spectral response for the detector. A new alternative DWELL design (InAs QDs placed in a GaAs QW) is investigated. In the standard design (InAs QDs placed in an In0.15Ga0.85As QW) the average strain is very high and many DWELL layers cannot be grown without introducing dislocations, which will lower the performance of the detectors.

CHAPTER 1. INTRODUCTION

Figure 1.3: Band structure of InAs/In0.15 Ga0.85 As DWELL

Figure 1.4: Various transitions in the dots-in-well (DWELL) QDIP

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Hence, the new design is for increasing the number of DWELL layers in order to increase the overall absorption, which in turn increases the photocurrent and detectivity. These three approaches have significantly improved the performance of DWELL detectors.

1.4

Multi-Band Radiation Detection

The rapid development of infrared (IR) detector technology, which primarily includes device physics, semiconductor material growth and characterization, and microelectronics, has led to new concepts like target recognition and tracking systems.1, 2 Among these concepts, multi-band radiation detection is being developed as an important tool to be employed in many practical applications. Detecting an objects infrared emission at multiple wavelengths can be used to eliminate background effects,3 and reconstruct the objects absolute temperature4 and unique features. This plays an important role in differentiating and identifying an object from its background. However, measuring multiple wavelength bands typically requires either multiple detectors or a single broad-band detector with a filter wheel coupled to it in order to filter incident radiation from different wavelength regions. Both of these techniques are associated with complicated detector assemblies, separate cooling systems, electronic components, and optical elements such as lenses, filters, and beam splitters. Consequently, such sensor systems (or imaging systems) involve fine optical alignments, which in turn require a sophisticated control mechanism hardware. These complications naturally increase the cost and the load of the sensor system, a problem which can be overcome by a single detector responding in multiple bands. The multi-spectral features obtained with multi-band detectors are processed using color fusion algorithms1 in order to extract signatures of the object with a maximum contrast. With the development of multi-band detector systems, there is an increased research1, 5 effort to develop image fusion techniques. Fay et al.1 have reported a color-fusion technique using a multi-sensor imagery system, which assembled four separate detectors operating in different wavelength regions. The major goal of my study is to investigate multi-band detection concepts and develop high performance multi-band detectors.

CHAPTER 1. INTRODUCTION

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At present, there are many applications where multi-band detectors are required. In land-mine detection3 the number of false positives can be reduced using multi-spectral approaches, allowing the the identification of real land-mine sites. Military applications include the use of multi-band detectors to detect muzzle flashes, which emit radiation in different wavelength regions,6 to locate the position of enemy troops and operating combat vehicles. Multi-band focal plane arrays (FPAs) responding in very-long-wavelength infrared (VLWIR) region (1430 m) can be used for space surveillance and space situational awareness,2 where observations of extremely faint objects against a dark background are required. Present missile-warning sensors are built focusing on the detection of ultraviolet (UV) emission by missile plume. However, with modern missiles, attempting to detect the plume is impractical due to its low UV emission. As a solution, IR emission7 of the plume can be used instead of UV. Then the detector system should be able to distinguish the missile plume against its complex background, avoiding possible false-alarms. Thus, a single band detector would not be a choice to achieve this. Using a two-color (or multi-color) detector, which operates in two IR bands where the missile plume emits radiation, the contrast between the missile plume and the background can be maximized. Moreover, a multi-band detector can be used as a remote thermometer4 where the objects radiation emission in the two wavelength bands is detected by a multi-band detector and the resulting two components of the photocurrent can be solved to extract the objects temperature.

1.5

Thesis Objective

The objectives are to develop theoretical models to well describe the electrical and optical properties of QDIP which can be used for device design optimization for better performance. Improving the device performance experimentally by fabricating and testing devices using combinations of different design parameters are very costly and time wasting. Hence, it is desirable to develop theoretical modeling based on the physical operating principals that can be used in characterization and optimizing the device performance through recommending the best design parameters suitable to achieve specific characteristics.

CHAPTER 1. INTRODUCTION

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The density of states of the QDs that gives both the discrete energy levels in the QDs in addition to the continuum states outside the QDs.

• The energy levels provided by the density of states give information about the possible energy transitions and therefore the operating wavelengths of the detector. • The corresponding calculated wavefunctions are used to calculate the dipole moment between different energy states which indicate the strength of the transition rates between the energy states and therefore gives information about the relative peak of the absorption co-efficient of the detector. • The effect of changing the shape and size of QDs has been studied to establish their effects on the operating wavelength and the corresponding value of the absorption co-efficient. The research work presented in this thesis has resulted in the following publications: 1. Soumitra Roy Joy, Golam Md. Imran Hossain, Tonmoy Kumar Bhowmick, Farseem Mannan Mohammedy, ”Influence of Quantum Dot Dimensions in a DWELL Photodetector on Absorption Co-efficient”, in Proc. Fifth Asia International Conference on Mathmematical Modeling and Computer Simulation (AMS), Indonesia, 2012, page no. 225-230. 2. Soumitra Roy Joy ; Tonmoy Kumar Bhowmick;Golam Md. Imran Hossain; Farseem Mannan Mohammedy, ”Effect of Asymmetric Well Quantum dots-in-a-well in-frared photodetectors on Density of States Using NEGF Formalism”, in Proc. International Conference on Solid State Devices and Materials Science, April 1-2, 2012, China, page no. 299-304.

1.6

Thesis Outline

The thesis contains five chapters. Chapter 1, above, discussed the different applications of infrared detectors in medical and environmental sensing. The QDIP is a

CHAPTER 1. INTRODUCTION

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promising technology with advantages over current technologies based on QWIP and MCT. A literature review of QDIP modeling, including classical and semiclassical methods, has been discussed in addition to an overview of the NEGF modeling that has been developed in the thesis. Chapter 2 gives a review of the NEGF. The different Green’s functions, the self energy, and scattering functions are presented as they will be used intensively in the following chapters of the thesis. Some applications of Green’s function have been presented. Also a comparative view between NEGF and Monte Carlo method of analyzing device is rendered, incorporating their respective advantages, approximation etc. Chapter 3 shows the development of theoretical modeling to obtain the DOS of the QDIP. The localized DOS is obtained from the retarded Green’s function. The retarded Green’s function is obtained numerically by solving the governing kinetic equation using the method of finite differences. The model was applied to calculate the DOS of DWELL (Dot in a Well) structure. Chapter 4 shows the development of theoretical modeling to calculate the dipole moment and absorption co-efficient of the QDIP. The first order dipole approximation and the Fermi-golden rule were used to model the interaction with light. The bound states of the QDs have been obtained by solving the eigenvalue problem of the QD Hamiltonian, while the continuum states have been obtained using the retarded Green’s function. The model has been applied to the QDWELL structure. The effect of changing the shape and size of QDs on the calculated absorption co-efficient has been studied using the DWELL structure. Finally Chapter 5 concludes the thesis with the major findings and the recommended improvements and extensions for future research.

Chapter 2 Non-equilibrium Green’s functions 2.1

Introduction

A proper treatment of quantum transport is one of the most difficult problems to deal with in solid state theory. While there have been many models and concepts developed to deal with particular aspects of quantum transport, the most general and rigorous theoretical framework is provided by the so-called non-equilibrium Green’s function theory (NEGF) developed by Keldysh and in a slightly different form independently by Kadanoff and Baym . It took an amazing 40 years before this method became recognized and employed as the framework of choice for a quantitative and predictive analysis of carrier dynamics in semiconductor based nanostructures. There are probably several reasons why it took so long. First if all, there are only few devices where quantum mechanical effects and incoherent scattering effects play an equally important role and call for a fully quantum mechanical treatment. The most prominent examples are quantum cascade lasers (QCL) invented by Capasso et al. in 2004. Particularly in the THz regime. The physics of QCLs is controlled by a carefully balanced competition between coherent tunneling and incoherent phonon emission processes. Secondly, NEGF calculations for realistic devices are extremely demanding computationally and have only become feasible recently. One of the first detailed implementations

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12

of NEGF to semiconductors and semiconductor devices has been developed by Lake and coworkers and applied to NEMO-I D, a sophisticated simulation tool for multi-q uantum well devices such as resonant tunneling diodes . In their research, the authors laid down the framework for applying NEGF to semiconductors by deriving most scattering self-energies that are relevant for semiconductors. Another major step forward was provided by Wacker et al., who provided several in-depth NEGF studies of the carrier dynamics and optical propen ies of QCLs. Currently, an increasing number of groups are employing and expanding the NEGF formalism to study quantum transpon aspects of modem semiconductor nanostructures. Nevertheless, the fommlism still suffers from being considered somewhat obscure and difficult to grasp and handle. we will allermpt to give an overview of the basic elements of the fommlism for nonexperts that may help to develop a beuer feeling for strengths and weaknesses of NEGF. Let us emphasize that NEGF is a solution framework rather than a concrete method for calculating properties of open quantum devices. It ensures that the non-equilibrium carrier distribution in a device is consistently calculated with the energy, width and occupancy of its quantum mechanical eigenstates (scattering states. to be precise). Consentration of charge, momentum and energy is guaranteed only if the scattering self energies are calculated exactly and self consistently, i.e. using fully dressed Green’s functions and vertices in many-body terms. Thus. the unavoidable use of approximations requires much more effort and care than in other methods just to avoid artifacts such as a violation of current conservation within the device. The NEGF method is able to deal with explicitly timedependent as well as with stationary problems. However. Ihe calculation of the required set of 4 types of Green’s functions for time-dependent carrier dynamics is still pretty much Oul of reach for a quantitative prediction of realistic device structures. Therefore. we limit the present discussion of stationary problems where 2 types of Green’s functions suffice. Non equilibrium Greens function methods are regularly used to calculate current and charge densities in nanoscale (both molecular and semiconductor) conductors under bias. An overview of the theory of molecular electronics can be found in [12] and for semiconductor nanoscale devices see [16]. The aim of this text is to provide some intuitive explanations of one particle Greens functions in

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

13

a compact form together with derivations of the expressions for the current and the density matrix. It is not intended as a complete stand-alone tutorial, but rather as a complement to [13],[16],[4],[3].

2.2

Preliminary Concept

A proper treatment of quantum transport is one of the most difficult problems to deal with in solid state theory. With aggressive device scaling, quantum mechanical phenomenon has become prominent for these nano-scale devices and hence quantum treatment in modeling has become a necessity. Self-consistent Schrdinger-Poisson solver is generally used for nano devices at equilibrium or very close to equilibrium. One would start by solving Schrdinger’s wave equation for the eigenstates ψ −~2 2 ∇ ψ(r) − qV (r)ψ(r) = ψ(r) (2.1) 2m∗ The eigenstates are filled according to Fermi’s function, and their squared amplitudes give the charge density, taking into account the spin degeneracy: n(r) = −2 ∗

X

|ψi (r)|2 f ((i − EF )/kT )

(2.2)

i

where: f (x) = (1 + exp(x))−1

(2.3)

The calculated charge density is then used in Poisson’s equation: −∇.(s ∇(r)) = q(n(r) + N (r))

(2.4)

and the potential is substituted back into Equation (1) until convergence. Once the potential and charge profiles have been calculated, a suitable transport model is used. For devices far from equilibrium, the NEGF has become the method of choice.

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

14

For closed systems, (channel) the governing equation is [EI − H] {ψ} = 0

(2.5)

Where H is the unperturbed Hamiltonian of the system, E is the energy of electron, I is an identity matrix, ψ is the eigenstate or the electron wave function. If we consider an open system, we need to modify the governing equation as following [EI − H − Σ] {ψ} = {S}

(2.6)

Here, Σ is the self-energy of different processes such as the contacts or the scattering centers within the device. S is the source term which tells us how electrons are getting into the system from outside or the contact.

Figure 2.1: System Contact with Channel

So, the wave function can be calculate as {ψ} = [G] {S}

(2.7)

[G] = [EI − H − Σ]−1

(2.8)

And

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

15

G is called ”Green’s function” To calculate ψ , one has to formulate the Hamiltonian matrix, H; then insert the self-energy,Σ ; calculate the Green’s function, G and lastly multiply G with the source term. Rigorous definition and justification are covered in the next section.

2.3

Green’s functions

Discrete Schrodinger equation: H |ni = E |ni

(2.9)

We divide the Hamiltonian and wavefunction of the system into contact (H1,2, |ψ1,2 i) and device (Hd , |ψd i) subspaces: We define the Greens function1 : (E − H)G(E) = I

2.3.1

(2.10)

Why do we want to calculate the Greens function?

The Green’s function gives the response of a system to a constant perturbation |vi in the Schrodinger equation: H |ψi = E |ψi + |vi

(2.11)

The response to this perturbation is: (E − H) |ψi = − |vi

(2.12)

|ψi = −G(E) |vi

(2.13)

Why do we need the response to this type of perturbation? Well, it turns out that its usually easier (see next section) to calculate the Greens function than 1

Others may (and do) use the opposite sign.

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

16

solve the whole eigenvalue problem1 and most (all for the one-particle system) properties of the system can be calculated from the Greens function. e.g., the wavefunction of the contact |ψ2 i can be calculated if we know the wavefunction on the device |ψd i. From third row of Eq. 2: H2 |ψ2 i + τ2 |ψd i = E |ψ2 i →

(2.14)

(E − H2 ) |ψ2 i = τ2 |ψd i →

(2.15)

|ψ2 i = g2 (E)τ2 |ψd i →

(2.16)

where g2 is the Greens function of the isolated contact 2 ((E − H2 )g2 = I). It is important to note that since we have an infinite system, we obtain two types of solutions for the Greens functions 2 , the retarded and the advanced3 solutions corresponding to outgoing and incoming waves in the contacts. Notation: We will denote the retarded Greens function with G and the advanced with G+ (and maybe GR and GA occasionally).Here, CAPITAL G denotes the full Greens function and its sub-matrices G1 , Gd , G1d etc. Lowercase is used for the Greens functions of the isolated subsystems, e.g., (E − H2 )g2 = I. Note that by using the retarded Greens function of the isolated contact (g2 ) in Eq. 9 we obtain the solution corresponding to a outgoing wave in the contact. Using the advanced Greens function (g2+ ) would give the solution corresponding to an incoming wave.

2.3.2

Self-Energy

The reason for calculating the Greens function is that it is easier that solving the Schrodinger equation. Also, the Greens function of the device (Gd ) can be 1

Especially for infinite systems. When the energy coincides with energy band of the contacts there are two solutions corresponding to outgoing or incoming waves in the contacts. 3 In practice these two solutions are usually obtained by adding an imaginary part to the energy. By taking the limit to zero of the imaginary part one of the two solutions is obtained. If the limit → 0+ is taken the retarded solution is found, → 0− gives the advanced. This can be seen from the Fourier transform of the time dependent Greens function. 2

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

17

calculated separately without calculating the whole Greens function (G). From the definition of the Greens function we obtain: Selecting the three equations in the second column: (E − H1 )G1d − τ1 Gd = 0

(2.17)

−τ1+ G1d + (E − Hd )Gd − τ2 G2d = I

(2.18)

(E − H2 )G2d − τ2 Gd = 0

(2.19)

We can solve Eqs. 11 and 13 for G1d andG2d : G1d = G1 τ1 Gd

(2.20)

G2d = G2 τ2 Gd

(2.21)

substitution into Eq. 12 gives: −τ1 + g1 τ1 Gd + (E − Hd )Gd − τ2+ g2 τ2 Gd = I

(2.22)

from which (Gd ) is simple to find: Gd = (E − Hd − Σ1 − Σ2 )−1

(2.23)

where Σ1 = τ1+ g1 τ1 and Σ2 = τ2+ g2 τ2 are the so called self-energies. Loosely one can say that the effect of the contacts on the device is to add the self-energies to the device Hamiltonian since when we calculate the Greens function on the device we just calculate the Greens function for the effective Hamiltonian Hef f ective = Hd + Σ1 + Σ2 .However, we should keep in mind that we can only do this when we calculate the Greens function. The eigen-values and -vectors of this effective Hamiltonian are not quantities we can interpret easily. For normal contacts, the surface Greens functions g1 and g2 used to calculate the self-energies are usually calculated using the periodicity of the contacts, this method is described in detail in appendix B of [4] and in section 3 of [16].

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

2.3.3

18

The spectral function

Another important use of the Greens function is the spectral function: A = i(G − G+ )

(2.24)

which gives the DOS and all solutions to the Schrodinger equation! To see this we first note that for any perturbation |vi we get two solutions  R ( ψ and ψ A ) to the perturbed Schrodinger equation: (E − H) |ψi) = |−vi

(2.25)

from the advanced and retarded Greens functions: R ψ = −G |vi

(2.26)

A ψ = −G+ |vi

(2.27)

  The difference of these solutions( ψ R − ψ A = (G − G+ ) |vi = −iA |vi) is a solution to the Schrodinger equation:

  (E − H)( ψ R − ψ A ) = (E − H)(G − G+ ) |vi = (I − I) |vi = 0

(2.28)

which means that |ψi = A |vi is a solution to the Schrodinger equation for any vector |vi. To show that the spectral function actually gives all solutions to the Schrodinger equation is a little bit more complicated and we need the expansion of the Greens function in the eigenbasis: G=

X |ki hk| 1 = E + iδ − H E + iδ − k k

(2.29)

where the δ is the small imaginary part (see footnote 3),|vi s are all eigenvectors1 to H with the corresponding eigenvalues k . Expanding the spectral function 1

Normalized!

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

19

in the eigenbasis gives: A = i(

A=i

X

1 1 − ) E + iδ − H E − iδ − H

|ki hk| (

k

A=

X

1 1 − ) E + iδ − k E − iδ − k

|ki hk|

k

2δ (E − k )2 + δ 2

(2.30)

(2.31)

(2.32)

where δ is our infinitesimal imaginary part of the energy. Letting δ go to zero gives: A = 2π

X

δ(E − k ) |ki hk|

(2.33)

k

(here δ(E −k ) is the delta function) which can be seen since (E−2δ 2 2 goes to k ) +δ zero everywhere but at E = k ,integrating over E (with a test function) gives the 2πδ(E − k ) factor. Eq. 27 shows that the spectral function gives us all solutions to the Schrodinger equation.

2.4

Response to an incoming wave

In the non-equilibrium case, reservoirs with different chemical potentials will inject electrons and occupy the states corresponding to incoming waves in the contacts. Therefore, we want to find the solutions corresponding to these incoming waves. Consider contact 1 isolated from the other contacts and the device. At a certain energy we have solutions corresponding to an incoming wave that is totally reflected at the end of the contact. We will denote these solutions with |ψ1,n i where 1 is the contact number and n is a quantum number (we may have several modes in the contacts). We can find all these solutions from the spectral function a1 of the isolated contact (as described above). Connecting the contacts to the device we can calculate the wavefunction on the whole system caused by the incoming wave in contact 1. To do this we note

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

20

that a wavefunction should be of the form

2.5

Charge density matrix

In the non equilibrium case we are often interested in two quantities: the current and the charge density matrix. Lets start with the charge density (which allows us to use a self-consistent scheme to describe charging). The charge density matrix is defined as: ρ=

X

f (k, µ) |ψk i hψk |

(2.34)

k

where the sum runs over all states with the occupation number f (Ek , µ) (pure density matrix) (note the similarity with the spectral function A, in equilibrium you find the density matrix from A and not as described below). In our case, the occupation number is determined by the reservoirs filling the incoming waves in the contacts such that: 1

f (Ek , µ1 ) =

1+e

Ek −µ1 kB T

(2.35)

is the Fermi-Dirac function with the chemical potential µ1 and temperature (T) of the reservoir responsible for injecting the electrons into the contacts. The wavefunction on the device given by an incoming wave in contact 1 (see Eq. 32) is: |ψd,k i = Gd τ1+ |ψ1,k i

(2.36)

Adding up all states from contact 1 gives:

Z

∞

ρd [contact1] =

dE E=−∞

X k

f (E, µ1 )δ(E − Ek ) |ψd,k i hψd,k |

(2.37)

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

2.6

21

Boundary conditions

In quantum transport, the treatment of boundary conditions requires significantly more care than in classical physics due to the nonlocality of quantum mechanics.[20],[11],[23],[17]. A common problem is to define Ohmic leads. In the context of NEGF, we may define leads as Ohmic if the current is controlled by the interior of the device rather than by serial resistances or interface states. This definition has several implications. First, there must be a smooth transition in the density of states of the leads and the device. Secondly, the same scattering mechanisms must act within the leads and the device. Thirdly, the carrier distribution within the leads must be a suitably accelerated Fermi distribution to reflect current conservation. All of these conditions are necessary to avoid quantum mechanical reflections and pile up of charge at the interface.[21],[24] This is illustrated in Figure . Figure shows a full NEGF calculation of the carrier density in a homogeneously n-doped piece of GaAs for zero bias as a function of position. Zero bias implies the quantum mechanical current from left to right to equal the current from right to left. If the leads are treated ballistically, the carriers accumulate near the device boundaries due to the resistance they meet within the medium. Only by employing the same scattering mechanisms and self-energies everywhere, the formalism yields a constant carrier density throughout the system. Figure illustrates the calculated carrier density in a biased n++-i-n+ GaAs structure. If the carrier distribution within the leads is assumed to be in equilibrium, one obtains an artificial pinch-off behavior; there is a depletion of carriers near the source side and an accumulation near the drain side of the device.

2.7

The semiclassical limit

Does the NEGF formalism reduce to the semiclassical limit when quantum effects play no role? [22],[15] For a realistic device structure, this is difficult to prove rigorously but the following example provides practical evidence that, indeed, the answer is affirmative. Consider a symmetric n-i-n GaAs diode at room temperature and for zero bias. Since such a device contains neither quantum wells nor barriers, one expects the

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

22

semi-classical Boltzmann equation to adequately describe the carrier density, assuming one includes impurity and phonon scattering in the standard fashion and includes electron-electron scattering at least within the Hartree approximation via the Poisson equation. We have performed a charge self-consistent NEGF calculation that takes into account the same type of scattering mechanisms. As shown in Figure , the NEGF electron density mimics faithfully the semiclassical results.

Figure 2.2: (Color online) Left: Calculated carrier density for a homogeneously doped semiconductor at zero bias, attached to leads with an equilibrium distribution. The expected homogeneous density is obtained only if the leads include the same type of scattering self energies as the device (red curve). Otherwise one obtains artificial charge accumulation (blue curve). Right: GaAs n-i-n structure at room temperature under bias with asymmetric doping profile as indicated by the grey lines.Once a current is flowing. the charge distribution within the leads must be a suitably shifted Fermi distribution that reflects global current conservation (results shown by thc red curve). Otherwise, NEGF calculations yield artificial pinch-off effects (blue lines)

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

23

Figure 2.3: Comparison of fully self-consistent NEGF and semiclassical carrier dynamics calculations for a standard GaAs resistor (50 nm n-i-n structure) at zero bias and room temperature

Figure 2.4: (Color onlinc) Carrier dynamics calculation for 50 nm n-i-n structure at room tcmperature with a 12 nm InGaAs quantum well as intrinsic zone attached to field-free GaAs leads of the same n-density.The applicd voltage is 150 mV across thc structure.Red curve: calculalion in terms of charge-self-consistent semiclassical Boltzmann equation.Blue curve: Calculation in terms of strictly ballistic NEGF, equivalent to thc solution of Schrodinger equation of open system. Black curve: Fully self-consistent NEGF calculation

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

2.8

24

Quantum well structures: Why NEGF is essential

A nice illustration of the power of NEGF can be obtained by calculating the local carrier distribution in a biased n-i-n structure that contains a quantum well. This is illustrated in Figure for a n-i-n structure with a 12 nm InGaAs quantum well in the intrinsic zone. The applied voltage across the 50 nm structure is 150 mV and we show 3 results. The red curve shows the semiclassical Boltzmann results and yields the well-known accumulation of charge near the quantum well barriers. This method completely ignores the existence of quantum mechanical bound states and reflects the classical Thomas-Fermi density. The blue curve illustrates another limiting case, namely the solution of the Schrdinger equation in the absence of any scattering. This reflects a strictly coherent, energy conserving, ballistic transport. In this case, the carrier density within the device is fully determined by the overlap of the lead wave functions with the device. Since there are no lead carriers below the GaAs band edge, the quantum well states in the intrinsic region remain unoccupied. Thus, the electron density within the quantum well only stems from (continuum state type) lead electrons which explains the oscillatory density in the intrinsic region. Finally, the black curve represents the full NEGF calculation. The inelastic scattering processes lead to a capture of carriers into the quantum well states and lead to a carrier density in the intrinsic zone that lies in between the semiclassical and the strictly ballistic quantum mechanical calculations. This result can be further illustrated by plotting the energy resolved density for the case of zero bias. Figure depicts this density for a strictly ballistic calculation that actually represents a NEGF calculation in the limit of all impurity, phonon, and interface scattering self energies set equal to zero. Note that the Poisson equation is still solved self-consistently with the Schrdinger equation even in this case. Figure , on the other hand, shows a complete NEGF calculation that clearly illustrates the carrier capture into the first and second bound state of the quantum well. Due to the charging effects caused by the capture, the bottom of the quantum well raises in energy so that both states actually form resonances that slightly overlap with the lead states.

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

25

Figure 2.5: Contour graph of calculated energy resolved electron density for 50 nm n-i-n structure at room temperature with a 12 nm lnGaAs quantum well as intrinsic zone for zero applied bias. The density scale is the analogous to the one in Fig. 5. but for lower doping. (a) Strictly ballistic NEGF calculation (no scattering included). (b) Fully self-consistent NEGF calculation

Figure 2.6: Contour graph of calculated energy resolved electron density for 50 nm n-i-n structure at room temperature with a 12 nm InGaAs quantum well as intrinsic zone for zero applied bias. (a) Fully self consistent NEGF calculation. (b) NEGF calculation with coupling

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

2.9

26

Limits and simplifications in NEGF

As mentioned in the introduction, the NEGF formalism guarantees conservation of important physical principles only if all Greens functions are evaluated exactly from a many-body standpoint. It is a very significant weakness of the method that even plausible approximations can fail badly. Generally, it is difficult to introduce simplications that do not violate basic conservation laws. As an example, we discuss the so-called decoupling approximation.[10] The NEGF formalism couples the energy of states with their occupation via 4 coupled integrodifferential equations. If the carrier density is not too high, it seems plausible to decouple the equations of and . This approximation can lead to a violation of Paulis principle, however, since there is no mechanism that prevents the occurrence of over-occupied states (i.e. states with occupancy higher than permitted by the Pauli principle).[10] To exemplify this situation, we consider the same GaAs-InGaAs-GaAs n-i-n structure as before, but with a slightly higher carrier concentration in the n-region. Figure shows the energy resolved carrier density of the n-i-n structure for zero bias, as calculated by the full NEGF approach. By contrast, Figure shows the result of the decoupling of Greens functions, leading to unphysically high occupation of the lowest bound state.

2.10

THz quantum cascade lasers: a classics for NEGF

An important question that we have not addressed so far is whether fully selfconsistent NEGF calculations actually agree with experiment. We have applied this formalism to GaAs/AlGaAs THz QCLs and included impurity, phonon, interface roughness scattering in the self-consistent Born approximation. In addition, the electron-electron scattering has been included both in the Hartree approximation as well as within the so-called GW approximation. For details, we refer to [18]. Importantly, the calculations contain no fitting parameter. Figure 8 depicts the calculated current-voltage characteristics of such a QCL for a particular sheet doping density, together with experimental data.[2] As one see, theory and

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

27

experiment agree very well with one another up to the voltage where lasing starts and both thermal as well as hot electron effects become relevant that have not been included in the calculations.

Figure 2.7: Comparison between experimental (Ref. [27]) and calculated (Ref. ([13]) current-voltage characteristics for AIGaAs/GaAs quantum cascade structure

2.11

Comparison between NEGF and Monte Carlo

2.11.1

Monte-Carlo

Monte Carlo simulation performs calculation by building models of possible results by substituting a range of values-a probability distribution-for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values. By using probability distributions, variables can have different probabilities of different outcomes occurring. Probability distributions are a much

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

28

more realistic way of describing uncertainty in variables of a device performance analysis. Through the Non-Equilibrium Green’s Function (NEGF) formalism, quantumscale device simulation can be performed with the inclusion of electron-phonon scattering. However, the simulation of realistically sized devices under the NEGF formalism typically requires prohibitive amounts of memory and computation time. Two of the most demanding computational problems for NEGF simulation involve mathematical operations with structured matrices called semiseparable matrices. Nonequilibrium Green’s function method is a very general scheme to include coherent quantum mechanics and incoherent scattering with phonons and device imperfections self-consistently. However, it is numerically very demanding and cannot be used for systematic device parameter scans. For this reason, we also implement the approximate, numerically e?cient ensemble Monte Carlo method and assess its applicability on the above mentioned transport problems. A research paper [2] shows that the approximate treatment of coherent tunneling and leakage into continuum states limits the applicability of the EMC method on transport regimes that are dominated by incoherent scattering. When all device states that contribute to transport are clearly non-degenerate, results of the current density obtained by the EMC method quantitatively agree with NEGF results and experiment. Also the simulated spectral gain pro?le is in good agreement for both methods. This is in particular important because the numerical load of NEGF calculations exceeds the load of the EMC method tremendously and typically prohibits a systematic improvement of QCL designs.

2.12

Why nonequilibrium Green functions?

Let us briefly describe its main features: • The method has as its main ingredient the Green function, which is a function of two spacetime coordinates. From knowledge of this function one can calculate time-dependent expectation values such as currents and densities, electron addition and the total energy of the system.

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

29

• In the absence of external fields the nonequilibrium Green function method reduces to the equilibrium Green function method which has had imporatnt applications in quantum chemistry. • Nonequilibrium Green functions can be applied to both extended and finite systems. • The nonequilibrium Green function can handle strong external fields nonperturbatively. The electron-electron interactions are taken into account by infinite summations. • The approximations within the nonequilibrium Green function method can be chosen such that macroscopic conservation laws as those of particle number, momentum and angular momentum are automatically satisfied • Dissipative processes and memory effects in transport that occur due to electron-electron interactions and coupling of electronic to nuclear vibrations can be clearly diagrammatically analyzed

2.13

Conclusions

The NEGF formalism provides the framework of choice for consistent carrier dynamics calculations of open nanosystems where quantum effects and incoherent scattering play a comparable role. When implemented with care, it reproduces the results of the semiclassical Boltzmann equation in the limit where quantum effects such as resonant tunneling and interference are unimportant. By definition, it also reproduces the solutions of the Schrdinger or Lippmann-Schwinger equation when inelastic scattering is turned off. A disadvantage of the method is the fact that charge and current conservation, and even Paulis principle are not automatically satisfied once seemingly reasonable approximations are introduced. Scattering vertices must necessarily be taken into account to infinite order, for example, to strictly obey current conservation and it is difficult to achieve a numerically satisfactory convergence. Approximations are unavoidable, though, once one seeks predictions for realistic nano-devices, simply due to the

CHAPTER 2. NON-EQUILIBRIUM GREEN’S FUNCTIONS

30

excessive numerical effort required to solve the full set of Keldysh equations selfconsistently. In fact, it will take some time before quantitative NEGF solutions for time-dependent quantum transport calculations become numerically feasible. In comparison with semiclassical calculations, much more effort is required to properly take into account the physics of contacts and the contact-device coupling. This is a consequence of the nonlocal nature of quantum mechanics and the nature of scattering solutions in open quantum systems.

Chapter 3 QDIPs density of states modeling 3.1

DWELL QDIP Structure

A theoretical model of pyramidal-shaped InAs quantum dots placed in an InGaAs quantum well, which is buried in a GaAs matrix, is shown in [[7]]. The model of the DWELL is based on a Bessel function expansion of the wave function. The model can estimate the ground state of the quantum dot. For the higher states in the quantum well, the model has to be modified to account for the free motion of electrons perpendicular to the growth direction. The DWELL detector grown by Molecular Beam Epitaxy, reported in Ref [1], consists of a ten-period active region of 6nm In0.15 Ga0.85 As, 2.4 ML of InAs, 6 nm In0.15 Ga0.85 As, and 50 nm GaAs, as shown in Fig. 3.1. The quantum dots are placed in the In0.15 Ga0.85 As quantum well which is in turn surrounded by the GaAs region. The TEM image of the DWELL heterostructure is shown in following Fig 3.2. [1] The darkest region is the InAs quantum dot. The quantum dots are situated in the upper half of the quantum well and have a conical shape whose base dimension is of 11 nm and height is of 6nm. The QD material InAs is deposited over the substrate and due to the lattice mismatch between deposited material and substrate, the strain is built up gradually. After a critical thickness (2.4 ML) is reached, the two-dimensional growth changes into a three-dimensional one and

31

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

32

Figure 3.1: Cross-section schematic of a 10 layer InAs/InGaAs quantum dot in a well detector,

Figure 3.2: Cross-section TEM image of a single QD layer of DWELL [14]

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

33

dislocation free QD islands begin to grow.

Figure 3.3: Cross-section TEM image of an InAs/InGaAs DWELL Heterostructures [14]

Figure 3.4: Theoretical Modeling of DWELL QDIP Structure

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

34

For numerical analysis we have modeled the DWELL device as shown in Fig 3.1. 60% of the band gap difference between InAs and GaAs is counted as the conduction band offset .The band offsets calculated are 477 meV between InAs and In0.15 Ga0.85 As and 93 meV between In0.15 Ga0.85 As and GaAs. The conduction band edge of the In0.15Ga0.85As is selected as reference energy level. A linear interpolation between two binary values is used to calculate the effective masses in the different materials. The effective masses used for GaAs, InAs and In0.15 Ga0.85 As are 0.067, 0.027and 0.061 (in terms of electron mass) respectively.

Figure 3.5: Conduction band offsets and energy levels of QDWELL [14]

For analysis, the device is thought of consisting of array of identical cylinders, where each cylinder contains one quantum dot. To calculate Hamiltonian of the device, a cross section along the cylinder axis is taken and disintegrated into a large number of equally spaced grids. The finite difference method is used to solve the differential equation governing Green’s function. The retarded Green’s function of the system is defined as 0

0

0

0

[E − Hop − Σr (E)]Gr (x, x , y, y , E) = δ(x − x )δ(y − y )

(3.1)

Here E is the total energy of electron,Σr is the self energy and Hop , the Hamiltonian operator of the system, is given by Hop = −∇.

~2 ∇ + V (x, y) 2m(x, y)

(3.2)

Here V(x,y) is the potential energy seen by the electron and m(x,y) is the

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

35

Figure 3.6: Cross section of a cylinder is disintegrated into a number of grids

effective mass. The diagonal elements of spectral function is given by diag(A(x, y, E)) = −2Im[Gr (x, y; E)]

(3.3)

The density of states, which is the number of states per unit energy per unit volume, is given by N (E) =

3.2

1 T r(A(x, y; E)) 2π

(3.4)

Numerical Analysis

The quantum dot photodetectors under our analysis have an estimated dot density of 5 × 1010 cm2 , and the average spacing between two adjacent dots is about 60nm. Due to this relatively large distance, in our simplified model of the quantum dot photo-detector, the neighboring dots are assumed to be vertically and laterally decoupled, and a quantum dot is modeled so as to be surrounded by semi-infinite contact composed of InGaAs and GaAs layer, and InAs wetting layer. The contact can be thought of being a continuation of cylinder radius and

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

36

the quantum dot exists at the center of the cylinder. Modeling quantum dot photo-detectors in such a way gives the benefit of exploiting the property of translational invariance of the contact. The Hamiltonian matrix for the device, which is tridiagonal and Hermitian, is formed by finite difference method and is given by



αx1

β1

0

...

    H=   

β1

αx2 β2 .. .

...

0 .. . 0

..



0 .. .

.

. . . . . . βN x−1

βN x−1 αx(N x)

       

Here,

     αx (i) =    

Uy1 + 2tx(i) + 2ty1 −ty1 0 −ty1 Uy2 + 2tx(i) + 2ty2 −ty2 .. .. . . −ty2 ···

0     β( i) =   

3.3

−tx1,i

0

0 .. .

−tx2,i

0

···

··· ···

0 .. . .. ... . −txN y,i

··· 0

−tyN y−1       

0 0 .. . −tyN y−1 Uy(N y) + 2tx(i) + 2tyN y

(3.6)

Calculation of Self Energy:

To illustrate the self-energy calculation which accounts for the device leads, we consider the effect of coupling the active device Hamiltonian to the drain. The infinite Hamiltonian and its Green’s function can be partitioned as follows: In the above equation, the subscript ‘lead’ indicates infinite block of matrices



     (3.5   

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

37

(H matrix and G matrix). The matrix block we only care about is Gdevice as we are not interested in the Green’s function within the Lead or Source. Gdevice can be expressed in terms of known quantities as follows: Gdevice = [EI − hdevice − Σr ]−1

(3.7)

Where the self energy term is

 −1   0 0 · · · EI − αN x+1 −β 0 · · · · 0 −β       0 0 · · ·    −β EI − αN x+2 −β · 0 0       P      (3.8) ·  0 −β · · · r =  ·          · 0 · ·   ·    −β · · · · · · For evaluating the matrix product in above equation, we only need the first block of the inverse of the infnite matrix associated with the Lead. Moreover, the diagonal blocks of this infinite matrix are repeated due to translational invariance within the Lead. 

αN x = αN x+1 = αN x+2 =

(3.9)

Using this property, and partitioning the matrix, a close form of the matrix for the first block of the inverse (denoted by gc ) of the infinite matrix, can be obtained as I = gc [EI − αN x+1 −c β]

(3.10)

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

38

Once gc hasbeensolved, wehavetheself energyterm 

X

r

    =   

τ gc τ + 0 0 0 .. . .. . 0 ···

··· ··· ··· ··· ... .. . ···

0 0 .. . .. . · · · τ gc τ +

        

(3.11)

tx is the coupling energy between adjacent grid points along x direction, and is given by tx =

~2 2m∗x a2

(3.12)

gc is the retarded green’s function of a unit cell of the contact, and is solved from the recursive relation gc−1 = ((E + iη)I − Hc −c τ + )

(3.13)

It is noteworthy that only the last vertical slice of the device couples to the Lead. Therefore, the self-energy for the Lead has a single nonzero block that perturbs the last diagonal block of the device Hamiltonian. To solve for gc , a basis transformation has to be performed. The eigenvectors of EI −α diagonalize gc simultaneously. Therefore we change the basis from 2D real space to a basis that is composed of the eigenvectors of EI − α (equivalent to a mode-space transformation at the boundary). This reduces the equation related to gc to a set of decoupled quadratic equations that can be solved for the diagonal entries gc , in the transformed representation. It should be noted that each of these equations results in two roots. The root representing outgoing waves is selected as we are ultimately interested in obtaining the retarded Green’s function for the device. An inverse basis transformation is then applied to evaluate gc in 2D real space. A similar procedure is invoked to solve for the self-energy part associated with the Lead of other side. The final size of the self-energy matrix is (Nx × Ny )2 for the real-space solution and (Nx )2 for the decoupled mode-space solution.

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

3.4

39

Computational Challenges

The device that we have dealt with is of GaAs/InGaAs/InAs heterostructure. For numerical analysis in FDM (Finite Difference Method), we have disintegrated the device structure into a large number of grids. For instance, a 40nm by 30nm cross section of the device was divided into near about 30000 grids. A pictorial representation of the device with grid division is as follows:

Figure 3.7: Device structure with grid division

The actual number of grid is too enormous to show in picture. In simulation, any two neighboring grids, (either in x direction or in y direction) are separated by 0.2nm distance. Thus in the entire device, there were a total of 30,000 grids. 30nm Along X axis - number of grid per row is 0.2nm = 150 grids 40nm Along Y axis - number of grid per row is 0.2nm = 200 grids Hence total no. of grid in the whole device = 200 x 150 = 30,000 We will justify the reason of taking such a mammoth number of grids although the device somewhere later, but for instance, let’s take a look at what are the problems associated with calculation that involve such a large number of grid points. In finite difference method, the matrix size of the Hamiltonian of a device is proportional to the square of the total number of grid point representing the entire device. For

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

40

example, if we divide the device in 6 grids, the corresponding Hamiltonian will be of 6 x 6 size. Each diagonal element of the Hamiltonian will correspond to a unique grid points. An illustration of the same is as follows:

Figure 3.8: Device cross section

Corresponding 

g1   a  · · ·  H= · · ·   0  0

a 0 ··· g2 b · · · b g3 c c g4 d ··· ··· d ··· ··· ···

 ··· 0  ··· 0   · · · · · ·   · · · · · ·  g5 e   e g6

The diagonal value of the Hamiltonian corresponds to each of the grid point taken on the device cross section. However, the off-diagonal elements are appearing due to coupling between the near-most grid points. Now, for the actual device, on which we have taken a considerable number of grid point in order to compute its property as precisely as possible, will yield a Hamiltonian having size of 30,000 x 30,000. This is an incredibly large matrix which will require a huge memory space. In fact, in a computer with moderate configuration, a MATLAB program can’t declare a matrix which exceeds 10,000 x 10,000 in size, let alone any operation we can make with it. The built in memory reserved for a MATLAB program overflows while storing the data of such an enormous matrix. The numerical analysis is thus stymied at the beginning of our thesis. We had to find a way out to circumvent the problem of memory run-out, and that constituted

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

41

how we confronted our very first challenge.

3.5

How to confront the challenges

There are several ways out to solve the problem of memory run-out. We will discuss them one by one, and also have a glance on the associated limitation that each of the way poses on our ability to analyze the device appropriately.

3.5.1

Reduce the number of grid

This is probably the very first (and the most naive, however!) idea that may come across one’s mind in such a case of limited computer memory. But it should be kept in mind that the separation between adjacent grids would have to be such that even the thinnest region of the device cross section is to be covered by sufficient number of grid points. In our device, the thinnest region is the wetting layer of InAs, whose thickness is of 0.5 nm. Hence, if grid separation is more than 0.2 nm, the wetting layer will contain insufficient number of grid, thereby the effect that the wetting layer exerts on the device characteristics will not be properly reflected in the simulated model.

3.5.2

Non-uniform distribution of grid point

The non-uniform distribution of grid point throughout the device can be a smart and intelligent way to make an economical use of available memory space while incorporating the effect of all the region simultaneously. In fact, Comsol simulator provides us with the feature of taking dense mesh points in some region whereas less dense or less concentrated mesh points in some other region as per the requirement of the user. However, in numerical analysis method performed in MATLAB, we have to provide with formula that rightly fits for non-uniform distribution of grid point. In contrast, the formula we could have laid our hand on was derived for uniform distribution of grid point. We, thus, were left with the choice of formulating a new equation for our own pursuit, or leave this way of solution as an unviable one.

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

3.5.3

42

Disintegrating a large matrix into several smaller matrices

Segregating a large matrix into several smaller ones can be another potential solution to our problem. Our mathematical analysis frequently needed matrix inversion operation, which is considered one of the most tedious and memory expansive job for a calculating machine. Since the matrix we dealt with is incredibly large in size, a direct command of matrix inversion (i.e. inv (Matrix)) would definitely end up with memory overflow. An alternative way of inverting a matrix is as follows: Let, A is a matrix, and B is the inverse of the matrix A. So, A*B= I, where I is an identity matrix of same size as A or B is. If a command in MATLAB is given as follows: B=A/I; Then MATLAB actually performs a solution of linear equation system which is much less hectic and time saving process. For instance, if A= "

2 7 1 5

#

Then A−1 can be calculated as follows: A−1 = B = "

# b1 b2 b3 b4

From here, we can write: 2 7 1 5

! ×

b1 b2 b3 b4

!

b1 b3

!

=

1 0 0 1

=

1 0

So 2 7 1 5

! ×

That is, 2 b1 +7 b2 = 1 And, b1 + 5 b3 = 0 And also

!

!

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

2 7 1 5

! ×

b2 b4

! =

0 1

43

!

That is, 2 b2 + 7 b4 = 0 And , b2 + 5 b4 = 1 Thus, instead of finding the inverse matrix altogether, we can individually find every element of the inverse matrix and then arrange all the calculated elements in order.

3.5.4

Dealing with Sparse Matrix other than conventional matrix

Sparse matrix are those, whose elements are mostly zero. A zero dominated sparse matrix can easily be manipulated to save memory since its zero elements do not need a separate memory space to occupy. The Hamiltonian that is concerned in our device simulation is a sparse matrix. It contains non-zero elements along the diagonal and off-diagonal position. The remaining elements are zero (unless any special circumstances appear, like: boundary condition). In fact, the sparse matrix springs off in almost all device simulation, since we usually consider a tight binding model of the device (a model where wave function of electron at a certain location is only influenced by nearmost grid points, coupling occurs only between neighboring grid points)

3.5.5

Using all the processors in a computer in a parallel process

By default, a MATLAB program engage only one core in a computer for computational purpose at a time. But special arrangement and coding can be done so as to compel all the available processors simultaneously and in a parallel way, which saves time to a great deal. A detail description of How self-parallelism can be done in a MATLAB program can be viewed from the following website: a) Parallel Computing Toolbox Perform parallel computations on multicore computers, GPUs, and computer clusters http://www.mathworks.com/products/parallel-computing/ b) Multicore - Parallel processing on multiple cores

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

44

http://www.mathworks.com/matlabcentral/fileexchange/13775

3.6

Result- Our Findings

A group of researchers led by Professor Sanjay Krishna reported their Spectral response measurements performed on the DWELL detectors with a Nicolet 870 FTIR spectrometer and a Keithley 428 current-amplifier. [1] Figure 3.9 shows the spectral response obtained from six detectors in which the width of the bottom InGaAs well was varied from 10 to 60 angstrom. The top InGaAs cap layer was unchanged to minimize changes in the dimensions of the dots. As expected, the operating wavelength of the detector showed a monotonic red shift from 7.2 to 11 µm from samples A to F. This is significant since it provides us with a method of controlling the operating wavelength of a QD detector. Note that in the sample with the largest well, there is a broad shoulder on the shorter wavelength side, which is possibly due to a transition from the ground state in the dot to a higher lying state in the InGaAs quantum well. Professor Krishna and his collaborators obtained the responsivity from a 15stack quantum DWELL detector. The measured responsivity was divided by a factor of 4 to account for the scattering in the substrate. Far-infrared (FIR) spectral response measurements were also undertaken in collaboration with Perera’s group [9]. The data were obtained using a Perkin-Elmer system 2000 FTIR with two sets of beamsplitters and windows and were corrected by background spectra. The resulting three-color response is shown in figure 3.10. The first two MIR peaks, i.e. 10 and 5 µm, have previously been observed by the same researchers group on similar detector structures [9]. The researchers group believes that the peak at 10∆m(124meV < ∆Ec) is probably a transition from a bound state in the dot to a bound state in the quantum well, whereas the peak around 5µm(250meV > ∆Ec) is possibly a transition from a state in the dot to a quasibound state close to the top of the well as shown in the inset to figure . A FIR peak centred around 25µm was also observed in these detectors and is shown in figure 3.11. The researchers group believes that this peak could possibly be due to transitions between two states in the QD since the calculated energy spacing between the dot levels is about 50-60 meV (20 − 25µm).

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

45

Figure 3.9: Progressive red shift in the peak operating wavelength of the detector as the width of the bottom InGaAs is increased from 10 to 60 angstrom. The spectre has been vertically displaced for clarity. [7]

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

46

Figure 3.10: Multicolour response in the mid-wave, long-wave and very longwavelength regimes with the associated transitions in the inset.[11]

Figure 3.11: The very long-wave infrared (VLWIR) response was observed till 80 K [23]

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

3.7

47

Comparing Experimental results with simulated ones

The reported structure in [8] has three color response with peaks at wavelengths of 5, 11, and 25m. The corresponding energy transitions E due to photon absorption at these wavelengths are approximately 250, 113 and 50 meV. hc eV λe

(3.14)

1243 meV λ(µm)

(3.15)

E=

E≈

Our calculated DOS for DWELL structure with bottom quantum well of 6nm (symmetric DWELL structure) is shown in Fig. 3.1. From the calculated DOS, we get E0, E1, E2, E3 (corresponding to each abrupt rise in DOS profile) as -270.3meV,-28meV, 64meV, and 112meV and results in E1 - E0, E2 - E1 and E3 - E2 as 242.3meV (≈ 250meV ), 92meV (≈ 124meV ) and 48meV (≈ 50meV ). Thus it explains the tri-band operation satisfactorily.

Figure 3.12: DOS profile for bottom well width=6nm

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

48

The experimentally measured spectral response of the asymmetric DWELL with increasing bottom well width shows a monotonic red shift [Ref: Quantum dots-in-a-well infrared photodetectors, Sanjay Krishna]. This result can be intuitively expected because increasing Quantum Well width means decreasing the confinement for electron and hence the neighboring energy levels comes closer, thus leading to smaller wavelength of photon absorption.

Figure 3.13: Effect of changing well width on DOS

A rigorous theoretical analysis of DOS for various Well width structure yields the same expected result. From figure 3.13, we see, the width of Quantum Well hardly has any prominent effect on the position of ground state energy, though it significantly changes other levels, causing a monotonic shift of DOS towards lower energy level with gradually increasing Well width. An interesting observation is that, the higher energy levels are getting closer to ground state at a rate faster than lower energy level, thus reducing the relative distance between any two energy levels we consider for transition. The corresponding energy levels are shown in table 1 and their differences are shown (in meV) in table 2.

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

49

Figure 3.14: Effect of changing well width on DOS

Energy in meV 3nm Well 4nm Well 6nm Well E0

-270.3

-270.3

-270.3

E1

-23

-26

-28

E2

79

74

64

Energy difference in meV E1 − E0

E2 − E0

E2 − E1

3 nm Well

247.3

349.3

102

4 nm Well

244.3

344.3

100

6 nm Well

242.3

334.3

92

It is a point of interest to see how different sites of the device contribute to different energy states. Figure 3.15 shows the potential profile for the DWELL model with 6 nm bottom half Quantum Well. From figure 3.16, we see, at the ground level (E= -278meV), energy state distribution takes a triangular form reveals the fact that at this bottom-most energy level, quantum state is provided solely by the quantum dot. Though at E = -62 meV, provision of energy state by the device is of minimal value (of order ≈ 10−8 ) and hence has no significant number of state in practice,

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

Figure 3.15: Potential Profile

Figure 3.16: LDOS at E=-270.3208mev

50

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

51

we nevertheless add here a state distribution (Figure 3.16) to point out the curious fact that as we are departing more and more from ground state energy, energy state are getting spread wider and wider from their previous condense location at quantum dot site. The previous triangle shape state distribution now assumes an almost dumble like shape.

Figure 3.17: LDOS at E=-61.8225mev

From figure 3.18, we can see, at the first continuum energy level (E= -15meV), the density of state is concentrated at wetting layer of InAs Figure 3.19) reveals that, at second continuum energy level (E= 77meV), the energy states resides prominently at InGaAs Quantum Well sites.

CHAPTER 3. QDIPS DENSITY OF STATES MODELING

Figure 3.18: LDOS at E=-15mev

Figure 3.19: LDOS at E=77mev

52

Chapter 4 Dipole Moment and Absorption Coefficient 4.1

Transitional Dipole Moment

The Transition dipole moment or Transition moment, usually denoted µnm for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the transition dipole moment is a complex vector quantity that includes the phase factors associated with the two states. An oscillating electric or magnetic moment can be induced in an atom or molecular entity by an electromagnetic wave. Its interaction with the electromagnetic field is resonant if the frequency of the latter corresponds to the energy difference between the initial and final states of a transition (∆E = hυ). The amplitude of this moment is referred to as the transition moment. Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the square of the magnitude gives the strength of the interaction due to the distribution of charge within the system. The SI unit of the transition dipole moment is the Coulomb-meter (Cm); a more conveniently sized unit is the Debye (D). The interaction energy, U, between a system of charged particles and an elec-

53

4. Dipole Moment and Absorption Coefficient

54

tric field, E, is given by: U =µ·E

(4.1)

The dipole moment is defined for a collection of charges by µ=

X

qi ri

(4.2)

i

Where ri is the position vector of charged particle i. The expectation value of the interaction energy is Z hU i =

  µ.Eˆ µn dτ ψn∗ −ˆ

(4.3)

If we assume that the magnitude of the electric field is constant over the length of the molecule (and that ψ is finite only over the length of the molecule) we can write hU i = hµi .E

(4.4)

where Z hµi =

ψn∗ µ ˆψn dτ

(4.5)

i.e. the strength of interaction between a distribution of charges and an electric field depends on the dipole moment of the charge distribution. In order to obtain the strength of interaction that causes a transition between two states, the transition dipole moment is used rather than the dipole moment. For a transition between and initial state, ψi , to a final state ψf , the transition dipole moment integral is. Z µf i =

ψf∗ µ ˆψi dτ

(4.6)

the probability for a transition (as measured by the absorption coefficient) is proportional to µ∗f i · µf i µf i may be positive, negative or imaginary. If µf i then the interaction energy is

4. Dipole Moment and Absorption Coefficient

55

zero and no transition occurs - the transition is said to be electric dipole forbidden. Conversely, if µf i islarge, thenthetransitionprobabilityandabsorptioncoef f icientarelarge. R 2 The intensity of the transition is therefore proportional to ψk∗ µ ˆψj dτ The dipole moment operator for an electron in one dimension is Z µf i = −e

ψf∗ xψi dx = −e hψf |x| ψi i

(4.7)

Notes on transitional (dipole) moment: 1. The absorption probability for linearly polarized light is proportional to the cosine square of the angle between the electric vector of the electromagnetic wave and fi; light absorption will be maximized if they are parallel, and no absorption will occur if they are perpendicular. 2. It is frequently said that a transition is polarized along the direction of its transition moment and this direction is called the polarization direction of the transition. 3. In the case of a doubly degenerate final state f, each of the two components at the same energy has a transition moment and the two moments define a plane. The transition is then said to be polarized in that plane, which also defines its polarizationdirection(s). This is typically the case for some of the transitions in highly symmetrical molecules.

4.2

Optical Transition using Fermi’s Golden Rule

Let us consider 1 a semiconductor illuminated by light. The interaction between photons and electrons in the semiconductor can be described by the Hamiltonian [5] H=

1 (p − eA)2 + V (r) 2m0

(4.8)

Where m0 is the free electron mass, e = −ke| for electrons, A is the vector potential accounting for the presence of the electromagnetic field, and V(r) is the periodic crystal potential. The above Hamiltonian can be understood because 1

One of the authors, Mr. Tonmoy Kumar Bhowmick was inspired by his thesis supervisor Dr. Farseem Mannan Mohammedy to take the course Semiconductor Device Theory from which he learnt a great deal about semiconductor physics. He is grateful to his thesis supervisor for his inpiration and encouragement

4. Dipole Moment and Absorption Coefficient

56

taking the partial derivatives of H with respect to the momentum variable p and the position vector r will lead to the classical equation of motion described by the Lorentz force equation in the presence of an electric and a magnetic field.

4.3

The Electron-Photon Interaction Hamiltonian

The Hamiltonian can be expanded into e e2 A2 p2 0 + V (r) − (p · A + A · p) + = Ho + H H= 2m0 2m0 2m0

(4.9)

Where Ho is the unperturbed Hamiltonian and H’ is considered as a perturbation due to light: H0 = 0

p2 + V (r) 2m0

H =−

e A·p m0

(4.10)

(4.11)

The Coulomb gauge ∇·A=0

(4.12)

Has been used such that p.A=A.p, noting that p = (~/i) ∇. The last term is much smaller than the terms linear in A, since absolute value of eA is much smaller than absolute p for most practical optical field intensities. This can be checked using e2 A2 2m0

p = ~k ≈ ~π/a0

(4.13)

Where a0 5.5angstrom is the lattice constant. Assuming the vector potential for the optical electric field of the form A = eˆA0 cos(kop · r − ωt)

(4.14)

4. Dipole Moment and Absorption Coefficient

57

A0 A0 ikop ·r −iωt e e + eˆ e−ikop ·r e+iωt (4.15) 2 2 Where kop is the wave vector, ω is the optical angular frequency and e is a unit vector in the direction of the optical electric field, we have A = eˆ

E(r, t) = −

δA δt

(4.16)

E(r, t) = −ˆ eωA0 sin(kop · r − ωt)

H(r, t) =

1 ∇×A µ

1 H(r, t) = − kop × eˆA0 sin(kop · r − ωt) µ

(4.17)

(4.18)

(4.19)

since the scalar potential vanishes for the optical field. The interaction Hamiltonian H’(r, t) can be written as 0

H (r, t) = − 0

0

e A(r, t) · p m0 0

H (r, t) = H (r)e−iωt + H + (r)e−iωt

(4.20)

(4.21)

0

where H (r, t) is 0

H (r, t) = −

4.4

eA0 eikop ·r eˆ · p 2m0

(4.22)

Transition Rate due to Electron-Photon Interaction

The transition rate for the absorption of a photon, assuming an electron is initially at state Ea, is given by Fermi’s golden rule and has been derived in Section 3.7

4. Dipole Moment and Absorption Coefficient

58

using the time-dependent perturbation theory: 2π D 0 E 2 (4.23) b H a δ(Eb − Ea − ~ω) ~ Where Eb > Ea has been assumed. Similarly the transition rate for the emission of photon if an electron is initially at state Eb is Wabs =

2π D 0 E 2 (4.24) a H b δ(Ea − Eb + ~ω) ~ The total upward transition rate per unit volume (s-1cm-3) in the crystal, taking into account the probability that state ’a’ is occupied and state ’b’ is empty, is Wabs =

Ra→b =

2 X X 2π V k k ~ a

0 2 H ba δ(Ea − Eb − ~ω)fa (1 − fb )

(4.25)

b

Where we sum over the initial and final states and assume that the FermiDirac distribution fa =

1

(4.26) 1+ is the probability that the state ’a’ is occupied. A similar expression holds for fb , where Ea is replaced by Eb , and (1 − fb ) is the probability that the state ’b’ is empty. The prefactor 2 takes into account the sum over spins, and the matrix 0 element Hba is given by e(Ea −Ef )/kB T

D 0 E 0 Hba = b H (r) a Z

0

Hba =

0

ψb∗ (r)H (r)ψa (r)d3 r

(4.27)

(4.28)

With |H 0 ba| = |H 0 ab|. The downward transition rate per unit volume (s-1cm3) in the crystal can be written: Rb→a

2 X X 2π 0 2 = Hba δ(Ea − Eb + ~ω)fb (1 − fa ) V k k ~ a

(4.29)

b

Noting the even property of the delta function, δ(−x) = δ(x), the net upward

4. Dipole Moment and Absorption Coefficient

59

transition rate per unit volume can be written as R = Ra→b − Rb→a

R=

2 X X 2π 0 2 Hba δ(Ea − Eb − ~ω)(fa − fb ) V k k ~ a

4.5

(4.30)

(4.31)

b

Optical Absorption Coefficient

The absorption coefficient α(1/cm) in the crystal is the fraction of photons absorbed per unit distance: α=

no. of photons absorbed per unit volume per second no. of photons injected per unit area per second

(4.32)

The injected number of photons per unit area per second is the optical intensity S (W/cm2) divided by the energy of a photon ~ω; therefore, α(~ω) =

α(~ω) =

R S/~ω

~ω 2 X X 2π |ˆ e · pba |2 δ(Ea − Eb − ~ω)(fa − fb ) 2 2 nr ω A0 /2µc V k k ~ a

(4.33)

(4.34)

b

Using the dipole (long wavelength) approximation that A(r) = Aeik.r ≈ A We find that the matrix elements can be written in terms of the momentum matrix element: µba = e hb |r| ai = erba

(4.35)

Hba = − hb |er · E| ai = −µba · E

(4.36)

0

p = m0

d m0 r= (rH0 − H0 r) dt i~

(4.37)

4. Dipole Moment and Absorption Coefficient

  −e Hba = b A(r) · p a m0 0

0

Hba = − 0

60

(4.38)

e A · hb |rH0 − H0 r| ai i~

(4.39)

e(Ea − Eb ) A · hb |r| ai i~

(4.40)

Hba = − 0

Hba = −iωA · hb |er| ai 0

Hba = µba · E

(4.41)

(4.42)

The absorption coefficient becomes

α(~ω) =

πω 2 X X |ˆ e · µba |2 δ(Ea − Eb − ~ω)(fa − fb ) nr c0 V k k a

(4.43)

b

Where Eb − Ea = ~ω has been used, and E = +iωA for the first term in A(r, t) with exp(−iωt) dependence In terms of the dipole moment, we write the absorption coefficient as

α(~ω) =

πω 2 X X |ˆ e · µba |2 δ(Ea − Eb − ~ω)(fa − fb ) nr c0 V k k a

(4.44)

b

We can see that the factors containing A2o are canceled since the linear optical absorption coefficient is independent of the optical intensity, which is proportional to A2o . When the scattering relaxiation is included, the delta function may be replaced by a Lorentzian function with a linewidth Γ. δ(Ea − Eb − ~ω) −→

Γ/2π (Ea − Eb − ~ω)2 + (Γ/2)2

(4.45)

Where a factor π has been included such that the area under the function is

4. Dipole Moment and Absorption Coefficient

61

properly normalized: Z δ(Ea − Eb − ~ω)d(~ω) = 1

4.6

(4.46)

Wavefunction Calculation

To calculate the dipole moment in (4.2), the wavefunctions corresponding to the initial and final states of the transitions should be obtained. The wavefunctions corresponding to the discrete energy states in the QDs are localized inside the QDs. Therefore, assuming zero boundary conditions on a virtual cylinder surrounding the QD, the bound wavefunctions and the corresponding discrete energy eigenvalues can be calculated. In the continuum states above the QD barrier, the assumption of zero boundary conditions is not suitable for obtaining the wavefunctions in this part of the DOS. Instead, the method of the Green’s function is used to calculate the wavefunctions in this part of the spectrum. A self-energy term is added to the Hamiltonian at the cylinder surface to simulate the proper boundary conditions representing the coupling with the surroundings outside the cylinder. The bound state wavefunctions in the quantum dots are calculated by solving the eigenvalue problem: [EI − H] ϕb = 0

(4.47)

which gives all the eigenstates and the corresponding eigenvalues and HD is the Hamiltonian matrix of the isolated cylinder cross section that contains the Quantum Dot. This procedure is suitable to calculate the bound states since they are localized inside the quantum dot and hence we can assume zero boundary condition at the cylinder radius. To get the wave-function in the continuum part of the spectrum, the Green’s function method has been used to simulate the free motion of electrons above the quantum dot and to add an additional self-energy term at the cylindrical surface to simulate the open boundary condition. The retarded Green’s function gives the response at any point due to a unit excitation at any other point. So the Hamiltonian of the isolated cylinder is modified by adding a self-energy term to simulate the interaction with the surroundings in

4. Dipole Moment and Absorption Coefficient

62

addition to the excitation term in the R.H.S. The wavefunctions [Ref: S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University, Press, 1995)] satisfy the equation h

EI − HD −

X i r

ϕc = S

(4.48)

where S is the excitation term due to propagating waves incident from the lead at the cylinder’s surface. The retarded Green’s function of the system in the cylindrical representation and in the matrix form is given by h

EI − HD −

X i r

0

0

GD = δ(x − x )δ(y − y )

(4.49)

From the above equation, the wavefunction can be related to the Green’s function by the relation: h X i−1 r ϕc = EI − HD − S = GD S

(4.50)

If we write the self energy r

Σ =

fm |χm i χm

(4.51)

(fn∗ − fn ) |χm i

(4.52)

X

D

0

m

then the excitation S is given by S=

X m

which gives the contribution from all propagating modes at same total energy E. In normal incidence operation, the electric field is polarized in the in-plane directions, that is, x or y directions. Therefore for an arbitrary polarized electric field in the xy plane, the dipole moment will be given by µbc = eˆ hψc |r| ψb i

(4.53)

E D µbc = eˆ ϕc xˆi + yˆj ϕb

(4.54)

which indicates the selection rules for allowed optical transitions. The elec-

4. Dipole Moment and Absorption Coefficient

63

trons can be photo-excited from the bound states of quantum dot to the continuum states lying in quantum well or barrier.

4.7

Our Findings and Results

The reported heterostructure in [19], cited as sample B (AlGaAs/InGaAs/InAs), when compared with sample A (AlGaAs/GaAs/InAlGaAs/InAs), has exhibited a significant improvement in photocurrent response for both S-plane (in plane) and P plane (45 deg to Growth plane) polarized light, which, the authors like to attribute to the improved Height to Base ratio (H/B) of Quantum Dot in Sample B than in sample A. In a bid to find out whether this improvement is predominantly caused by Height to Base ratio improvement or by some other reason (since the two samples are of different composition), we have chosen a heterostructure reported in [8]. In this specimen, The self-assembled growth of quantum dots in the Stranski-Krastanov (SK) growth mode grows Quantum Dots having height of 6.5nm and base of 11nm on 2.4 ML wetting layer. In theoretical simulation of absorption coefficient, we have changed the base dimension of quantum dot, keeping its height fixed at 6.5nm. The experimentally measured polarized spectral response of the DDWELL sample B with H/B=8nm/12nm=0.67, shows a blue shift of peak response over DDWELL sample A with H/B= 5nm/17nm= 0.29 in case of both S and P polarized light. However, we believe, this shift is prominently caused by compositional variation of the two samples, and therefore, other than wavelength shift, are primarily interested in verifying the claim of magnitude of spectral response improvement due to H to B ratio improvement in this paper. This result may be intuitively expected because increasing Quantum Dot Base dimension means decreasing the zero dimensional confinement for electron and thus decreasing atom-like behavior of QD structure. A rigorous theoretical analysis of transitional dipole moment for standard Quantum dot dimension cited in [8] with 6.5nm Height and 11nm Base yields the figure 4.2. This figure shows two dominant peak and one pseudo-dominant peak. However, not all the peak be necessarily existent in experimental result of photocurrent response, since Fermi’s Golden Rule will eliminate the possibility

4. Dipole Moment and Absorption Coefficient

64

of any transition from even to even or odd to odd ? function. The calculated transitional dipole moment is later used to estimate absorption coefficient at 77K. We considered inhomogeneous broadening (Lorentzian broadening) is dominant in absorption phenomena. The curve of absorption coefficient in Fig 4.2 exhibits not only a smoothened function but also the elimination of the pseudo appearance of a peak in dipole moment curve.

Figure 4.1: Dipole moment versus wavelength for S polarized incident light in a standard dot-in-a-well infrared photodetector. QD has Height/Base=6.5nm/11nm

A relative comparison among absorption coefficient of QDIP for various Height to Base ratio is graphically shown in Fig 4.3 and Fig ??. Fig 4.3, applicable for S polarized light, indicates, peak of absorption coefficient (and henceforth, probably spectral response) is vividly highest among all three H/B value taken into account. However, it does not experience a monotonic fall in maxima as H/B value is declined. Fig 4.4, applicable for P polarized incident light gives us a rather straightforward notion. It indicates a clear improvement in absorption coefficient with the rise of Height to Base ratio.

4. Dipole Moment and Absorption Coefficient

65

Figure 4.2: Absorption coefficient versus wavelength for S polarized incident light in a standard dot-in-a-well infrared photodetector. QD has Height/Base=6.5nm/11nm

Figure 4.3: Comparison of absorption coefficient of different hypothetical QDIPs having various Height to Base ratio. Incident Light is S polarized (in plane incidence)

4. Dipole Moment and Absorption Coefficient

66

However, in both case of polarization, the maxima of absorption coefficient invariably shifts towards red end with gradual rise in Height to Base ratio. Table below indicates the relative magnitude of maxima of absorption coefficient, for S and P polarization. Notably, since the base diameter of QD is always larger than the height [6], the polarization dependent infrared absorption is anisotropic and s -polarized (in-plane) absorption is always less than the vertical polarized absorption.

Figure 4.4: Comparison of absorption coefficient of different hypothetical QDIPs having various Height to Base ratio. Incident Light is P polarized (45 degree incidence to growth plane)

4.8

Conclusion

Here, we have pointed out the interesting phenomenon of how the quantum dot dimensions exert impact in determining DWELLs absorption co-efficient. This analysis will eventually shed some light on the relative change of magnitude of the photocurrent response in conjunction with change of QD dimensions. The InAs/InGaAs/GaAs DWELL structure is taken and its quantum dot base is var-

4. Dipole Moment and Absorption Coefficient

67

Figure 4.5: Relative change of absorption peak with dot dimension change. Absorption peak found for 7nm QD base is taken as unity, when incident light is S polarized.

Figure 4.6: Relative change of absorption peak with dot dimension change. Absorption peak found for 7nm QD base is taken as unity, when incident light is P polarized.

4. Dipole Moment and Absorption Coefficient

68

ied while its height is kept fixed at standard value (6.5nm). In each case, absorption co-efficient of the device is calculated resulting from both S and P polarized incident light. The experimentally found fact of responsivity improvement with quantum dot height to base ratio improvement also matches with our numerically simulated result within a certain range of observation. The parabolic variation of absorption maximum with change in QD height to base ratio for S polarized light can likely be attributed to the race between two possible factors: QD volume and QD dimension ratio. However, whether absorption maximum will change in a similar parabolic way for P polarized light as well beyond our observed range or will retain its linear shape remains a matter of further investigation.

Chapter 5 Conclusions A summary of the main contributions of this research about DWELL modeling and optimization is presented in this chapter. Future research recommendations are provided for improvement and upgrading of the models developed achieving more accurate device characterization which would lead to improved device design.

5.1

CONCLUSION

In our thesis, we have explored some interesting and important behavior of Quantum dot-in-a-well detector by simulation and compared with the available experimental data. DWELL detector has one wonderful feature. Its peak wavelength response can be tuned in two ways: a) by changing the bias voltage, and b) by changing the dimension of quantum well in the heterostructure. It implies that a DWELL detector is tunable both before and after fabrication. In the very first phase, we have investigated how the change in thickness of quantum well brings change in Density of Quantum State. In fact, we have varied the dimension of bottom half well keeping the upper half well fixed in thickness so as to house the quantum dots promptly. A gradual increase of the width of bottom half well results in a gradual reduction in distance between any two neighboring energy levels. In particular, the position of ground energy level remains unaffected by the change of quantum well dimension, whereas the higher energy levels shifts towards the ground level, and hence lessening the inter

69

CHAPTER 5. CONCLUSIONS

70

sub-band gap. This observation eventually substantiates the fact that, if we increase the height of quantum well, we may expect to obtain a red shift of peak wavelength response. However, this observation may experience an exception in continual red shift of wavelength, since an availability of quantum state does not necessarily prove the existence of an electron in that state. Occupation of state by an electron is also dependent on Fermi function that we apparently have not taken into account in this investigation. In the second phase, we have pointed out the interesting phenomenon of how the quantum dot dimensions exert impact in determining DWELL’s absorption co-efficient. This analysis will eventually shed some light on the relative change of magnitude of the photocurrent response in conjunction with change of QD dimensions. The InAs/InGaAs/GaAs DWELL structure is taken and its quantum dot base is varied while its height is kept fixed at standard value (6.5nm). In each case, absorption co-efficient of the device is calculated resulting from both S and P polarized incident light. The experimentally found fact of responsivity improvement with quantum dot height to base ratio improvement also matches with our numerically simulated result within a certain range of observation. The parabolic variation of absorption maximum with change in QD height to base ratio for S polarized light can likely be attributed to the race between two possible factors: QD volume and QD dimension ratio. However, whether absorption maximum will change in a similar parabolic way for P polarized light as well beyond our observed range or will retain its linear shape remains a matter of further investigation. Noteworthy, this work can have future extension to find out some more properties inherently related to DWELL detectors’ mechanism. In example, the calculation of transmission co-efficient, together with previously calculated absorption co-efficient will give us responsivity of the detector. Besides, the dark current behavior with variation in temperature and voltage bias will be a very crucial point to explore. In addition, we are planning to find a feasible way out to incorporate the effect of coupling between two neighboring quantum dots in our mathematical model in a bid to bring the simulated result closer to the experimentally observed behavior of a DWELL photo-detector.

CHAPTER 5. CONCLUSIONS

5.2

71

FUTURE WORK

The present research lays the foundation for developing quantum transport models based on NEGF to describe the main characteristics of DWELLs. The goal of these models is to understand the physical operations of these nano-devices in order to have an insight of how the detector design can be improved such that it gives the optimized performance efficiently fulfilling the required characteristics. There is a trade off between the accuracy of the model and its complexity. Therefore, for good results a compromise between the accuracy and complexity is necessary. More research is required to improve and upgrade the models developed in the thesis to make them more accurate and to calculate more physical quantities required in fully characterizing the DWELLs. Some of these recommendations will be described in this section

5.2.1

INTERMIXING AND STRAIN EFFECT

The models developed in the thesis are based on a single band effective mass approximation. The DWELLs modeled are unipolar devices and the values of the conduction band offsets and electron effective masses are used to solve the differential equation of the Greens function. As described in chapter 3, all points in the lattice grid associated to the QD have single values for the potential energy and electron effective mass and the same thing for the QW and barrier regions. This means that no intermixing is assumed between the QD material and the material corresponding to the surrounding barriers. For example, for InAs QDs grown on GaAs, the model assumed no intermixing between In and Ga to get a simplified potential energy and effective mass profiles. The real situation is that intermixing between different heterostructure materials does occur which will modify the values of the band offsets and effective masses . Also the strain field distribution in the QD region will affect the values of band offsets and effective masses such that they will have a specific distribution in the QD and the surrounding regions. The numerical model developed in the thesis is general and can accept any input potential energy and effective masses profile if known. So finding out the exact compositions of the QD and the surrounding barrier in

CHAPTER 5. CONCLUSIONS

72

addition to the effect of the strain field distribution on the values of the conduction band offsets and effective masses will allow the model to describe a more realistic configuration of the QD region and hence giving better results.

5.2.2

NON-UNIFORMITY OF QDS SIZE, AND DISTRIBUTION

One of the assumptions used in the model to simplify the calculations is that it actually describes a uniform distribution of QDs in the layers of the detector in addition to using the average values of height and diameter of QDs in the calculations. This gives symmetry that allows, for example, calculating the portion of the current that passes through one QD and then the total current is the QD density times this portion of the current. In addition it simplifies the calculations of the DOS, energy eigenvalues and the corresponding wavefunctions. The model used in the thesis considered the uncertainty in the shape and size of QDs by using average values for the height and diameter of the QDs by adding a small imaginary part to the electron energy in the equation. Adding this small imaginary part causes broadening to the energy levels such that they have a Gaussian shape instead of sharp delta functions obtained when all QDs have the same shape and size. A better description of the random shape and size of QDs can be done through a sensitivity analysis technique described in . This technique gives an estimate about the rate of change of a specific physical quantity as a function of one of the dimensional parameters, for example. In this way there is no need of multiple simulations to scan all possible values of heights and diameters of QDs and instead, the average values can be used one time in the simulation and then the sensitivity analysis will give the corresponding simulation values at other values of QD sizes. So including this technique may save time of repeating the simulations in case other QDs need to be tested.

CHAPTER 5. CONCLUSIONS

5.2.3

73

INTERACTION WITH THE LATTICE

The NEGF model gives a straightforward way of including interactions in the system whether with the surrounding contacts or internally in the QDs and the surrounding barriers. The types of interactions to be included and the degree of approximations to be used determine both the accuracy and the complexity of the calculations. The interactions with the lattice are modeled in its simplest form. The self-energy due to lattice interactions is assumed to be constant independent on energy and position and corresponding to localized elastic scattering of a constant scattering rate. This simple form of self-energy does not require a selfconsistent solutions between the Greens functions and the self-energy as in the case of considering the self-consistent first order Born approximation for modeling electron phonon interactions. A realistic description of the interaction with the lattice requires including self-energies corresponding to interactions between electrons and longitudinal and acoustic phonons, but a self-consistent solution is needed in this case. Assuming constant and uniform scattering rates simplifies the calculations but it might not be valid for high applied biases and high temperature. The electron-electron interaction is considered qualitatively in the model through a self-consistent solution between the electron density and the potential energy as shown in chapter 5. A better description of electron-electron interaction can be done through the Hartree-Fock approximation to calculate the self-energy due to the charges of the QD and surrounding QDs.

5.2.4

TRANSIENT RESPONSE

Furthermore, there is one important characteristic of DWELL that has not been discussed in this work which might be useful for some applications. The transient response of the DWELL should be modeled for fluorescent applications where the time decay of the received pulse is important. The models in the thesis focus on DC applications such as thermal imaging and focal plane arrays where the signal is in steady state. For time varying signal, the transient response for the responsivity is needed. Some techniques to model the transient response in QWIPs have been shown in using Monte Carlo methods. For an accurate

CHAPTER 5. CONCLUSIONS

74

description of the transient response in a quantum system like a DWELL, the real time Greens functions can be used to calculate the responsivity as a function of time . Further research is needed to solve the problem without adding a lot of complexity due to the increase of the dimensions of the problem by adding an extra time axis. One simple way would be to obtain a circuit model for the DWELL in order to estimate a time constant RC for the detector. The time constant could be obtained by calculating the resistance of the barrier material between the QD layers and the equivalent capacitance between the contacts of the detector taking into consideration the QD capacitance in the layers.

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