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To our loving parents.

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Acknowledgements

We would like to acknowledge the various supports that we have re-ceived throughout my research work and in writing this dissertation.

We would like to thank our supervisor, Dr. Farseem Mannan Mo-hammedy, for trusting us with this work and giving us the oppor-tunity to be involved in such a sophisticated and interesting topicof research. We would like to thank him for proposing many of theideas presented here, which we was able to study in depth. Prof. Mo-hammedyhas been nothing short of a great mentor to us, offering hisguidance 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.

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Abstract

Quantum dot in a well (DWELL) infrared photodetectors have emergedas a promising technology in the mid- and far-infrared (3 25m) formedical and environmental sensing that have a lot of advantages overcurrent technologies based on Mercury Cadmium Telluride (MCT)and quantum well (QW) infrared photodetectors (QWIPs). In addi-tion to the uniform and stable surface growth of III/V semiconductorssuitable for large area focal plane applications and thermal imaging,the three dimension connement in QDs allow sensitivity to normalincidence, high responsivity, low darkcurrent and high operating tem-perature. The growth, processing and characterizations of these de-

tectors are costly and take a lot of time. So, developing theoreticalmodels based on the physical operating principals will be so useful incharacterizing and optimizing the device performance.

Theoretical models based on non-equilibrium Greens functions havebeen developed to electrically and optically characterize different struc-tures of DWELL s. The advantage of the model over the previousdeveloped classical and semiclassical models is that it fairly describesquantum transport phenomenon playing a signicant role in the per-

formance of such nano-devices and considers the microscopic devicestructure including the shape and size of QDs, heterostructure devicestructure and doping density. The model calculates the density of states from which all possible energy transitions can be obtained andhence obtains the operating wavelengths for intersubband transitions.The responsivity due to intersubband transitions is estimated and theeffect of having different sizes and different height-to-diameter ratioQDs can be obtained for optimization.

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

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Contents

Contents v

List of Figures viii

Nomenclature x

1 Introduction 11.1 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Quantum Dot Infrared Photodetectors (QDIPs) . . . . . . . . . . 3

1.3 Importance of DWELL Design . . . . . . . . . . . . . . . . . . . . 51.4 Multi-Band Radiation Detection . . . . . . . . . . . . . . . . . . . 71.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Non-equilibrium Greens functions 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Preliminary Concept . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Greens functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Why do we want to calculate the Greens function? . . . . 152.3.2 Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 The spectral function . . . . . . . . . . . . . . . . . . . . . 18

2.4 Response to an incoming wave . . . . . . . . . . . . . . . . . . . . 192.5 Charge density matrix . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 212.7 The semiclassical limit . . . . . . . . . . . . . . . . . . . . . . . . 21

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CONTENTS vi

2.8 Quantum well structures: Why NEGF is essential . . . . . . . . . 242.9 Limits and simplications in NEGF . . . . . . . . . . . . . . . . . 262.10 THz quantum cascade lasers: a classics for NEGF . . . . . . . . . 262.11 Comparison between NEGF and Monte -Carlo . . . . . . . . . . . 27

2.11.1 Monte-Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 272.12 Why nonequilibrium Green functions? . . . . . . . . . . . . . . . 282.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 QDIPs density of states modeling 313.1 DWELL QDIP Structure . . . . . . . . . . . . . . . . . . . . . . . 313.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Calculation of Self Energy: . . . . . . . . . . . . . . . . . . . . . . 363.4 Computational Challenges . . . . . . . . . . . . . . . . . . . . . . 393.5 How to confront the challenges . . . . . . . . . . . . . . . . . . . . 41

3.5.1 Reduce the number of grid . . . . . . . . . . . . . . . . . . 413.5.2 Non-uniform distribution of grid point . . . . . . . . . . . 413.5.3 Disintegrating a large matrix into several smaller matrices 42

3.5.4 Dealing with Sparse Matrix other than conventional matrix 433.5.5 Using all the processors in a computer in a parallel process 43

3.6 Result- Our Findings . . . . . . . . . . . . . . . . . . . . . . . . . 443.7 Comparing Experimental results with simulated ones . . . . . . . 47

4 Dipole Moment and Absorption Coefficient 534.1 Transitional Dipole Moment . . . . . . . . . . . . . . . . . . . . . 534.2 Optical Transition using Fermis Golden Rule . . . . . . . . . . . 554.3 The Electron-Photon Interaction Hamiltonian . . . . . . . . . . . 564.4 Transition Rate due to Electron-Photon Interaction . . . . . . . . 574.5 Optical Absorption Coefficient . . . . . . . . . . . . . . . . . . . . 594.6 Wavefunction Calculation . . . . . . . . . . . . . . . . . . . . . . 614.7 Our Findings and Results . . . . . . . . . . . . . . . . . . . . . . 634.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Conclusions 695.1 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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CONTENTS vii

5.2 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2.1 INTERMIXING AND STRAIN EFFECT . . . . . . . . . 715.2.2 NON-UNIFORMITY OF QDS SIZE, AND DISTRIBUTION 725.2.3 INTERACTION WITH THE LATTICE . . . . . . . . . . 735.2.4 TRANSIENT RESPONSE . . . . . . . . . . . . . . . . . . 73

References 75

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List of Figures

1.1 Band structure of a GaAs/AlGaAs Quantum well . . . . . . . . . 31.2 Density of states, bandstructure and carrier distribution for (a)

bulk, (b) quantum well, (c) quantum wire and (d) quantum dots. 41.3 Band structure of . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Various transitions in the dots-in-well (DWELL) QDIP . . . . . . 6

2.1 System Contact with Channel . . . . . . . . . . . . . . . . . . . . 142.2 (Color online) Left: Calculated carrier density for a homogeneously

doped semiconductor at zero bias, attached to leads with an equi-librium distribution. The expected homogeneous density is ob-tained only if the leads include the same type of scattering self energies as the device (red curve). Otherwise one obtains articialcharge accumulation (blue curve). Right: GaAs n-i-n structure atroom temperature under bias with asymmetric doping prole asindicated by the grey lines.Once a current is owing. the chargedistribution within the leads must be a suitably shifted Fermi dis-tribution that reects global current conservation (results shownby thc red curve). Otherwise, NEGF calculations yield articialpinch-off effects (blue lines) . . . . . . . . . . . . . . . . . . . . . 22

2.3 Comparison of fully self-consistent NEGF and semiclassical carrierdynamics calculations for a standard GaAs resistor (50 nm n-i-nstructure) at zero bias and room temperature . . . . . . . . . . . 23

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LIST OF FIGURES ix

2.4 (Color onlinc) Carrier dynamics calculation for 50 nm n-i-n struc-ture at room tcmperature with a 12 nm InGaAs quantum wellas intrinsic zone attached to eld-free GaAs leads of the same n-density.The applicd voltage is 150 mV across thc structure.Redcurve: calculalion in terms of charge-self-consistent semiclassicalBoltzmann equation.Blue curve: Calculation in terms of strictlyballistic NEGF, equivalent to thc solution of Schrodinger equationof open system. Black curve: Fully self-consistent NEGF calculation 23

2.5 Contour graph of calculated energy resolved electron density for50 nm n-i-n structure at room temperature with a 12 nm lnGaAsquantum well as intrinsic zone for zero applied bias. The densityscale 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 . . . . . . . . . . . . . 25

2.6 Contour graph of calculated energy resolved electron density for50 nm n-i-n structure at room temperature with a 12 nm InGaAsquantum well as intrinsic zone for zero applied bias. (a) Fully self consistent NEGF calculation. (b) NEGF calculation with coupling 25

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

3.1 Cross-section schematic of a 10 layer InAs/InGaAs quantum dotin a well detector, . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Cross-section TEM image of a single QD layer of DWELL [14] . . 32

3.3 Cross-section TEM image of an InAs/InGaAs DWELL Heterostruc-tures [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Theoretical Modeling of DWELL QDIP Structure . . . . . . . . . 333.5 Conduction band offsets and energy levels of QDWELL [14] . . . 343.6 Cross section of a cylinder is disintegrated into a number of grids 353.7 Device structure with grid division . . . . . . . . . . . . . . . . . 393.8 Device cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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LIST OF FIGURES x

3.9 Progressive red shift in the peak operating wavelength of the de-tector as the width of the bottom InGaAs is increased from 10 to60 angstrom. The spectre has been vertically displaced for clarity.[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Multicolour response in the mid-wave, long-wave and very long-wavelength regimes with the associated transitions in the inset.[ 11] 46

3.11 The very long-wave infrared (VLWIR) response was observed till80 K [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.12 DOS prole for bottom well width=6nm . . . . . . . . . . . . . . 473.13 Effect of changing well width on DOS . . . . . . . . . . . . . . . . 483.14 Effect of changing well width on DOS . . . . . . . . . . . . . . . . 493.15 Potential Prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.16 LDOS at E=-270.3208mev . . . . . . . . . . . . . . . . . . . . . . 503.17 LDOS at E=-61.8225mev . . . . . . . . . . . . . . . . . . . . . . . 513.18 LDOS at E=-15mev . . . . . . . . . . . . . . . . . . . . . . . . . 523.19 LDOS at E=77mev . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Dipole moment versus wavelength for S polarized incident light in astandard dot-in-a-well infrared photodetector. QD has Height/Base=6.5nm/11nm 64

4.2 Absorption coefficient versus wavelength for S polarized incidentlight in a standard dot-in-a-well infrared photodetector. QD hasHeight/Base=6.5nm/11nm . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Comparison of absorption coefficient of different hypothetical QDIPshaving various Height to Base ratio. Incident Light is S polarized(in plane incidence) . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Comparison of absorption coefficient of different hypothetical QDIPshaving various Height to Base ratio. Incident Light is P polarized(45 degree incidence to growth plane) . . . . . . . . . . . . . . . . 66

4.5 Relative change of absorption peak with dot dimension change.Absorption peak found for 7nm QD base is taken as unity, whenincident light is S polarized. . . . . . . . . . . . . . . . . . . . . . 67

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LIST OF FIGURES xi

4.6 Relative change of absorption peak with dot dimension change.Absorption peak found for 7nm QD base is taken as unity, whenincident light is P polarized. . . . . . . . . . . . . . . . . . . . . . 67

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Chapter 1

IntroductionHigh performance Infrared photodetectors in the mid- and far-infrared (3- 25 m)wavelength range have attracted much interest due to their important sensingapplications . They are used in medical and environmental sensing, optical com-munications, thermal imaging, night vision cameras, and missile tracking andrecognition. It is required to obtain a technology that gives high performance

at high operating temperature and with low cost. Current technologies based onMercury Cadmium Telluride (MCT) and quantum well (QW) infrared detectors(QWIPs) have some disadvantages that lower the overall performance of the sens-ing devices. The MCTs epitaxial growth problems limit the manufacturing yieldof large area focal plane arrays (FPAs) applications. QWIPs do not support nor-mal incidence detections and so need complicated optical coupling in addition tothe requirement of operating at very low temperature . The advance in epitaxialgrowth of heterostructure semiconductors allows for the fabrication of devices atnano scale dimensions. These nano-devices have new physical operating princi-ples and novel performance characteristics. Quantum dots (QDs) grown by theself-assembled epitaxial technique have attracted much interest in recent years forlaser and photodetector applications. In addition to the low cost, stable and uni-form surface epitaxial growth of the III/V semiconductors, suitable for large areaFPAs application in thermal imaging, the three-dimensional connement in QDshas many advantages such as the intrinsic sensitivity to normal-incidence lightwhich simplies the optical conguration for any application, reduced electron-phonon scattering, long-lived excited states, low dark current and high temper-

1

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CHAPTER 1. INTRODUCTION 2

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

1.1 Quantum Dots

A quantum well (QW) is a thin layer which can conne electrons or holes inthe dimension perpendicular to the layer surface, while the movement with inthe layer is not restricted . A QW is formed when a lower band gap materialis sandwiched between higher band gap materials. Figure illustrates the bandstructure of a GaAs/AlGaAs11 based QW . As seen in the gure, the electron is

conned in the z-direction, or normal to the surface of the layer, by the QW. Asa layer thickness approaches de-Broglie wavelength (i.e. about 10 nm), quantumeffects can be seen. Therefore, usually the QW thickness is in the order of 1 to 15nm. Similarly, in the quantum dot the carrier is conned in all three dimensions.

The change in connement can be better understood if we compare the densityof states for bulk (0-dimension), QW (1-dimension), Quantum dash or wire (2-dimensions) and QD (3-dimensions). As seen in Figure, the density of statesfunction 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 stepfunction. For the quantum wire the density of states further decreases comparedto the QW. For QDs, the density of states decreases compared to a quantumwire 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 lowdensity of states and small size of the dots means that fewer carriers are needed toinvert the carrier population, which results in low threshold current density andhigh characteristic temperature when incorporated as active region in a laser. Interms of detector, the absorption of the dots can be easily saturated due to the

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

nite density of states.

1.2 Quantum Dot Infrared Photodetectors (QDIPs)

The three-dimensional connement of QDs helps in localization of carriers re-ducing the thermionic emission which in turn lowers the dark current . Theintersubband energy level spacing in the QDs is greater than the phonon energyand, therefore, reduces the phonon scattering, which is a dominant scatteringmechanism in bulk and QWs. This is the reason for long carrier relaxation times

in QDIPs, which in turn increases the photoconductive gain. The responsivityand detectivity are also increased due to the increase in gain and photocurrent. In addition, QDIPs are sensitive to normal incidence radiation, which is notpossible in QWIPs, due to polarization selection rules, and requires specializedgratings to direct the radiation into the detector. The QDs are normally dopedto about than 1-2 electrons per dot in order to prevent carriers from occupyingthe excited state which will increase the dark current. The thickness of the bar-riers surrounding the quantum dots, the doping concentration, and the number

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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.

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of quantum dot layers are important parameters to consider while designing aQDIP.

QDIPs suffer from low QE due to low absorption cross section resulting fromlow density of QDs and nite spacing between the dots . This thesis discussesdifferent ways to improve the quantum efficiency and other performance param-eters like responsivity, detectivity, operating temperature, of QDIPs based ondots-in-well (DWELL) design.

1.3 Importance of DWELL DesignIn the DWELL design, the active region consists of 2.4 MLs of InAs QDs placedin 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 Figure1.6. As seen in the gure, there is a large conduction band offset (i.e. 250 meV)between the ground electronic state of the InAs QD and the conduction bandedge of the GaAs barrier, which reduces the thermionic emission and therefore

low dark current . Due to low dark current, QDIPs are expected to have higheroperating temperatures than QWIPs. The total band offset is calculated fromphotoluminescence (PL) spectrum and using the 60-40split. The excited state isobtained from spectral response and theoretical modeling . The spacing betweenthe 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-to-bound transitions from the ground electronic state of the InAs QD to the In0.15Ga0.85AsQW and bound-to-continuum transition from the ground electronic state of theInAs QD to a state in the GaAs barrier as illustrated in Figure . Dependingon the bias, one of the transitions is observed in the spectral response for thedetector.

A new alternative DWELL design (InAs QDs placed in a GaAs QW) is in-vestigated. 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 with-out introducing dislocations, which will lower the performance of the detectors.

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Figure 1.3: Band structure of InAs/In 0.15Ga 0.85As 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 orderto increase the overall absorption, which in turn increases the photocurrent anddetectivity. These three approaches have signicantly improved the performanceof DWELL detectors.

1.4 Multi-Band Radiation Detection

The rapid development of infrared (IR) detector technology, which primarily in-

cludes device physics, semiconductor material growth and characterization, andmicroelectronics, has led to new concepts like target recognition and trackingsystems.1, 2 Among these concepts, multi-band radiation detection is being de-veloped as an important tool to be employed in many practical applications. De-tecting an objects infrared emission at multiple wavelengths can be used to elimi-nate background effects,3 and reconstruct the objects absolute temperature4 andunique features. This plays an important role in differentiating and identifyingan object from its background. However, measuring multiple wavelength bands

typically requires either multiple detectors or a single broad-band detector with alter wheel coupled to it in order to lter incident radiation from different wave-length regions. Both of these techniques are associated with complicated detectorassemblies, separate cooling systems, electronic components, and optical elementssuch as lenses, lters, and beam splitters. Consequently, such sensor systems (orimaging systems) involve ne optical alignments, which in turn require a sophis-ticated control mechanism hardware. These complications naturally increase thecost and the load of the sensor system, a problem which can be overcome bya single detector responding in multiple bands. The multi-spectral features ob-tained with multi-band detectors are processed using color fusion algorithms1 inorder to extract signatures of the object with a maximum contrast. With thedevelopment of multi-band detector systems, there is an increased research1, 5effort to develop image fusion techniques. Fay et al.1 have reported a color-fusiontechnique using a multi-sensor imagery system, which assembled four separatedetectors operating in different wavelength regions. The major goal of my studyis to investigate multi-band detection concepts and develop high performancemulti-band detectors.

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At present, there are many applications where multi-band detectors are re-quired. In land-mine detection3 the number of false positives can be reducedusing multi-spectral approaches, allowing the the identication of real land-minesites. Military applications include the use of multi-band detectors to detect muz-zle ashes, which emit radiation in different wavelength regions,6 to locate theposition of enemy troops and operating combat vehicles. Multi-band focal planearrays (FPAs) responding in very-long-wavelength infrared (VLWIR) region (14-30 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 todetect the plume is impractical due to its low UV emission. As a solution, IRemission7 of the plume can be used instead of UV. Then the detector systemshould 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. Usinga two-color (or multi-color) detector, which operates in two IR bands where themissile plume emits radiation, the contrast between the missile plume and thebackground can be maximized. Moreover, a multi-band detector can be used as aremote thermometer4 where the objects radiation emission in the two wavelengthbands 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 electricaland optical properties of QDIP which can be used for device design optimizationfor better performance. Improving the device performance experimentally byfabricating and testing devices using combinations of different design parametersare very costly and time wasting. Hence, it is desirable to develop theoreticalmodeling based on the physical operating principals that can be used in char-acterization and optimizing the device performance through recommending thebest design parameters suitable to achieve specic characteristics.

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

The energy levels provided by the density of states give information aboutthe possible energy transitions and therefore the operating wavelengths of the detector.

The corresponding calculated wavefunctions are used to calculate the dipolemoment between different energy states which indicate the strength of thetransition rates between the energy states and therefore gives informationabout 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 estab-lish their effects on the operating wavelength and the corresponding valueof the absorption co-efficient. The research work presented in this thesishas resulted in the following publications:

1. Soumitra Roy Joy, Golam Md. Imran Hossain, Tonmoy Kumar Bhowmick,Farseem Mannan Mohammedy, Inuence of Quantum Dot Dimensions ina DWELL Photodetector on Absorption Co-efficient, in Proc. Fifth AsiaInternational Conference on Mathmematical Modeling and Computer Sim-ulation (AMS), Indonesia, 2012, page no. 225-230.

2. Soumitra Roy Joy ; Tonmoy Kumar Bhowmick;Golam Md. Imran Hos-sain; Farseem Mannan Mohammedy, Effect of Asymmetric Well Quantumdots-in-a-well in-frared photodetectors on Density of States Using NEGFFormalism, in Proc. International Conference on Solid State Devices andMaterials Science, April 1-2, 2012, China, page no. 299-304.

1.6 Thesis Outline

The thesis contains ve chapters. Chapter 1, above, discussed the different appli-cations of infrared detectors in medical and environmental sensing. The QDIP is a

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promising technology with advantages over current technologies based on QWIPand MCT. A literature review of QDIP modeling, including classical and semi-classical methods, has been discussed in addition to an overview of the NEGFmodeling that has been developed in the thesis.

Chapter 2 gives a review of the NEGF. The different Greens functions, theself energy, and scattering functions are presented as they will be used intensivelyin the following chapters of the thesis. Some applications of Greens functionhave 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 DOSof the QDIP. The localized DOS is obtained from the retarded Greens function.The retarded Greens function is obtained numerically by solving the governingkinetic equation using the method of nite differences. The model was appliedto calculate the DOS of DWELL (Dot in a Well) structure.

Chapter 4 shows the development of theoretical modeling to calculate thedipole moment and absorption co-efficient of the QDIP. The rst order dipoleapproximation and the Fermi-golden rule were used to model the interaction withlight. The bound states of the QDs have been obtained by solving the eigenvalueproblem of the QD Hamiltonian, while the continuum states have been obtainedusing the retarded Greens function. The model has been applied to the QDWELLstructure. The effect of changing the shape and size of QDs on the calculatedabsorption co-efficient has been studied using the DWELL structure.

Finally Chapter 5 concludes the thesis with the major ndings and the rec-ommended improvements and extensions for future research.

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Chapter 2

Non-equilibrium Greensfunctions

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 conceptsdeveloped to deal with particular aspects of quantum transport, the most generaland rigorous theoretical framework is provided by the so-called non-equilibriumGreens function theory (NEGF) developed by Keldysh and in a slightly differentform independently by Kadanoff and Baym . It took an amazing 40 years beforethis method became recognized and employed as the framework of choice for aquantitative and predictive analysis of carrier dynamics in semiconductor basednanostructures. There are probably several reasons why it took so long. First if

all, there are only few devices where quantum mechanical effects and incoherentscattering effects play an equally important role and call for a fully quantum me-chanical 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 betweencoherent tunneling and incoherent phonon emission processes. Secondly, NEGFcalculations for realistic devices are extremely demanding computationally andhave only become feasible recently. One of the rst detailed implementations

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CHAPTER 2. NON-EQUILIBRIUM GREENS FUNCTIONS 12

of NEGF to semiconductors and semiconductor devices has been developed byLake and coworkers and applied to NEMO-I D, a sophisticated simulation tool formulti-q uantum well devices such as resonant tunneling diodes . In their research,the authors laid down the framework for applying NEGF to semiconductors byderiving most scattering self-energies that are relevant for semiconductors. An-other major step forward was provided by Wacker et al., who provided severalin-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 semiconductornanostructures. Nevertheless, the fommlism still suffers from being consideredsomewhat obscure and difficult to grasp and handle. we will allermpt to give anoverview of the basic elements of the fommlism for nonexperts that may help todevelop a beuer feeling for strengths and weaknesses of NEGF.

Let us emphasize that NEGF is a solution framework rather than a concretemethod for calculating properties of open quantum devices. It ensures that thenon-equilibrium carrier distribution in a device is consistently calculated with theenergy, width and occupancy of its quantum mechanical eigenstates (scatteringstates. to be precise). Consentration of charge, momentum and energy is guaran-teed only if the scattering self energies are calculated exactly and self consistently,i.e. using fully dressed Greens functions and vertices in many-body terms. Thus.the unavoidable use of approximations requires much more effort and care than inother methods just to avoid artifacts such as a violation of current conservationwithin the device. The NEGF method is able to deal with explicitly timede-pendent as well as with stationary problems. However. Ihe calculation of therequired set of 4 types of Greens functions for time-dependent carrier dynamicsis still pretty much Oul of reach for a quantitative prediction of realistic devicestructures. Therefore. we limit the present discussion of stationary problemswhere 2 types of Greens functions suffice.

Non equilibrium Greens function methods are regularly used to calculate cur-rent and charge densities in nanoscale (both molecular and semiconductor) con-ductors under bias. An overview of the theory of molecular electronics can befound in [12] and for semiconductor nanoscale devices see [16]. The aim of thistext is to provide some intuitive explanations of one particle Greens functions in

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

2.2 Preliminary Concept

A proper treatment of quantum transport is one of the most difficult problemsto deal with in solid state theory. With aggressive device scaling, quantum me-

chanical phenomenon has become prominent for these nano-scale devices andhence quantum treatment in modeling has become a necessity. Self-consistentSchrdinger-Poisson solver is generally used for nano devices at equilibrium orvery close to equilibrium. One would start by solving Schrdingers wave equationfor the eigenstates

2

2m

2(r ) qV (r )(r ) = (r ) (2.1)

The eigenstates are lled according to Fermis function, and their squared

amplitudes give the charge density, taking into account the spin degeneracy:

n(r ) = 2 i

| i (r )|2 f (( i E F )/kT ) (2.2)

where:

f (x) = (1 + exp(x)) 1 (2.3)

The calculated charge density is then used in Poissons 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 proles have been calculated, a suitable transportmodel is used.

For devices far from equilibrium, the NEGF has become the method of choice.

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For closed systems, (channel) the governing equation is

[EI H ]{} = 0 (2.5)

Where H is the unperturbed Hamiltonian of the system, E is the energy of electron, I is an identity matrix, is the eigenstate or the electron wave function.If we consider an open system, we need to modify the governing equation asfollowing

[EI H ] {} = {S } (2.6)

Here, is the self-energy of different processes such as the contacts or thescattering centers within the device. S is the source term which tells us howelectrons 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

{} = [G]{S } (2.7)

And

[G] = [EI H ] 1 (2.8)

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G is called Greens function To calculate , one has to formulate the Hamil-tonian matrix, H; then insert the self-energy, ; calculate the Greens function, Gand lastly multiply G with the source term. Rigorous denition and justicationare covered in the next section.

2.3 Greens functions

Discrete Schrodinger equation:

H |n = E |n (2.9)

We divide the Hamiltonian and wavefunction of the system into contact(H 1,2, |1,2 ) and device (H d, |d ) subspaces:

We dene the Greens function 1:

(E H )G(E ) = I (2.10)

2.3.1 Why do we want to calculate the Greens function?

The Greens function gives the response of a system to a constant perturbation|v in the Schrodinger equation:

H | = E | + |v (2.11)

The response to this perturbation is:

(E H ) | = | v (2.12)

| = G(E ) |v (2.13)

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

1 Others may (and do) use the opposite sign.

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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., thewavefunction of the contact |2 can be calculated if we know the wavefunctionon the device |d . From third row of Eq. 2:

H 2 |2 + 2 |d = E |2 (2.14)

(E H 2) |2 = 2 |d (2.15)

|2 = g2(E ) 2 |d (2.16)

where g2 is the Greens function of the isolated contact 2 (( E H 2)g2 = I ).It is important to note that since we have an innite system, we obtain two

types of solutions for the Greens functions 2, the retarded and the advanced 3

solutions corresponding to outgoing and incoming waves in the contacts.Notation: We will denote the retarded Greens function with G and the ad-

vanced with G+

(and maybe GR

and GA

occasionally).Here, CAPITAL G denotesthe full Greens function and its sub-matrices G1, Gd, G1d etc. Lowercase is usedfor the Greens functions of the isolated subsystems, e.g., ( E H 2)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 ( g+2 ) would give the solution correspondingto an incoming wave.

2.3.2 Self-EnergyThe reason for calculating the Greens function is that it is easier that solvingthe Schrodinger equation. Also, the Greens function of the device ( Gd) can be

1 Especially for innite systems.2 When the energy coincides with energy band of the contacts there are two solutions cor-

responding 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 canbe seen from the Fourier transform of the time dependent Greens function.

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calculated separately without calculating the whole Greens function (G). Fromthe denition of the Greens function we obtain:

Selecting the three equations in the second column:

(E H 1)G1d 1Gd = 0 (2.17)

+1 G1d + ( E H d)Gd 2G2d = I (2.18)

(E H 2)G2d 2Gd = 0 (2.19)

We can solve Eqs. 11 and 13 for G1dandG 2d :

G1d = G1 1Gd (2.20)

G2d = G2 2Gd (2.21)

substitution into Eq. 12 gives:

1 + g1 1Gd + ( E H d)Gd +2 g2 2Gd = I (2.22)

from which (Gd) is simple to nd:

Gd = ( E H d 1 2) 1 (2.23)

where 1 = +1 g1 1 and 2 = +2 g2 2 are the so called self-energies.

Loosely one can say that the effect of the contacts on the device is to addthe self-energies to the device Hamiltonian since when we calculate the Greensfunction on the device we just calculate the Greens function for the effectiveHamiltonian H effective = H d + 1 + 2.However, we should keep in mind that wecan 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 calculatethe self-energies are usually calculated using the periodicity of the contacts, thismethod is described in detail in appendix B of [ 4] and in section 3 of [16].

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2.3.3 The spectral functionAnother 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 rst note that for any perturbation |v we get two solutions

( R and A ) to the perturbed Schrodinger equation:

(E H ) | ) = | v (2.25)

from the advanced and retarded Greens functions:

R = G |v (2.26)

A = G+ |v (2.27)

The difference of these solutions( R A = ( G G+ ) |v = iA |v ) is asolution to the Schrodinger equation:

(E H )( R A ) = ( E H )(G G+ ) |v = ( I I ) |v = 0 (2.28)

which means that | = A |v is a solution to the Schrodinger equation forany vector |v .

To show that the spectral function actually gives all solutions to the Schrodingerequation is a little bit more complicated and we need the expansion of the Greensfunction in the eigenbasis:

G = 1

E + i H =

k

|k k|E + i k

(2.29)

where the is the small imaginary part (see footnote 3), |v s are all eigenvec-tors 1 to H with the corresponding eigenvalues k . Expanding the spectral function

1 Normalized!

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in the eigenbasis gives:

A = i( 1

E + i H

1E i H

) (2.30)

A = ik

|k k| ( 1

E + i k

1E i k

) (2.31)

A =k

|k k| 2

(E k)2 + 2 (2.32)

where is our innitesimal imaginary part of the energy. Letting go to zerogives:

A = 2k

(E k) |k k| (2.33)

(here (E k) is the delta function) which can be seen since 2(E k )2 + 2 goes tozero everywhere but at E = k ,integrating over E (with a test function) gives the2 (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 in- ject electrons and occupy the states corresponding to incoming waves in the con-tacts. Therefore, we want to nd the solutions corresponding to these incomingwaves.

Consider contact 1 isolated from the other contacts and the device. At acertain energy we have solutions corresponding to an incoming wave that is totallyreected at the end of the contact. We will denote these solutions with |1,nwhere 1 is the contact number and n is a quantum number (we may have severalmodes in the contacts). We can nd all these solutions from the spectral functiona1 of the isolated contact (as described above).

Connecting the contacts to the device we can calculate the wavefunction onthe whole system caused by the incoming wave in contact 1. To do this we note

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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 currentand the charge density matrix. Lets start with the charge density (which allowsus to use a self-consistent scheme to describe charging).

The charge density matrix is dened as:

=k

f (k, ) |k k | (2.34)

where the sum runs over all states with the occupation number f (E k , ) (puredensity matrix) (note the similarity with the spectral function A, in equilibriumyou nd the density matrix from A and not as described below). In our case, theoccupation number is determined by the reservoirs lling the incoming waves inthe contacts such that:

f (E k , 1) = 11 + e

E k 1k B T

(2.35)

is the Fermi-Dirac function with the chemical potential 1 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 (seeEq. 32) is:

|d,k = Gd +1 |1,k (2.36)

Adding up all states from contact 1 gives:

d[contact 1] =

E = dE

k

f (E, 1) (E E k) |d,k d,k | (2.37)

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2.6 Boundary conditionsIn quantum transport, the treatment of boundary conditions requires signi-cantly more care than in classical physics due to the nonlocality of quantummechanics.[20],[11],[23],[17]. A common problem is to dene Ohmic leads. In thecontext of NEGF, we may dene leads as Ohmic if the current is controlled bythe interior of the device rather than by serial resistances or interface states. Thisdenition has several implications. First, there must be a smooth transition inthe density of states of the leads and the device. Secondly, the same scatteringmechanisms must act within the leads and the device. Thirdly, the carrier dis-tribution within the leads must be a suitably accelerated Fermi distribution toreect current conservation. All of these conditions are necessary to avoid quan-tum mechanical reections and pile up of charge at the interface.[ 21],[24] This isillustrated in Figure . Figure shows a full NEGF calculation of the carrier densityin 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 thecurrent from right to left. If the leads are treated ballistically, the carriers ac-

cumulate near the device boundaries due to the resistance they meet within themedium. Only by employing the same scattering mechanisms and self-energieseverywhere, the formalism yields a constant carrier density throughout the sys-tem. Figure illustrates the calculated carrier density in a biased n++-i-n+ GaAsstructure. If the carrier distribution within the leads is assumed to be in equilib-rium, one obtains an articial pinch-off behavior; there is a depletion of carriersnear the source side and an accumulation near the drain side of the device.

2.7 The semiclassical limitDoes the NEGF formalism reduce to the semiclassical limit when quantum effectsplay no role? [22],[15] For a realistic device structure, this is difficult to proverigorously but the following example provides practical evidence that, indeed, theanswer 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

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semi-classical Boltzmann equation to adequately describe the carrier density, as-suming one includes impurity and phonon scattering in the standard fashion andincludes electron-electron scattering at least within the Hartree approximationvia the Poisson equation. We have performed a charge self-consistent NEGFcalculation that takes into account the same type of scattering mechanisms. Asshown in Figure , the NEGF electron density mimics faithfully the semiclassicalresults.

Figure 2.2: (Color online) Left: Calculated carrier density for a homogeneouslydoped semiconductor at zero bias, attached to leads with an equilibrium distri-bution. The expected homogeneous density is obtained only if the leads includethe same type of scattering self energies as the device (red curve). Otherwise oneobtains articial charge accumulation (blue curve). Right: GaAs n-i-n structureat room temperature under bias with asymmetric doping prole as indicated bythe grey lines.Once a current is owing. the charge distribution within the leadsmust be a suitably shifted Fermi distribution that reects global current con-servation (results shown by thc red curve). Otherwise, NEGF calculations yield

articial pinch-off effects (blue lines)

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

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

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2.8 Quantum well structures: Why NEGF is es-sential

A nice illustration of the power of NEGF can be obtained by calculating the localcarrier distribution in a biased n-i-n structure that contains a quantum well. Thisis illustrated in Figure for a n-i-n structure with a 12 nm InGaAs quantum wellin the intrinsic zone. The applied voltage across the 50 nm structure is 150mV and we show 3 results. The red curve shows the semiclassical Boltzmann

results and yields the well-known accumulation of charge near the quantum wellbarriers. This method completely ignores the existence of quantum mechanicalbound states and reects the classical Thomas-Fermi density. The blue curveillustrates another limiting case, namely the solution of the Schrdinger equation inthe absence of any scattering. This reects a strictly coherent, energy conserving,ballistic transport. In this case, the carrier density within the device is fullydetermined by the overlap of the lead wave functions with the device. Sincethere are no lead carriers below the GaAs band edge, the quantum well states

in the intrinsic region remain unoccupied. Thus, the electron density withinthe quantum well only stems from (continuum state type) lead electrons whichexplains the oscillatory density in the intrinsic region. Finally, the black curverepresents the full NEGF calculation. The inelastic scattering processes lead toa capture of carriers into the quantum well states and lead to a carrier density inthe intrinsic zone that lies in between the semiclassical and the strictly ballisticquantum mechanical calculations.

This result can be further illustrated by plotting the energy resolved densityfor 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 impu-rity, phonon, and interface scattering self energies set equal to zero. Note thatthe Poisson equation is still solved self-consistently with the Schrdinger equationeven in this case. Figure , on the other hand, shows a complete NEGF calculationthat clearly illustrates the carrier capture into the rst and second bound state of the quantum well. Due to the charging effects caused by the capture, the bottomof the quantum well raises in energy so that both states actually form resonancesthat slightly overlap with the lead states.

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Figure 2.5: Contour graph of calculated energy resolved electron density for 50nm n-i-n structure at room temperature with a 12 nm lnGaAs quantum well asintrinsic zone for zero applied bias. The density scale is the analogous to theone in Fig. 5. but for lower doping. (a) Strictly ballistic NEGF calculation (noscattering included). (b) Fully self-consistent NEGF calculation

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

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2.9 Limits and simplications in NEGFAs mentioned in the introduction, the NEGF formalism guarantees conserva-tion of important physical principles only if all Greens functions are evaluatedexactly from a many-body standpoint. It is a very signicant weakness of themethod that even plausible approximations can fail badly. Generally, it is diffi-cult to introduce simplications that do not violate basic conservation laws. Asan example, we discuss the so-called decoupling approximation.[ 10] The NEGFformalism couples the energy of states with their occupation via 4 coupled integro-differential equations. If the carrier density is not too high, it seems plausible todecouple the equations of and . This approximation can lead to a violation of Paulis principle, however, since there is no mechanism that prevents the occur-rence of over-occupied states (i.e. states with occupancy higher than permittedby the Pauli principle).[ 10] To exemplify this situation, we consider the sameGaAs-InGaAs-GaAs n-i-n structure as before, but with a slightly higher carrierconcentration in the n-region. Figure shows the energy resolved carrier densityof 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, leadingto unphysically high occupation of the lowest bound state.

2.10 THz quantum cascade lasers: a classics forNEGF

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

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experiment agree very well with one another up to the voltage where lasing startsand both thermal as well as hot electron effects become relevant that have notbeen included in the calculations.

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

2.11 Comparison between NEGF and Monte -Carlo

2.11.1 Monte-Carlo

Monte Carlo simulation performs calculation by building models of possible re-

sults by substituting a range of values-a probability distribution-for any factorthat has inherent uncertainty. It then calculates results over and over, each timeusing a different set of random values from the probability functions. Dependingupon the number of uncertainties and the ranges specied for them, a MonteCarlo simulation could involve thousands or tens of thousands of recalculationsbefore it is complete. Monte Carlo simulation produces distributions of possibleoutcome values. By using probability distributions, variables can have differentprobabilities of different outcomes occurring. Probability distributions are a much

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more realistic way of describing uncertainty in variables of a device performanceanalysis.

Through the Non-Equilibrium Greens Function (NEGF) formalism, quantum-scale device simulation can be performed with the inclusion of electron-phononscattering. However, the simulation of realistically sized devices under the NEGFformalism typically requires prohibitive amounts of memory and computationtime. Two of the most demanding computational problems for NEGF simulationinvolve mathematical operations with structured matrices called semiseparable

matrices.Nonequilibrium Greens function method is a very general scheme to include

coherent quantum mechanics and incoherent scattering with phonons and deviceimperfections self-consistently. However, it is numerically very demanding andcannot be used for systematic device parameter scans. For this reason, we alsoimplement the approximate, numerically e?cient ensemble Monte Carlo methodand assess its applicability on the above mentioned transport problems.

A research paper [2] shows that the approximate treatment of coherent tunnel-ing and leakage into continuum states limits the applicability of the EMC methodon transport regimes that are dominated by incoherent scattering. When all de-vice states that contribute to transport are clearly non-degenerate, results of thecurrent density obtained by the EMC method quantitatively agree with NEGFresults and experiment. Also the simulated spectral gain pro?le is in good agree-ment for both methods. This is in particular important because the numericalload of NEGF calculations exceeds the load of the EMC method tremendouslyand typically prohibits a systematic improvement of QCL designs.

2.12 Why nonequilibrium Green functions?

Let us briey describe its main features:

The method has as its main ingredient the Green function, which is a func-tion of two spacetime coordinates. From knowledge of this function one cancalculate time-dependent expectation values such as currents and densities,electron addition and the total energy of the system.

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In the absence of external elds the nonequilibrium Green function methodreduces to the equilibrium Green function method which has had imporatntapplications in quantum chemistry.

Nonequilibrium Green functions can be applied to both extended and nitesystems.

The nonequilibrium Green function can handle strong external elds non-perturbatively. The electron-electron interactions are taken into account by

innite summations.

The approximations within the nonequilibrium Green function method canbe chosen such that macroscopic conservation laws as those of particle num-ber, momentum and angular momentum are automatically satised

Dissipative processes and memory effects in transport that occur due toelectron-electron interactions and coupling of electronic to nuclear vibra-tions can be clearly diagrammatically analyzed

2.13 Conclusions

The NEGF formalism provides the framework of choice for consistent carrier dy-namics calculations of open nanosystems where quantum effects and incoherentscattering play a comparable role. When implemented with care, it reproducesthe results of the semiclassical Boltzmann equation in the limit where quantumeffects such as resonant tunneling and interference are unimportant. By deni-

tion, it also reproduces the solutions of the Schrdinger or Lippmann-Schwingerequation when inelastic scattering is turned off. A disadvantage of the methodis the fact that charge and current conservation, and even Paulis principle arenot automatically satised once seemingly reasonable approximations are intro-duced. Scattering vertices must necessarily be taken into account to inniteorder, for example, to strictly obey current conservation and it is difficult toachieve a numerically satisfactory convergence. Approximations are unavoidable,though, once one seeks predictions for realistic nano-devices, simply due to the

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

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Chapter 3

QDIPs density of states modeling

3.1 DWELL QDIP Structure

A theoretical model of pyramidal-shaped InAs quantum dots placed in an InGaAsquantum 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 statesin the quantum well, the model has to be modied to account for the free motionof 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 In 0.15Ga 0.85As, 2.4 ML of InAs, 6nm In 0.15Ga 0.85As, and 50 nm GaAs, as shown in Fig. 3.1. The quantum dotsare placed in the In 0.15Ga 0.85As quantum well which is in turn surrounded by theGaAs 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 basedimension is of 11 nm and height is of 6nm. The QD material InAs is depositedover the substrate and due to the lattice mismatch between deposited materialand 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

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CHAPTER 3. QDIPS DENSITY OF STATES MODELING 32

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

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

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dislocation free QD islands begin to grow.

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

Figure 3.4: Theoretical Modeling of DWELL QDIP Structure

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For numerical analysis we have modeled the DWELL device as shown in Fig3.1. 60% of the band gap difference between InAs and GaAs is counted as theconduction band offset .The band offsets calculated are 477 meV between InAsand I n 0.15Ga 0.85As and 93 meV between I n 0.15Ga 0.85As and GaAs. The conduc-tion band edge of the In0.15Ga0.85As is selected as reference energy level. Alinear interpolation between two binary values is used to calculate the effectivemasses in the different materials. The effective masses used for GaAs, InAs andIn 0.15Ga 0.85As 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 thedevice, a cross section along the cylinder axis is taken and disintegrated into alarge number of equally spaced grids. The nite difference method is used tosolve the differential equation governing Greens function. The retarded Greensfunction of the system is dened as

[E H op r (E )]Gr (x, x ,y,y , E ) = (x x ) (y y ) (3.1)

Here E is the total energy of electron, r is the self energy and H op, the Hamil-tonian operator of the system, is given by

H op = . 2

2m(x, y) + V (x, y) (3.2)

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

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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 unitvolume, is given by

N (E ) = 12

T r(A(x, y; E )) (3.4)

3.2 Numerical AnalysisThe quantum dot photodetectors under our analysis have an estimated dot den-sity of 5 1010cm2, and the average spacing between two adjacent dots is about60nm. Due to this relatively large distance, in our simplied model of the quan-tum dot photo-detector, the neighboring dots are assumed to be vertically andlaterally decoupled, and a quantum dot is modeled so as to be surrounded bysemi-innite contact composed of InGaAs and GaAs layer, and InAs wettinglayer. The contact can be thought of being a continuation of cylinder radius and

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the quantum dot exists at the center of the cylinder.Modeling quantum dot photo-detectors in such a way gives the benet of ex-

ploiting the property of translational invariance of the contact. The Hamiltonianmatrix for the device, which is tridiagonal and Hermitian, is formed by nitedifference method and is given by

H =

x1 1 0 . . . 0

1 x2 2 . . . ..

.0 .. .... .. . Nx 10 . . . . . . Nx 1 x(Nx )

Here,

x (i) =

U y1 + 2 tx (i) + 2 ty1 ty1 0 0 ty1 U y2 + 2 tx (i) + 2 ty2 ty2 0 0

... ty2. . . ...

tyNy 10 tyNy 1 U y(Ny ) + 2 tx (i) + 2 tyNy

(3.

( i) =

tx 1,i 0 0

0 tx 2,i...

... .. . ...

0 tx Ny,i

(3.6)

3.3 Calculation of Self Energy:

To illustrate the self-energy calculation which accounts for the device leads, weconsider the effect of coupling the active device Hamiltonian to the drain. Theinnite Hamiltonian and its Greens function can be partitioned as follows:

In the above equation, the subscript lead indicates innite block of matrices

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(H matrix and G matrix). The matrix block we only care about is Gdevice as weare not interested in the Greens function within the Lead or Source. Gdevice canbe expressed in terms of known quantities as follows:

Gdevice = [EI hdevice r ] 1 (3.7)

Where the self energy term is

r =

0 0 0 0

EI Nx +1 0 EI Nx +2 0 0

1 0

0 0

(3.8)

For evaluating the matrix product in above equation, we only need the rstblock of the inverse of the infnite matrix associated with the Lead. Moreover, the

diagonal blocks of this innite matrix are repeated due to translational invariancewithin the Lead.

Nx = Nx +1 = Nx +2 = (3.9)

Using this property, and partitioning the matrix, a close form of the matrixfor the rst block of the inverse (denoted by gc) of the innite matrix, can beobtained as

I = gc[EI Nx +1 c ] (3.10)

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Once gchasbeensolved,wehavetheselfenergyterm

r =

gc + 0 00 0 0... .. .

...... . . .

...0 gc +

(3.11)

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

is given by

tx = 2

2mxa2 (3.12)

gc is the retarded greens function of a unit cell of the contact, and is solvedfrom the recursive relation

g 1c = (( E + i)I H c c + ) (3.13)

It is noteworthy that only the last vertical slice of the device couples to theLead. Therefore, the self-energy for the Lead has a single nonzero block thatperturbs the last diagonal block of the device Hamiltonian. To solve for gc, abasis transformation has to be performed. The eigenvectors of EI diagonalizegc simultaneously. Therefore we change the basis from 2D real space to a basisthat is composed of the eigenvectors of EI (equivalent to a mode-spacetransformation at the boundary). This reduces the equation related to gc to aset of decoupled quadratic equations that can be solved for the diagonal entriesgc , in the transformed representation. It should be noted that each of theseequations results in two roots. The root representing outgoing waves is selectedas we are ultimately interested in obtaining the retarded Greens function for thedevice. An inverse basis transformation is then applied to evaluate gc in 2D realspace. A similar procedure is invoked to solve for the self-energy part associatedwith the Lead of other side. The nal size of the self-energy matrix is ( N x N y)2

for the real-space solution and ( N x )2for the decoupled mode-space solution.

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3.4 Computational ChallengesThe device that we have dealt with is of GaAs/InGaAs/InAs heterostructure.For numerical analysis in FDM (Finite Difference Method), we have disintegratedthe device structure into a large number of grids. For instance, a 40nm by 30nmcross section of the device was divided into near about 30000 grids. A pictorialrepresentation 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 separatedby 0.2nm distance. Thus in the entire device, there were a total of 30,000 grids.

Along X axis - number of grid per row is 30nm0.2nm = 150 gridsAlong Y axis - number of grid per row is 40nm0.2nm = 200 gridsHence 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 devicesomewhere later, but for instance, lets take a look at what are the problems asso-ciated with calculation that involve such a large number of grid points. In nitedifference method, the matrix size of the Hamiltonian of a device is proportionalto the square of the total number of grid point representing the entire device. For

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how we confronted our very rst challenge.

3.5 How to confront the challenges

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

3.5.1 Reduce the number of gridThis is probably the very rst (and the most naive, however!) idea that may comeacross ones mind in such a case of limited computer memory. But it should bekept in mind that the separation between adjacent grids would have to be suchthat even the thinnest region of the device cross section is to be covered bysufficient number of grid points.

In our device, the thinnest region is the wetting layer of InAs, whose thicknessis of 0.5 nm. Hence, if grid separation is more than 0.2 nm, the wetting layerwill contain insufficient number of grid, thereby the effect that the wetting layerexerts on the device characteristics will not be properly reected in the simulatedmodel.

3.5.2 Non-uniform distribution of grid point

The non-uniform distribution of grid point throughout the device can be a smartand intelligent way to make an economical use of available memory space whileincorporating the effect of all the region simultaneously. In fact, Comsol simu-lator provides us with the feature of taking dense mesh points in some regionwhereas less dense or less concentrated mesh points in some other region as perthe requirement of the user. However, in numerical analysis method performedin MATLAB, we have to provide with formula that rightly ts for non-uniformdistribution of grid point. In contrast, the formula we could have laid our handon was derived for uniform distribution of grid point. We, thus, were left withthe choice of formulating a new equation for our own pursuit, or leave this wayof solution as an unviable one.

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3.5.3 Disintegrating a large matrix into several smallermatrices

Segregating a large matrix into several smaller ones can be another potentialsolution to our problem. Our mathematical analysis frequently needed matrixinversion operation, which is considered one of the most tedious and memoryexpansive job for a calculating machine. Since the matrix we dealt with is incred-ibly large in size, a direct command of matrix inversion (i.e. inv (Matrix)) woulddenitely end up with memory overow. An alternative way of inverting a matrixis 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 MAT-LAB 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 71 5

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

b1 b2b3 b4

From here, we can write:

2 7

1 5

b1 b2

b3 b4 =

1 0

0 1So

2 71 5

b1b3

= 10

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

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2 71 5

b2b4

= 01

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

3.5.4 Dealing with Sparse Matrix other than conventional

matrixSparse matrix are those, whose elements are mostly zero. A zero dominated sparsematrix can easily be manipulated to save memory since its zero elements do notneed a separate memory space to occupy. The Hamiltonian that is concernedin our device simulation is a sparse matrix. It contains non-zero elements alongthe diagonal and off-diagonal position. The remaining elements are zero (unlessany special circumstances appear, like: boundary condition). In fact, the sparsematrix springs off in almost all device simulation, since we usually consider atight binding model of the device (a model where wave function of electron at acertain location is only inuenced by nearmost grid points, coupling occurs onlybetween neighboring grid points)

3.5.5 Using all the processors in a computer in a parallelprocess

By default, a MATLAB program engage only one core in a computer for com-

putational purpose at a time. But special arrangement and coding can be doneso 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 canbe done in a MATLAB program can be viewed from the following website:

a) Parallel Computing Toolbox Perform parallel computations on multicorecomputers, GPUs, and computer clusters

http://www.mathworks.com/products/parallel-computing/b) Multicore - Parallel processing on multiple cores

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

3.6 Result- Our Findings

A group of researchers led by Professor Sanjay Krishna reported their Spectralresponse measurements performed on the DWELL detectors with a Nicolet 870FTIR spectrometer and a Keithley 428 current-amplier. [1] Figure 3.9 shows thespectral 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 wasunchanged to minimize changes in the dimensions of the dots. As expected, theoperating wavelength of the detector showed a monotonic red shift from 7.2 to 11m from samples A to F. This is signicant since it provides us with a method of controlling the operating wavelength of a QD detector. Note that in the samplewith 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 higherlying state in the InGaAs quantum well.

Professor Krishna and his collaborators obtained the responsivity from a 15-stack quantum DWELL detector. The measured responsivity was divided by afactor of 4 to account for the scattering in the substrate. Far-infrared (FIR) spec-tral response measurements were also undertaken in collaboration with Pererasgroup [9]. The data were obtained using a Perkin-Elmer system 2000 FTIR withtwo sets of beamsplitters and windows and were corrected by background spectra.

The resulting three-color response is shown in gure 3.10. The rst twoMIR peaks, i.e. 10 and 5 m, have previously been observed by the same re-searchers group on similar detector structures [ 9]. The researchers group believesthat the peak at 10 m(124meV < Ec ) is probably a transition from a boundstate in the dot to a bound state in the quantum well, whereas the peak around5m(250meV > Ec ) is possibly a transition from a state in the dot to a quasi-bound state close to the top of the well as shown in the inset to gure . A FIRpeak centred around 25 m was also observed in these detectors and is shown ingure 3.11. The researchers group believes that this peak could possibly be dueto transitions between two states in the QD since the calculated energy spacingbetween the dot levels is about 50-60 meV (20 25m).

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

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

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

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3.7 Comparing Experimental results with sim-ulated 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 absorptionat these wavelengths are approximately 250, 113 and 50 meV.

E = hce

eV (3.14)

E 1243(m)

meV (3.15)

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 prole) 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 prole for bottom well width=6nm

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The experimentally measured spectral response of the asymmetric DWELLwith increasing bottom well width shows a monotonic red shift [Ref: Quantumdots-in-a-well infrared photodetectors, Sanjay Krishna]. This result can be intu-itively expected because increasing Quantum Well width means decreasing theconnement 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 yieldsthe same expected result. From gure 3.13, we see, the width of QuantumWell hardly has any prominent effect on the position of ground state energy,

though it signicantly changes other levels, causing a monotonic shift of DOStowards lower energy level with gradually increasing Well width. An interestingobservation is that, the higher energy levels are getting closer to ground state ata rate faster than lower energy level, thus reducing the relative distance betweenany two energy levels we consider for transition. The corresponding energy levelsare shown in table 1 and their differences are shown (in meV) in table 2.

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Figure 3.14: Effect of changing well width on DOS

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

E 0 -270.3 -270.3 -270.3E 1 -23 -26 -28E 2 79 74 64

Energy difference in meV E 1 E 0 E 2 E 0 E 2 E 13 nm Well 247.3 349.3 1024 nm Well 244.3 344.3 1006 nm Well 242.3 334.3 92

It is a point of interest to see how different sites of the device contribute todifferent energy states. Figure 3.15 shows the potential prole for the DWELLmodel with 6 nm bottom half Quantum Well.

From gure 3.16, we see, at the ground level (E= -278meV), energy statedistribution takes a triangular form reveals the fact that at this bottom-mostenergy level, quantum state is provided solely by the quantum dot.

Though at E = -62 meV, provision of energy state by the device is of minimalvalue (of order 10 8 ) and hence has no signicant number of state in practice,

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Figure 3.15: Potential Prole

Figure 3.16: LDOS at E=-270.3208mev

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

Figure 3.17: LDOS at E=-61.8225mev

From gure 3.18, we can see, at the rst 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), theenergy states resides prominently at InGaAs Quantum Well sites.

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Figure 3.18: LDOS at E=-15mev

Figure 3.19: LDOS at E=77mev

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Chapter 4

Dipole Moment and AbsorptionCoefficient

4.1 Transitional Dipole Moment

The Transition dipole moment or Transition moment, usually denoted nm for a

transition between an initial state, m, and a nal state, n, is the electric dipolemoment associated with the transition between the two states. In general thetransition dipole moment is a complex vector quantity that includes the phasefactors associated with the two states. An oscillating electric or magnetic mo-ment can be induced in an atom or molecular entity by an electromagnetic wave.Its interaction with the electromagnetic eld is resonant if the frequency of thelatter corresponds to the energy difference between the initial and nal statesof a transition ( E = h). The amplitude of this moment is referred to as the

transition moment. Its direction gives the polarization of the transition, whichdetermines how the system will interact with an electromagnetic wave of a givenpolarization, while the square of the magnitude gives the strength of the inter-action due to the distribution of charge within the system. The SI unit of thetransition dipole moment is the Coulomb-meter (Cm); a more conveniently sizedunit is the Debye (D).

The interaction energy, U, between a system of charged particles and an elec-

53

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4. Dipole Moment and Absorption Coefficient 54

tric eld, E, is given by:

U = E (4.1)

The dipole moment is dened for a collection of charges by

=i

q ir i (4.2)

Where ri is the position vector of charged particle i. The expectation value

of the interaction energy is

U = n . E n d (4.3)If we assume that the magnitude of the electric eld is constant over the length

of the molecule (and that is nite only over the length of the molecule) we canwrite

U = .E (4.4)

where

= n n d (4.5)i.e. the strength of interaction between a distribution of charges and an electric

eld 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, i , to a nal state f , the transitiondipole moment integral is.

f i = f id (4.6)the probability for a transition (as measured by the absorption coefficient) is

proportional to f i f if i may be positive, negative or imaginary. If f i then the interaction energy is

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4. Dipole Moment and Absorption Coefficient 55

zero and no transition occurs - the transition is said to be electric dipole forbidden.Conversely, if f i islarge, thenthetransitionprobabilityandabsorptioncoeff icientarelarge.

The intensity of the transition is therefore proportional to

k j d 2

The dipole moment operator for an electron in one dimension is

f i = e f x idx = e f |x| i (4.7)Notes on transitional (dipole) moment: 1. The absorption probability for lin-

early polarized light is proportional to the cosine square of the angle between theelectric vector of the electromagneti

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