PHONON TUNING AND RECYCLING IN PHOTONIC ENERGY CONVERSION: ATOMIC-STRUCTURE METRICS AND EXAMPLES. Jedo Kim

Transcription

1 PHONON TUNING AND RECYCLING IN PHOTONIC ENERGY CONVERSION: ATOMIC-STRUCTURE METRICS AND EXAMPLES by Jedo Kim A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in the University of Michigan 2011 Doctoral Committee: Professor Massoud Kaviany, Chair Professor Roy Clarke Professor Stephen R. Forrest Professor Roberto D. Merlin

2 Jedo Kim c 2011 All Rights Reserved

3 ABSTRACT PHONON TUNING AND RECYCLING IN PHOTONIC ENERGY CONVERSION: ATOMIC-STRUCTURE METRICS AND EXAMPLES by Jedo Kim Chair: Massoud Kaviany Phonon generation in photonic energy conversions is inevitable and till now these phonons have been removed as heat to achieve stable operation. In this study, through various phonon-assisted transition processes, we propose recycling these phonons thus reducing the heat losses. This phonon recycling increases the energy conversion efficiency and reduces the cooling load and its associated energy consumption. This theoretical work is done through the fundamentals pertaining to the atomiclevel carrier kinetics and the structural metrics of the phonon recycling in photonics. The photonic systems considered are the ion-doped laser, amorphous-silicon solar photovoltaic, and potential-barrier integrated light emitting diode. First the carriers (phonon, electron and photon) interaction kinetics in the anti-stokes cooling of solids is analyzed starting with the Fermi golden rule applied to the weak photon-electronphonon couplings. Then the influence of phonons (equilibrium and nonequilibrium) is established in the related emission, transport and absorption transitions. Finally these are used to optimize the phonon-assisted processes, and improve process performance. Ab-initio simulations are used to guide and complement the theoretical ii

4 treatments. Based on the harmonic oscillator assumption, a general guide is proposed for selection of the optimal host glass constituents for laser cooling. Using Li, Al, and Na bonds with F gives the largest cooling rate improvement up to 200 % over the currently used blends. It is predicted that the proposed phonon-assisted photon-absorbing (cooling) layer in ion-doped lasers carries away up to 35 % of the generated phonon as spontaneous photon emission, thus improving the laser portability. The low phonon energy Sn alloying proposed for the amorphous silicon solar photovoltaic, is predicted to enhance the current generation by up to 11 %. The integrated, potential-barrier layer proposed for recycling the phonons generated by nonradiative recombination processes in light emitting diodes, absorbs multiple phonons and is driven by external potential. We predict up to 30 % phonon recycling for a barrier height of 0.28 ev. In all examples, efficiency gains are predicted from the proposed phonon recycling. This study is an atomic-level oriented contribution to the thermal management (and its effects on efficiency) of energy conversion devices. iii

5 Acknowledgements Since starting my graduate studies here in Ann Arbor, I have been fortunate to meet many outstanding researchers, educators, labmates, and friends. Without their help and inputs, it would have been impossible to complete this work. I would like to specially thank Professor Massoud Kaviany, my advisor, for being a constant source of motivation and valuable discussions which lead to novel ideas. I am thankful to Professors Roy Clarke, Stephen R. Forrest, and Roberto D. Merlin, my Ph.D. Committee members, for their thesis evaluation and for valuable comments. I would like to express gratitude to Professors Stephen Rand, Jasprit Singh, and Kim Winick for discussions related to the carrier interactions and to lasers in general. I am thankful to Professor David Drabold from the University of Ohio, and Professors Steve Yalisove, and John Kieffer for discussions related to amorphous structures and fabrication of a-si alloy. I am also grateful to Dr. Mehmet Arik of the GE Global Research for helpful comments on LED. I would like to extend special thanks to all my current and the former and labmates; Xiulin Ruan, Baoling Huang, Gisuk Hwang, Dahye Min, Seungha Shin, and Hyoungchul Kim for their guidance and extensive discussions related to research and everyday life. Finally, I greatly appreciate my iv

6 parents constant support and patience during many years of education in Korea, Canada, and the United States. This work was supported by the NSF (Thermal Transport and Processes) grants CTS and CTS , and by the DOE grant to the Center for Solar and Thermal Energy Conversion, University of Michigan. v

7 Table of Contents Abstract ii Acknowledgements iv List of Figures x List of Tables xx Nomenclature xxii Abbreviations xxvii Chapter 1 1 Introduction Phonons in photonic energy conversion Fermi golden rule Phonon recycling Statements of objective and scope of thesis Kinetics of electron-phonon-photon couplings 9 vi

8 2.1 Absorption Single-photon absorption Phonon-assisted photon absorption Multiphonon absorption Transmission Optical and acoustic phonon transmission Phonon up and down conversion Emission Summary Phonon tuning in laser cooling of rare-earth ion doped solids Introduction Cooling rate Electron-Photon Interaction Charge-Displacement Approximation for μ ph-e Judd-Ofelt Theory Estimation of Transition Dipole Moment Electron-Lattice Interaction for Optical Phonon Electron-Phonon Coupling Phonon DOS Estimation Cooling Rate Optimal Photon Frequency Optimal Host Pairs Discussion vii

9 3.5.1 Transition Dipole Moment Optimization Limits in Laser Cooling of Solids Off-Resonance Transition Dipole Moment Time Scales for Laser Cooling of Solids Summary Phonon recycling in rare-earth ion-doped lasers Introduction Ion-doped Lasers and Anti-Stokes Cooling Kinetics Phonon Transmission and Up Conversion Nonequilibrium Phonon Distribution and Photon Re-absorption Improved Laser Efficiency with Phonon Recycling Summary Phonon-assisted enhanced absorption of alloyed amorphous silicon for solar photovoltaics Introduction Absorption Coefficient of a-si: H Photon-Electron-Phonon Interactions and Absorption Electronic Structure of a-si x X 1 x Phonon Density of States of a-si x X 1 x Phonon-Coupling Enhanced Absorption and Current Density Effects on Carrier Transport Summary viii

10 6 Phonon tuned thermionic cooling in light emitting diodes Introduction Integrated LED and BTC Layers Carrier kinetics in LED and BTCL Phonon recycling in LED Summary Summary and Future work Contributions Proposed Future Work Outlook Appendix 106 A Ab-initio calculations of f -orbital electron phonon interaction in laser cooling 106 A.1 Introduction A.2 Calculation A.3 Results and Discussion A.4 Summary Bibliography 122 ix

11 List of Figures 1.1 Summary of the phonon recycling introduced in photonic energy conversion systems. The input and output energy paths are shown with the light grey arrows, while the phonon energy path is shown in different shade. The photon, electron, and phonon energy conversion regimes are labeled and the carrier couplings are also shown Diagram of electron-phonon-photon interactions in phononic energy conversion systems. The various absorption, transmission, and emission kinetics are shown Material (atomic and MD) metrics of the photon-electron-phonon interactions in laser cooling of Yb +3 -doped solids. (a) Model for the optical phonon coupling with a bound electron followed by photon absorption. (b) Purely radiative emission process. The phonon side band transitions are also shown. (c) Purely nonradiative emission process. 24 x

12 3.2 (a) Dimensionless emission spectrum of Yb 3+ : ZBLANP at T =10K. The transitions ( 2 F 5/2 ) 4 2 (F 7/2 ) 0,1,2,3, from the first excited manifold to four ground level manifolds are extrapolated (Voigt profile) and are also shown. (b) Dimensionless absorption spectrum for the same [64] Typical normalized cooling rate, as a function of phonon energy, for a diatomic crystal using the Debye-Gaussian model of phonon density of state D p. The discrete normal modes of the complex do not correspond to the available phonon modes of the crystal resulting in a low cooling rate. γ a, absorption rate and the energy level oscillation are also shown Comparison between ϕ (r ) [43] and suggested simple relation ϕ = 12/r 2 for Yb. ϕ (r ) is approximated by 12/r 2 for r B /2 r r B The dipole factor g μ values for different rare-earth ions, with different initial and final electronic states. Transitions between similar sets of initial and final states have g μ values close to each other, showing dependence of g μ on the initial and final state wave functions Variation of electron energy with respect to the normal coordinate, at the doped-ion site, using an infinite square well model. (a) Ground state, and (b) excited state. (c) The configuration coordinate diagram for the process (a) Peak stretching mode frequency using diatomic molecular data for cation-fluoride pair xi

13 3.8 Variation of the structural parameter g s with respect to the atomic number, identifying some crystal structure grouping. The results are for C-F where C is the cation as shown Partial electron-optical phonon interaction potential using the infinitesquare-well model for cation-fluoride pair (a) Comparison between the MD simulation phonon DOS [87] and the Debye Gaussian model, for Y 2 O 3. (b) Comparison between the experimental results [38] for phonon DOS and the Debye-Gaussian model, for Fe 3 O (a)idealized (resonance) andarealistic (side bands) normalized cooling rate as a function of phonon energy, for Yb 3+ : Zr-F. The cooling efficiencies are also plotted as solid line. The experimental result [34] is also shown Dimensionless cooling rate as a function of atomic number, for discrete values of temperature for Yb 3+ : C-F where C stands for cation. Some elements are added for reference. Also, a fourth-order polynomial fit is shown to guide the eye. Note that semiconductors and rare-earth materials have been omitted xii

14 3.13 (a) Dimensionless cooling rate as a function of temperature, for Yb 3+ : Tl-F, Zr-F, Hf-F, Nb-F Fe-F, Mg-F, and Al-F.. The results are for ideal conditions, i.e., quantum efficiency of one and identical FCC structures. (b) Variation of the maximum, normalized cooling rate for various cation-fluoride (C-F) pairs, as a function of the phonon energy for Yb 3+ : C-F, where C represents the various elements in the periodic table. Note that semiconductors and rare-earth materials have been omitted. The dashed line is only intended to guide the eye Dimensionless cooling rate as a function of phonon energy for diatomic host Yb 3+ :CF 4 or blend host, BF 4. The cooling rates have been calculated using ideal conditions, i.e., quantum efficiency of one and FCC structure, and are linearly superimposed. Although the magnitude of the cooling rate is moderated, the absorption probability increases as the phonon spectrum broadens. The cooling rates for diatomic hosts are shown in dashed lines and exhibit less broadening (a) Variation of dimensionless cooling rate, absorption rate, and phonon DOS with respect to phonon energy. Maximum cooling rate can be achieved when the energy of maximum normal mode of the ion-ligand complex coincides with the cut-off phonon energy E p,c of the host and is the most available mode. (b) Multiple cooling peaks (increased cooling probabilities) are possible when multiple blends are present in the host and low symmetry is achieved at the ion site xiii

15 3.16 Variation of time constants as a function of temperature. The radiative relaxation time is the shortest followed by the phonon-assisted, photon absorption at low temperatures. Multi-phonon relaxation exhibits strong temperature dependence and is dominant at high temperatures Heat transfer mechanism of laser cooling of solids. The solid sample exchanges radiation heat with the surroundings through vacuum. The sample temperature will be determined by the balance between the external radiation (thermal load) and the cooling rate Phonon recycling in ion-doped lasing layer and its adjacent cooling layer. The energy-flow diagram, interactions among ions, photons and phonons. Photon and phonon energy flows are indicated with clear and grey arrows, respectively. The phonons emitted during lasing cycle are transmitted through the interface by acoustic and optical-phonon transmission and three-phonon upconversion due to an energy gradient Electronic energy levels of laser and cooling layer dopants and interactions with energy carriers. The corresponding transition rates are indicated with numbered boxes. The single-phonon rapid decay follows the absorption rate, while the mutiphonon decay is independent of the photon transitions xiv

16 4.3 (a) Overlap of phonon density of states (Debye-Gaussian model) for low (red) and high (blue) cut-off frequency crystals (not to scale). Filled areas under the curve are the occupied phonon states at a given temperature. Saturation of low frequency phonons in the cooling layer makes upconversion favorable when transmission of phonons are needed from the lasing layer. (b) The three phonon transfer mechanism are shown as letters (a) Phonon cooling performance as a function of temperature for phonon recycled ion-doped laser using three different cooling crystals. 1 cm in diameter and 1.5 mm thick Yb 3+ :YAG disk is used for the laser layer, and emission spectra of Yb 3+ :ZBLANP is used to calculate the spontaneous emission peak [99, 34, 63]. The second law limit of anti-stokes cooling is also shown. (b) Laser operation conditions as a function of temperature with (1 ) and without (1) phonon recycling. For 1, 12.8 W of heat is removed by phonon recycling, while producing W of spontaneous emission. When the same external cooling (40.3 W) is used, this reduced cooling load corresponds to a reduced laser crystal temperature and moves the operation condition to Entropy flow diagram for phonon recycling in ion-doped laser system. Second law requires every component in the system to satisfy the inequality. Individual components of the laser system are marked by numbers and corresponding inequalities using the heat transfer rates at source or sink temperatures xv

17 5.1 Absorption coefficient for a-si: H at 12.7 and 298 K [81]. At low temperatures, single photon absorption is dominant but significant temperature dependence is shown near room temperature. The dotted line shows qualitative enhancement of absorption when the phonon coupling is enhanced with no temperature change. The inset shows the energy processes involved in the transitions Conceptual rendering of photon-electron-phonon coupling in a-si-x alloy using isolated electron representation. The electron distributions for initial (i), final (f ), and phonon coupled (l) states are shown along with the photon and phonon energies Calculated electron density of states of a-si, a-si 0.5 Ge 0.5,anda-Si 0.5 Sn 0.5 with disordered 64-atom cells generated by the Wooten-Winer-Weaire method [113]. The bandgap E e,g and the Urbach tail are shown. A sample structure of a-si 0.5 X 0.5 is shown as an inset Ab-initio calculated phonon density of states of a-si, a-si 0.5 Ge 0.5,and a-si 0.5 Sn 0.5 with disordered 64-atom cells generated by the Wooten- Winer-Weaire method [113]. The experimental D p of a-si using neutron scattering is also shown [11] Normalized phonon-coupled current density using the peak phonon energies of a-si 0.5 Ge 0.5 and a-si 0.5 Sn 0.5. The scaled total phonon-coupled current density is shown as net enhancement using broken lines xvi

18 6.1 (a) Integrated GaN/InGaN LED layer and metal/algaas/gaas/metal BTCL. (b) The conduction electron energy diagram and related carrier transitions. (c) The energy flow diagram of such BTCL integrated LED chip Detailed carrier transitions are shown for (a) radiative and nonradiative transitions in LED, (b) phonon transmission in LED/BTCL boundary, and (c) phonon upconversion (d) multiphonon absorption in BTCL Variation of multiphonon transition rate with respect to number of phonons absorbed N p. The total LED transition rate γ LED = γ e-ph + γ e-p,srh+ γ e-p,auger and the acoustic phonon upconversion rates are also shown Variation of the energy conversion rate with respect to the number of phonons required to over come the potential barrier. The overall quantum efficiency is also shown xvii

19 A.1 A conceptual rendering of the ion-ligand complex under atomic displacement due to phonon excitation. The energy change of the electron using harmonic potential can be represented by the equivalent potential energy change of the complex. Ab-initio calculations show the change in the partial electronic bandstructure of the ion at the Γ point due to atomic displacement. The equilibrium and displaced bandstructures are indicated by solid and broken lines, respectively, and the corresponding electronic energies are marked on y axis (Ee o is for the equilibrium and E e is for the displaced) A.2 (a) shows the 2 x 2 x 2 supercell created for doping of Yb atom in CdF 2 host. (b) shows the Yb: CdF 2 complex constructed by removing outer ligands which is not expected to influence 4f electron shell of Yb. The simulated ligand movement is also shown by arrows pointing away from the central ion A.3 Electronic bandstructure of CdF 2 doped with Yb. The strongest partial electronic bandstructures of Yb are shown by broken lines. (a) Shows the equilibrium structure, and (b) shows the nonequilibrium structure with the breathing mode displacement A.4 One ligand displacement of the Yb: CdF 2 complex which simulates one quantum of longitudinal phonon propagation xviii

20 A.5 (a) Variations of the normalized cooling rate of Yb 3+ :Cd-Fwithrespect to absorbed phonon energy (variance in input photon energy). Broken line with higher ϕ e-p,o value found from ab-initio calculations and solid line is with lower ϕ e-p,o value found from the harmonic potential. (b) Variations of the normalized cooling rate of Yb 3+ :Cd-F, : Ca-F and : Mg-F with respect to absorbed phonon energy (variance in input photon energy). N is the number of atoms near the ion which contributes to the phonon modes xix

21 List of Tables 2.1 Summary of electron-phonon-photon interaction kinetics used in this study (a) absorption, (b) transmission, and (c) emission Energy eigen values and average radii of 4f electrons in rare-earth ions [31] Radial charge displacement ΔR e estimates and corresponding g μ values for some rare-earth elements doped in LaF 3 crystal, calculated using μ ph-e from Table Values of B i in the Judd-Ofelt transition dipole moment relation for some rare-earth ion in LaF 3 [15] Calculated values of radiative life time and electric dipole moment for some rare-earth ions in LaF 3 crystal [15]. The index of refraction n is between 1.57 and Magnitude of structural parameter g s for some crystalline solids Comparison of the phonon DOS peak energy E p,p values using the linear chain model, with the ion-ligand values, and the experimental values for some diatomic crystals xx

22 6.1 Various transition rates (s 1 ) for LED and BTCL using properties found in ref. [97, 112, 46, 102]. n e = cm 3 and T = 323 K are used throughout A.1 Lattice constants used in the ab-initio calculations and their relaxed values. The displacement of the ligands are also shown [80, 7] A.2 Comparison of the electron-phonon interaction potential between estimates based on harmonic potentials and and values calculated using ab-initio methods xxi

23 Nomenclature A ph,a photon absorption cross-sectional area A SHR Shockly-Read-Hall recombination constant A Auger c D E Auger recombination constant constant density of states energy e c f p, f ph g H electron charge phonon, photon distribution function constant Hamiltonian I ph L m spectral intensity length mass N p number of phonon n Q number density energy transfer xxii

24 Q 1 normal coordinate R Ṡ radial length energy conversion rate s ph,i polarization vector T temperature u p u ph phonon speed speed of light V volume Greek symbols α ph,a absorptance Δ Δ p difference displacement Δϕ e Δϕ b δ D (E) ɛ ɛ o electrical potential potential barrier height Dirac delta function strain vacuum permittivity η Γ γ γ G efficiency force constant transition rate Grüneisen parameter xxiii

25 λ κ μ ph-e wave length wave number dipole moment ρ density ρ r spectral reflectivity τ ϕ e-p,a relaxation time electron-acoustic phonon coupling ϕ e-p,o electron-optical phonon coupling σ ph,i absorption coefficient ψ Ω ω wave function solid angle frequency Subscripts A B a a b C C c anion ligand atom absorption alloy boundary cation cooling cutoff xxiv

26 c-c d e carrier-carrier dopant electron e, e effective electron e, g electron bandgap f i final initial i, im ionized impurity int interaction J e j e current current density L m laser intermediate n, im neutral impurity o P p equilibrium pump phonon p, c phonon cutoff p, p phonon peak ph ph, i sp photon incoming photon spontaneous xxv

27 tr transit Others average * dimensionless xxvi

28 Abbreviations BTCL DFT DOS DMM barrier transition cooling layer density functional theory density of states diffusion miss match (L)APW(linearized) augmented planewave LA LED LO MD SHR SPV TA TO longitudinal acoustic light emitting diode longitudinal optical molecular dynamics Shockley-Read-Hall solarphotovoltaic transverse acoustic transverse optical xxvii

29 Chapter 1 Introduction The focus of this research is theoretical treatment of phonon recycling in photonic processes, and here the motivation, means, and objective are given. 1.1 Phonons in photonic energy conversion In photonic energy conversion, phonons have been considered as mere byproducts created by irreversibility in the processes and seen as havoc which reduces the efficiency of such energy conversions. Indeed, photonic energy conversion processes are most efficient in the cryogenic temperatures, where large energy phonons are absent. However, the phonon influence is inevitable for most applications such as in lasers, SPV (solar photovoltaics), LED (light emitting diode). As the thermal management of such devices, the heat transfer analysis (conduction, convection, and radiation) focus on efficient phonon transport and removal using elaborate cooling systems and sophisticated heat conduction materials and geometries. However, there have been 1

30 a few processes which require phonon participation for efficient energy conversion [94, 19, 110]. One of them is laser cooling of solids, where phonon energy is used to promote the ground state electrons to the excited state via photon having energy lower than the resonance energy required for that transition (an anti-stokes process) [96, 74, 45, 29, 8]. The excited electron falls back to the ground state emitting a photon with the resonance frequency. The net phonon energy removed from the process achieves cooling and the phonon spectrum of the host can be tuned for optimal efficiency using careful selection of host atoms (this will be shown in Chapter 3). The difficulty in analyzing some key photonic processes where phonons play key roles in the energy conversion is the lack of systematic approach when the focus shifts from the electrons to phonons. Especially, the nonequilibrium phonons which are produced during the energy conversion processes are generally considered to quickly thermalize to equilibrium state, i.e., reduce the phonon population to the equilibrium distribution at some equilibrium temperature T. Recent investigations show that after time-resolved X-ray diffuse scattering, the crystals remain out of equilibrium much longer than anticipated (up to nanoseconds) [108], which suggest significant nonequilibrium phonon availability. Although there exists many computational methods to treat each carrier (photon, electron, or phonon) separately (electrodynamics for photons, quantum treatment for electrons, and classical molecular dynamics for phonon), an integrated theoretical treatment examining the kinetics of interactions among the carriers is necessary for assessing the potential of nonequilibrium phonon participation in energy conversion processes. 2

31 1.2 Fermi golden rule To treat the interactions among the carriers through the transition rates γ i,f,the Fermi golden rule (FGR) is used and is written as [28] γ i,f = f γ i-f = 2π M i-f 2 δ D (E f E i ), (1.1) f where M i-f is the interaction matrix element, δ D is the Dirac delta function, E i and E f are the initial and final energies of the carrier. Depending on the transition process, the interaction matrix element differ and can be expanded with the first, second, or higher order perturbation theory. The photon-electron-phonon interaction is treated with the second-order perturbation theory expanding the interaction matrix according to the non-zero elements resulting from application of the operators [28, 87, 52]]. The transition rates developed using the FGR are then used in the analyses of various energy conversion rates Ṡi-f through Ṡ i-f = ω j n j γ i-f, (1.2) where ω j is the energy absorbed or released during the transition and n j is the carrier concentration. 3

32 1.3 Phonon recycling The phonons which are generated during the photonic energy conversion processes can be recycled to improve the efficiency of photonic systems and reduce the cooling load (heat removed to maintain the stability of the system). A summary of the examples in this thesis is shown in Fig. 1.1 along with the conceptual rendering of the carrier couplings. The phonon-assisted absorption (cooling) layer is introduced in the ion-doped laser crystals by using the excess pumping light to drive the anti-stokes cooling cycle. By increasing the dopant density in the cooling layer, absorption of the phonons results in a lower operating temperature for the laser, thus predicting improved laser emission. In the second example with the amorphous silicon solar photovoltaics (a-si SPV), a significant contribution to the absorption coefficient near the bandgap is due to the phonon-assisted transition. By enhancing the phonon participation, we predict improved absorption coefficient and electron current generation. In the last example, the potential barriers is integrated adjacent to the light emitting diode (LED) layer to absorb the phonons before they are injected to the quantum well (QW) for the radiative decay. The phonons absorbed thus exit the cycle through the photon emission, thus reducing the heat loss of the system. The phonons which are absorbed in these proposed systems are both equilibrium and nonequilibrium phonons and make these phonon recycling (active internal cooling) possible, and we optimize these effects by addressing the related materials metrics. The photon-electron interaction introduces the dot product between the electronic transition dipole moment coupling constant μ e-ph (integral of the electron wave function overlap integral be- 4

33 Principal Carriers Phonon Recycling Q p Heat Loss Interactions Phonon (p) a-si Solar Photovoltaic Electron (e) Phonon- Assisted Photon Absorption Phonon Recycling Phonon Absorption LED Potential Barrier Nonradiative Emission Phonon-Electron Couplings φ e-p,a and φ' e-p,o Phonon- Coupled State l Phonon ħω p ψ e,l Ligand Atom Phton-Electron Couplings m B m B ψ e,i 2 ψ e Γ A-B m A Optically Active Atom Γ A-B Solar Spectrum Photon (ph) Pump Laser Ion-Doped Laser Radiative Emission ψ e,f ψ e,i Final State f Electron Overlap μ e = ψ e,f e c r ψ e,i Initial State i m A Optically Active Atom Figure 1.1: Summary of the phonon recycling introduced in photonic energy conversion systems. The input and output energy paths are shown with the light grey arrows, while the phonon energy path is shown in different shade. The photon, electron, and phonon energy conversion regimes are labeled and the carrier couplings are also shown. tween two electronic states) and polarized photon vector s ph,i and is shown in Fig The phonon-electron interaction is through deformation potential scattering, where the acoustic phonon-electron coupling potential ϕ e-p,a and the optical phononelectron coupling potential ϕ e-p,o appear through the atomic displacement and strain caused by the lattice vibrations. 5

34 1.4 Statements of objective and scope of thesis The major objective of this work is to develop a fundamental theoretical approach to the phonon recycling in photonic systems and to demonstrate its potential in a few specific energy conversion systems, under the weak interactions and the independent carrier assumptions. The thesis is organized as follows. Chapter 2 introduces the various carrier interactions used throughout this thesis. Using the FGR, the phonon transport theory, and the simplified decay relationships, the transition rates for the absorption, transmission, and emission processes are modeled. In Chapter 3, the material metrics for the optimal laser cooling of the ion-doped solids are derived using the atomic and molecular dynamic properties of the constituents. The anti-stokes process is modeled as the optical phonon coupling with the bound electron of rare-earth ion, followed by a photon absorption. The transition dipole moment is estimated using a simplified charge displacement model and the Judd-Ofelt theory of rare-earth ions, both suggesting that transitions with large energy level differences and similar angular momentum states should be used. The electron-phonon coupling is interpreted as the derivative of the electronic energy with respect to the displacement of the nearest neighbor ligands whose stretching mode frequency is approximated using the available molecular data. The more appropriate Debye-Gaussian model is used for the phonon density of states of the diatomic crystal. Then, the FGR is used for the photon-induced, phonon-assisted electronic transition probability and applied to the cooling rate equation by defining a phonon-assisted 6

35 transition dipole moment. Based on the material metrics, an example blend is investigated for its cooling performance and a general guide is proposed for selection of optimal laser cooling hosts. Furthermore, the cooling rate limits are discussed and three distinct characteristic times are identified with the photon-induced, phononassisted transition time controlling the overall cooling rate. Chapter 4 proposes the phonon recycling in ion-doped lasers system by introducing a high phonon cutoff energy material for the anti-stokes emission. The recycling of the phonon is expected to result in a lower operating temperature and a higher efficiency. The transmitted optical and upconverted acoustic phonons cross the heterogeneous boundary and contribute to larger population of the optical phonon and increase the second-order cooling transition. Optimization of phonon spectra makes this upconversion favorable. The theoretical study and the predicted quantitative efficiency results are done for the phonon recycling in Yb 3+ :YAGlasingandYb 3+ : ZrF 2 cooling layers. We also examine the thermodynamic limits. In Chapter 5, using the observed temperature dependence of the a-si: H photon absorption spectrum and the weak-phonon interaction second-order transition theory, the phonon coupling-enhanced photon absorption is predicted for a-si-ge and a-si-sn alloys. The ab-initio calculated electron and phonon properties of the alloyed amorphous phase show minimally altered electronic states and significant red-shifted phonon energies. This phonon shift enhances the optical phonon-coupled photon absorption, thus resulting in an increased current generation near the optical bandedge. We find that this enhancement favors the low-energy optical phonon modes, thus making the soft-bond forming Sn the choice alloying element. 7

36 In Chapter 6, we propose and analyze the recycling of optical phonons emitted by nonradiative decay, which is a major thermal management concern for the high-power LED, by introducing an integrated, heterogeneous multiphonon-absorption barrier cooling layer. We address the theoretical treatment of photon-electron-phonon interaction kinetics for the optimal number of the absorbed phonons (i.e., barrier height). We consider a GaN/InGaN LED with a metal/algaas/gaas/metal potential barrier, discuss the energy conversion rates, and show reduction in cooling requirement. Chapter 7 summaries the highlights the finding of this study and proposes future directions for the related research. In appendix A, results of ab-initio calculation of electron-phonon interaction in laser cooling are reported and compared with the values estimated using analytical harmonic potential approximation. Electronic bandstructure changes due to normal mode ligand displacement is used to find the energy change of f orbital optically active electron at the gamma point. The band structure calculations show that the partial bandstructure of Yb is flat, indicating limited interaction with the neighboring atoms. The electron-phonon interaction potential calculated is within a factor of two (larger) of the value using the harmonic potentials and the difference is attributed to the necessary approximations of each method. Single ligand displacement, which simulates one phonon propagation, is also used and the resulting electron-phonon interaction potentials are in agreement with those from normal mode vibration of the entire complex. Using these new interaction potentials, a larger cooling rates of the ion-host pair is calculated. 8

37 Chapter 2 Kinetics of electron-phonon-photon couplings The various carrier interaction processes are shown in Fig The corresponding electronic energy transitions are shown along with absorbed or emitted photons and phonons associated with these transitions. Absorptions include the single-photon absorption, phonon-assisted photon absorption, and multiphonon absorption. The phonon transmissions across the interface are both optical and acoustic phonon transmission and the interfacial optical-acoustic and acoustic-optical up- and downconversions. Emissions include single-photon emission, multiphonon emission, the Shockley- Read-Hall recombination, and the Auger recombination. 9

38 (a) Absorption Photon Absorption Phonon-assisted Photon Absorption Multiphonon Absorption ħω ph. γ e-ph ħω ph. γ e-ph ħω p,o. γ p-e ħω p,o Interfacial Optical Phonon Transmission (b) Transmission Interfacial Acoustic Phonon Transmission ħω p,o ħω p,a. γ tr, O Interfacial Optical-Acoustic Downconversion. γ tr, A Interfacial Acoustic-Optical Upconversion ħω p,o. γ p-p,o A+A ħω p,a ħω p,a ħω p,o. γ p-p,a+a O (c) Emission Photon Emission Multi-phonon Emission Recombination ħω ph ħω p,o ħω p,o ħω p,o. γ e-ph. γ e-p. γ e-p,srh. γ e-p,auger Shockley-Read-Hall Auger Figure 2.1: Diagram of electron-phonon-photon interactions in phononic energy conversion systems. The various absorption, transmission, and emission kinetics are shown. 10

39 2.1 Absorption Single-photon absorption The single-photon absorption rate is a first-order transition photon-electron coupling. It is represented by the electric dipole-moment transition approximation between the two states having a non-zero matrix element in the electric dipole moment integral. The total Hamiltonian of the system is H=H e +H ph +H ph-e. (2.1) The first term is H e = ω e,g a a, (2.2) which is the Hamiltonian of the ion electronic levels, where ω e,g is the energy difference between the optically active energy levels of the dopant ion, and a (a) isthe creation (annihilation) operator of an electronic excitation. The second term is H ph = ω ph,i c c, (2.3) which is the electromagnetic laser field Hamiltonian, where ω ph,i is the incoming photon frequency and c (c) is the creation (annihilation) operator of a photon. The 11

40 third term is H ph-e = s ph,i μ ph-e( ω ph,i 2ɛ o V )1/2 (c + c)(a + a), (2.4) which is the photon-electron interaction Hamiltonian where s ph,i is the polarization vector of photon, μ ph-e is the electric dipole moment vector of electronic transition, ɛ o is the vacuum permittivity, and V is the interaction volume. Now, the FGR is used to calculate the total transition rate γ ph-e as [84] γ ph-e = f γ i-f = 2π M f,i 2 D ph f = 2π M f,i 2 κ 2 dω V, (2.5) u ph (2π) 3 where M f,i is the interaction matrix element and D ph is the photon density of states given by κ 2 dωdκ V/(2π) 3 de ph with de ph = u ph dκ, Ω is the solid angle, κ is the wave number, and u ph is speed of light. Since the two states of the electron have a non-zero matrix element, the transition allowed is first-order and the interaction matrix element M f,i is expanded as M f,i,1st = f H int i = ϕ f,f ph H ph-e ϕ i,f ph +1, (2.6) where f ph is the photon distribution function. Then using Eqs. (2.4) and (2.5) the 12

41 transition rate becomes γ ph-e = 1 τ ph-e = ω3 e,g μ 2 3πɛ o u 3 ph-e, (2.7) ph which is the definition of the Einstein A Ein coefficient (we have used ω = κu ph ) Phonon-assisted photon absorption The phonon-assisted photon absorption is essential for the phonon recycling in lasers and for the photon absorption near the bandedge of a-si SPV. To account for the transition rate for the photon-induced, local vibrational mode assisted electronic transition rate of the optically active atom, the electron-phonon interaction Hamiltonian in the cooling rate equation, as described in [28, 87], is modified to include the electron-optical phonon interaction. Similar to the radiative transition, the Hamiltonian of the entire system is H=H e +H p +H ph +H ph-e +H e-p. (2.8) The first term is the electron Hamiltonian defined before, the second term is H p = p ω p b b, (2.9) which is the phonon field Hamiltonian, where ω p is the phonon frequency and b (b) is the creation (annihilation) operator of a phonon in mode p. The third and fourth 13

42 terms are the photon field and electron-photon interaction Hamiltonian, respectively, defined before. The fifth term is the electron-phonon interaction Hamiltonian described as distortion of the ligand ions affecting the crystal field. Such a distortion is a function of local strain, therefore, we expand the crystal field potential in powers of such strain. The local strain is defined by the strain term ɛ i,j as ɛ i,j = 1 2 ( d i x j + d j x i ) (i, j =1, 2, 3). (2.10) For simplicity we use ɛ d x x=0, (2.11) i.e., not taking into any account the anisotropy of the elastic waves. The origin of the coordinate is the point in the lattice where the ion nucleus is located. The derivative of the displacement d is set to be the normal coordinate, Q q = d x x =0 =( ) 1/2 (b + b ), (2.12) 2m AC ω p where q specifies a particular mode. When a particular mode of vibration dominates the electron-phonon interaction, we express the electron-phonon interaction Hamiltonian in terms of the normal coordinates. Using the optical deformation potential 14

43 theory, the crystal field Hamiltonian term becomes [83] H c =H o + ϕ e-p,o Q q. (2.13) The higher-order terms have been neglected. Then the interaction Hamiltonian term becomes H e-p = ϕ e-p,oq q = ϕ e-p,o ( ) 1/2 (b q + b q ), (2.14) 2m AC ω p where ϕ e-p,o is the electron-phonon coupling potential, ω p is the phonon frequency, and m AC is the effective mass of the oscillating atoms. Then the second-order term of the perturbation expansion gives the transition (absorption) rate γ ph-e-p (the Fermi golden rule) as γ ph-e-p = f γ i-f = 2π M f,i 2 δ D (E f E i ). (2.15) f The M f,i matrix admits a second-order perturbative expansion as M fi,2nd = m m f H int m m H int i E T e,i ET e,m ψ f,f ph,f p H ph-e ψ m,f ph +1,f p ψ m,f ph +1,f p H e-p ψ i,f ph +1,f p +1. E i (E m ω p ) (2.16) 15

44 Note that when the perturbation goes to zero (i.e., the initial and final energies are identical) we recover the unperturbed state. When the interaction matrix is rewritten using the Hamiltonian expansion, the transition rate becomes γ ph-e-p = π 2m AC (s ph,i μ ph-e) 2 ɛ o ϕ 2 e-p,o D p (E p )f o p (E p) E 3 p ω ph,i D ph, (2.17) V where D p (E p ) is the phonon density of states of phonon having energy E p, f p is the Bose-Einstein distribution function, ω ph,i is the incoming photon frequency, and D ph is the photon density of states which integrates to unity, since the incoming laser light is assumed to be monochromatic and one-photon interaction is assumed. Note that D p (E p ) is the phonon density of states for the normal (local) modes which may be different from that for the bulk host material Multiphonon absorption The relation for single-optical phonon absorption rate in semiconductors is derived using the FGR and is given as [100] γ p-e = ϕ 2 e-p,o m3/2 fp onp (E p ), (2.18) 2 1/2 π 3 ρω p,o e,e Ee 1/2 where ϕ e-p,o is the optical phonon deformation potential, m e,e is the reduced electron mass, E e is the electron energy, and N p is the number of phonons absorbed. The parabolic band assumption is made for the electronic density of states near the bandedge and exponential dependence (of the phonon distribution function) for the 16

45 multiphonon transition processes is used which is commonly used for multiphonon processes [62, 85]. 2.2 Transmission Optical and acoustic phonon transmission Acoustic phonon transmission rate can be estimated by dividing transmission length (L) by the acoustic velocity and the transmission coefficient is predicted by the diffusion mismatch model (DMM) using the difference in the mode velocities across the interface [106, 50]. The DMM phonon boundary transmission coefficient is approximated as τ b,1-2 = α α u 2 p,2,α u 2 p,1,α + j u 2 p,2,α (2.19) where u p,1,α and u p,2,α are the various phonon mode (α) velocities in materials 1 and 2. The optical phonon velocities are usually small thus does not travel far before it decays to lower energy modes, however, in highly mass mismatched systems such as InN and GaN, the optical phonon life times may be much longer which allows for optical phonon propagation [115, 16] Phonon up and down conversion At materials interfaces, depending on the mismatch of the cutoff frequencies of the phonons, the phonon up- and downconversion can exist at high occupation numbers 17

46 [55]. This rate can be calculated by the three-phonon interaction model given as [103] γ p-p = M p-p 2 R 8πρ 3 ω u 7 p,a u2 p,a 2 ω3 p,o p,o [f o p (ω p,a ) ][f o p (ω p,a ) ][f o p (ω p,o )+ 1 2 ± 1 2 ], M p-p 2 =4ρ 2 γ 2 G u2 p,a u2 p,o, R = 2 3 1/2 Γ AA Γ CC Γ AA +Γ CC, (2.20) where ρ is the density, u p,a and u p,o are the acoustic and optical phonon velocities respectively, ω p,a and ω p,o are the acoustic and optical phonon frequencies respectively, fp o is the phonon distribution function, γ G is the Grüneisen parameter, and Γ AA and Γ CC are the force constants which can be estimated by material metrics [46]. The Debye-Einstein phonon density of states model is used and the two acoustic phonons are assumed to be identical. 2.3 Emission The spontaneous radiative emission is formulated similar to the photon absorption derived in Section and considered to be equal when dealing with a nondegenerate system when no stimulated emission is present [98]. The multiphonon emission for isolated ions are commonly modeled using a semi-empirical model [62, 85] γ e-p = γ e-ph(t =0)(f o p +1)Np, (2.21) where γ e-p(t = 0) is the radiative decay rate at T = 0 K and the highest available 18

47 phonon energy is used for the fitting parameter. The dominant nonradiative decays in semiconductors are the Shockley-Read-Hall (SHR) and Auger recombination [97]. The SHR recombination is due direct multiphonon decay at the defect sites, while the Auger recombination is due to transfer of energy to a conduction electron followed by a multiphonon decay. These nonradiative recombinations are commonly presented with simple expressions (e.g., for GaN [75]). The total nonradiative transition rate γ e-p depends on the carrier concentration (SHR is given as a constant coefficient, while the Auger has a quadratic dependence) as γ e-p = A SHR + A Auger n 2 e, (2.22) where A SHR and A Auger are the SRH and Auger recombination coefficients respectively. Note that we have used units of s 1. The phonon frequencies generated by these nonradiative transitions depend on the highest available phonon frequency in the lattice and decay to the lower frequencies via phonon-phonon interaction processes [57]. 2.4 Summary The electron-phonon-photon interaction kinetics which will be used in the subsequent chapters have been summarized in Table 2.1. To analyze the energy conversion mechanisms in a system, one needs to identify various carrier interaction kinetics present in the system and identify the dominate process (mechanism). Then ac- 19

48 cording to the parameters affecting the bottleneck process, one begins to select the materials which optimize the system efficiency. In-depth derivations and assumptions used in this thesis will be presented in the subsequent chapters. 20

49 Table 2.1: Summary of electron-phonon-photon interaction kinetics used in this study (a) absorption, (b) transmission, and (c) emission. (a) Absorption Photon absorption [84] γ ph-e = ω3 e,g 3πɛ o u 3 ph μ 2 ph-e Phonon-assisted photon absorption [28, 87, 52] γ ph-e-p = π (s ph,i μ ph-e) 2 ϕ 2 D p (E p )fp o (E p ) e-p,o 2m AC ɛ o Phonon absorption [100] γ p-e = ϕ 2 e-p,o m3/2 e,e Ee 1/2 fp o (E p ) 2 1/2 π 3 ρω p,o (b) Transmission Optical and acoustic phonon transmission [106, 50] α u 2 p,2,α γ b,1-2 = τ b,1-2 γ p,tr = u 2 p,1,α + α j u 2 p,2,α E 3 p Acoustic-optical up and down conversion [103] γ p-p = M p-p 2 R 8πρ 3 ωp,aω 2 p,o 3 u 7 p,a u2 p,o (c) Emission Multi-phonon emission [62, 85] γ e-p = γ e-ph(t =0)(fp o +1)Np L u p ω ph,i D ph V [fp o (ω p,a ) ][f p o (ω p,a ) ][f p o (ω p,o )+ 1 2 ± 1 2 ], M p-p 2 =4ρ 2 γgu 2 2 p,au 2 p,o, R = 2 Γ AA Γ CC 3 1/2 Γ AA +Γ CC Recombination [97, 75] γ e-p = A SHR + A Auger n 2 e 21

50 Chapter 3 Phonon tuning in laser cooling of rare-earth ion doped solids 3.1 Introduction Laser cooling of solids is achieved when a photon having a lower frequency (energy) excites an electron with assistance from a phonon and this is followed by a single photon emission with a mean frequency higher than that of the incident photon (i.e., anti-stokes process). Pringsheim recognized the possibility of the anti-stokes process which was later confirmed experimentally by Epstein et al. [25] in the first successful experiment of laser cooling of solids. The entropy aspect of laser cooling of solids is discussed in [90] and a general review is given in [88]. Host materials including, ZBLAN (ZrF 4 -BaF 2 -LaF 3 -AlF 3 -NaF) [39], ZBLANP (ZBLAN-PbF 2 )[34],YAG (Y 3 Al 5 O 12 ), CNBZn (CdF 2 -CdCl 2 -NaF-BaF 2 -BaCl 2 -ZnF 2 ), and KPb 2 Cl 5 [27] have been doped with various rare-earth ions, e.g., Yb 3+,Er 3+,andTm 3+, and success- 22

51 fully laser cooled. In contrast to experimental successes, the theoretical extension of the anti-stokes process has not advanced far. Thus, the theoretical approach to laser cooling of solids has not allowed for predictive selection of materials for efficient cooling. Recent theoretical analyses address localized electrons [24] and strong electronphonon coupling [89]. However, further investigation of the interaction amongst the three carriers, namely, photon, electron, and phonon, are needed. Here we propose material selection metrics using simplified theoretical models for the three-carrier interactions, and suggest possible improvements in the materials selection. Figures 3.1(a) to (c) demonstrate the materials, i.e atomic, and molecular dynamics (MD), metrics of the photon-electron-phonon interactions in laser cooling of Yb 3+ -doped solids. The photon-induced, phonon-assisted absorption process is modeled as a phonon absorption, followed by a photon absorption. Three steps are identified (a) phonon-assisted absorption, (b) radiative decay, and (c) non-radiative (purely phonon) decay. These are designated by their kinetics represented by phononassisted transition time τ ph-e-p, purely radiative decay τ e-ph, and purely phonon decay τ e-p where subscripts ph, e and p represent photon, electron and phonon respectively. As shown in Fig. 3.1(a), the electron (in ion) oscillates between the ground level manifolds due to continuous excitation by phonons (thermal vibrations). At the same time, due to changes in the position of the immediate neighboring atom, the electronic wave function is altered resulting in oscillations of the energy levels within the manifold. This reflects the observed thermal broadening. Since the energy levels of the manifolds are quantized and discrete, only allowed phonons corresponding to the energy level difference are able to promote electrons between these levels. Conversely, 23

52 Ground State Electron Level of Yb 3+ 2 F 7/ F 7/2 2 F 5/2 Ligands Cation (C) Anion (A) Yb 3+ (I) m I E 01 =E p Vibration Waves ω p (Phonon) Excited Yb3+ Ion Electron Cloud Domain Yb Q Δ E14 =hω ph, i m C m A Yb 3+ Ion Electron Cloud Domain Γ IA Q 1 Γ AC (a) Photon-Induced, Phonon Assisted, Electronic Transition in Weak Electron-Phonon Coupling Regime Q Lattice Relaxation (Under Energy Minimization) Phonon Side Bands F 2 7/2 F 5/2 2 F 7/2 2 F 5/ Yb 3+ Resonance Transition Δ E04 =hω ph e Q Yb 3+ ΔE 04 = N p ћω p Q Q Q ω ph,e (b) Radiative Transition (Resonance) ω p (c) Multi-Phonon Non-Radiative Transition in Strong Electron-Phonon Coupling Regime Figure 3.1: Material (atomic and MD) metrics of the photon-electron-phonon interactions in laser cooling of Yb +3 -doped solids. (a) Model for the optical phonon coupling with a bound electron followed by photon absorption. (b) Purely radiative emission process. The phonon side band transitions are also shown. (c) Purely nonradiative emission process. 24

53 when the energy spacings of the oscillating ground state manifold matches the available phonon modes, the electron is promoted to a higher energy level within the manifold. Note that the these energy levels should exist within the limits defined by the Heisenberg uncertainty theorem, which is related to the natural broadening of these energy levels [101]. In the vicinity of the doped ions, the available local modes are characterized by the normal modes of the ion-ligand complex, which have attributes of optical phonon due to the breathing mode. The displacement due to longitudinal optical phonon is represented by Δ p. For the rare-earth ions, the optically active f n shell electron is well localized (within the atomic spacing of the ion-ligand complex), thus the short-wavelength phonons become important. The electron interaction with optical phonons will be discussed in the subsequent sections. When a photon is introduced, the photon encounters an electron in oscillation and promotes it to the excited level. This process is not only a function of the availability of phonons, but also a function of the transition dipole moment (electron transition overlap integral). Therefore, we examine the transition dipole moment as well as phonon availability. When the electron is promoted to the excited state, it has multiple paths through which it can decay back to the ground state: purely radiative, purely nonradiative, and vibronic transitions. Purely radiative decay process is shown in Fig. 3.1(b). The electron directly decays to the lowest lying ground level by emitting a photon having a frequency equal to its resonance transition frequency. This decay process is normally the strongest transition at moderate temperatures. Figure 3.1(c) shows the purely nonradiative process in which the electron is de-excited by emitting multiple phonons simultaneously. This process dominates at high temperatures. Nevertheless, for the 25

54 (a) E* ph,e ( 2 F 5/2 ) 4 ( 2 F 7/2 ) 0 ( 2 F 5/2 ) 4 ( 2 F 7/2 ) 1 Yb 3+ : ZBLANP ( 2 F 5/2 ) 4 ( 2 F 7/2 ) 2 Total Intensity Individual Intensity (Voigt Profile) ( 2 F 5/2 ) 4 ( 2 F 7/2 ) λ (nm) (b) 1.0 Yb 3+ : ZBLANP 0.8 Total Intensity Individual Intensity (Voigt Profile) 0.6 ( 2 F 7/2 ) 0 ( 2 F 5/2 ) 4 E* ph.a 0.4 ( 2 F 7/2 ) 0 ( 2 F 5/2 ) ( 2 F 7/2 ) 0 ( 2 F 5/2 ) λ (nm) Figure 3.2: (a) Dimensionless emission spectrum of Yb 3+ : ZBLANP at T =10K. The transitions ( 2 F 5/2 ) 4 2 (F 7/2 ) 0,1,2,3, from the first excited manifold to four ground level manifolds are extrapolated (Voigt profile) and are also shown. (b) Dimensionless absorption spectrum for the same [64]. rare-earth ions in a crystal, the purely nonradiative decay is suppressed, because of the weak electron-phonon interaction strength due to the localization of optically active f n electron shell. This is one of the motivations of using rare earth doped solids. Lastly, the electron also can be de-excited by a vibronic process, i.e., emitting a photon and a phonon that is, the opposite process of photon-induced, phonon-assisted electron absorption. The emission spectrum (dimensionless Eph,e )ofyb3+ : ZBLANP is shown in Fig. 3.2(a) [64]. Apart from the strongest resonance transition ( 2 F 5/2 ) 4 2 (F 7/2 ) 0,there 26

55 exist three phonon side-band transitions designated as (1,2,3). When the mean absorbed phonon energy is larger than the average emitted phonon (the three vibronic transitions), cooling occurs. The conditions for cooling/heating are ω ph,i ω ph,e = ω p < 0 cooling (anti-stokes process) (3.1) ω ph,i ω ph,e = ω p > 0 heating (Stokes process), (3.2) where ω ph,i is the incoming photon frequency, ω ph,e is the average emission photon frequency, and ω p is the phonon frequency. The absorption spectrum (dimensionless Eph,a )ofyb3+ : ZBLANP in Fig. 3.2(b) shows that there are also three sub-levels in the excited state manifold. The spectrum indicates that the absorption intensity of the phonon assisted transition can be an order of magnitude lower than the resonance transition. This reflects small second-order transition rate which involves all three carriers compared to that of the first-order process rate. One reason for such low transition rate is the mismatch between normal modes of the ion-ligand complex and the maximum available normal mode energy of the crystal, resulting in a low cooling rate. Figure 3.3 shows a typical variation of the normalized cooling rate for a diatomic crystal with respect to phonon energy. The discrete energy level of the ground electronic state oscillates due to presence of phonons. The absorption rate is proportional to the product of this discrete energy level and the available phonon modes within the lattice. Since this absorption rate is directly proportional to the cooling rate, cooling peak is observed. 27

56 . S ph-e-p Q ph,i Heating. γ ph,a Phonon D p (E p ) D p (Ep) Normal Modes Cooling Zero Phonon Line Energy Level Oscillation. E p,c S ph-e-p Q ph,i E p Figure 3.3: Typical normalized cooling rate, as a function of phonon energy, for a diatomic crystal using the Debye-Gaussian model of phonon density of state D p.the discrete normal modes of the complex do not correspond to the available phonon modes of the crystal resulting in a low cooling rate. γ a, absorption rate and the energy level oscillation are also shown Cooling rate Using the transition rates developed in chapter 2, the cooling rate of the anti- Stokes process can be developed using the atomic quantities. The energy equation is written as Ṡ ph-e-p = Ṡph,a Ṡph,e, (3.3) where Ṡph-e-p is the cooling rate, and Ṡph,a and Ṡph,e are the absorption and emission rates respectively. Then it is possible to express the cooling rate in terms of the absorption power as Ṡ ph-e-p = ω ph,i γ ph-e-p =(1 ω ph,e ω ph,i V γ e-ph γ ph-e-p n d dv s ω ph,e γ ph-e n d dv. V ) ω ph,i γ ph-e-p n d dv s. (3.4) V 28

57 At steady state, we assume that the absorption rate is equal to the sum of the radiative and non-radiative decay rates, i.e., γ ph-e-p = γ e-ph + γ e-p, and then the quantum efficiency is defined as η e-ph = γ e-ph γ ph-e-p = γ e-ph. (3.5) γ e-ph + γ e-p Note that this is an idealized quantum efficiency for the purpose of this theoretical analysis which does not consider the effects of defects, re-absorption, surface contamination, energy transfer and etc. Empirical study of contaminates on quantum efficiency can be found in [40]. Then Eq. (3.4) becomes Ṡ ph-e-p =(1 ω ph,e η e-ph)a ph,a n d u ph e ph,i dv ω ph,i V =(1 ω ph,e η e-ph)σ ph,a u ph ω ph,i n ph,i V, (3.6) ω ph,i where A ph,a is the absorption cross-sectional area which is defined as A ph,a = ω ph,i γ ph-e-p/u ph ω ph,i n ph,i, n d is the dopant concentration, σ ph,a is the absorption coefficient (which is the product of A ph,a and n d ), and n ph,i is the number of photons per unit volume. The spectral absorptance α ph,i is related to the absorption coefficient (for optically-thin solid) as, α ph,i =1 exp( σ ph,a L) σ ph,a L, σ ph,a L 1, (3.7) 29

58 and Ṡph,a = α ph,i Q ph,i,whereq ph,i is the incident laser power. Then Eq. (3.6) becomes Ṡ ph-e-p = α ph,λ,i Q ph,i (1 ω ph,e ω ph,i η e-ph). (3.8) Using the the definition of absorptance given in Eq. (3.6) we have Ṡ ph-e-p = π 2ɛ o m AC μ ph-e 2 ϕ 2 e-p,o D p (E p )f o p (E p ) E 3 p ω ph,i n d L (1 ω ph,e η e-ph)q ph,i, (3.9) u ph ω ph,i for the optical phonon absorption. We have used the spatial average of the transition dipole moment, which couples with the incoming polarization, as (s ph,i μ ph-e) 2 = (μ ph-e/3 1 2 ) 2 = μ ph-e 2 From Eq. (3.9), the cooling rate is a function of atomic and MD quantities, including ϕ e-p,o, E p,andd p (E p ). 3.2 Electron-Photon Interaction The electron-photon interaction is characterized by an electric dipole transition from the initial state to the final state. In order to develop material selection guidelines for choosing the rare-earth ion, we perform an order of magnitude estimation by first introducing a simplified transition dipole moment estimation based on the Hatree-Fock integrals. Then we use the Judd-Ofelt theory (semi-empirical) for comparison and also use this, more accurate calculation of the transition dipole moment in the cooling rates. 30

59 3.2.1 Charge-Displacement Approximation for μ ph-e For the rare-earth compounds, the multiplicity of the f states and related properties such as the magnetic moment and spectra, indicate that for most of the rare-earth group it is a good approximation to consider f electrons as atomic-like orbitals [51]. The calculation of a single particle 4f-electron wave function in atom shows that even if these states are occupied beyond the 5s, 5p, and6s orbitals, the charge distribution of the 4f electron is such that most of it is inside the sphere of maximum charge density of the 5s and 5p levels. This is attributed to the dominant role played by the effective f potential ϕ 4f (r), a radial potential well confining the 4f wave function to a small region of space. The resultant wave function ψ o 4f is atomic-like. When the rare earth ion is in a solid, the boundary conditions and potential well change, mainly in the outer most parts of the cell, through the superposition of the atomic potential wells. The change in the potential results in a change in the electron wave function to ψ 4f, which can be written as the sum of the atomic-like wave function and a polarization wave function ψ 4f. ψ 4f is taken orthogonal to ψo 4f, has a non-atomic character, and has contribution outside the sphere where 5s, 5p, and 5d electrons have their maximum charge density [51]. So, we write ψ 4f = a n ψ o 4f +(1 a 2 n) 1/2 ψ 4f, (3.10) where ψ 4f, ψ o 4f,andψ 4f are normalized and a n depends on the total amount of f character for a given configuration 4f n. The lower the average energy of the f-electron band in the f n configuration for that atom in crystals, the larger the average a n.in 31

60 Table 3.1: Energy eigen values and average radii of 4f electrons in rare-earth ions [31]. z Element E 4f,eV R 4f, Å 59 Pr Nd Pm Sm Eu Tb Dy Ho Er Tm Yb general, a n = a n (E e ), and a 2 n (E e,1) <a 2 n (E e,2), for E e,1 <E e,2 within an f band. We assume the ground state to be composed of the lowest energy state and take a 2 n 1. This allows us to approximate the wave function of the 4f electron in crystals by an atomic-like wave function. The atomic-like wave functions of 4f electrons for the rare-earth ions have been calculated numerically by solving the Hartree-Fock-Slater equation and using the self-consistent field method [31], which is expressed as [ d2 l(l +1) + dr 2 r 2 P nl (r) = 1 4π + ϕ (r)]p nl (r) =E nlp nl (r), 2π π 0 0 rψnl o (r) sinθ dθ dφ. (3.11) From this, the total energy of 4f electrons is found and is given in Table 3.1. To estimate μ ph-e from the energy eigenvalues listed in Table 3.1, we examine the 32

61 definition of the transition dipole moment μ ph-e = ψ f e cxψ i dv. (3.12) The transition dipole moment μ ph-e is approximated as electronic charge times the net displacement ΔR e,if of the optically active electron during the transition, or μ ph-e g μ e c ΔR e,if, (3.13) where g μ is a dipole factor which depends on the shapes of the wave functions (initial and final states) and a n, is a measure of atomic-like behavior of the 4f electron in the condensed state. In the neighborhood of the average 4f electron radius R 4f, we can approximate the potential function ϕ (r) l(l+1)/r 2 (Fig. 3.4) for all rare-earth elements of interest. Here, potential function ϕ (r) =2m e r 2 B ϕ(r)/ 2, E =2m e r 2 B E/ 2,andr = r/r B is a dimensionless parameter, where the r B is the Bohr radius. Then we write the dimensionless transition energy as ΔE e,g = E e,f E e,i = 1 2 [ϕ (R e,i ) ϕ (R e,f )] 6( 1 R 2 e,i E e,f + E e,i 2E 4f 1 ) Re,f 2 1 R 2 e,i + 1 R 2 e,f 2 1 R4f 2. (3.14) We assume E e,f + E e,i 2E 4f,toestimateΔR e = R f R i. Table 3.1 lists ΔR e and 33

62 7 6 Yb log φ* φ*(r) 12 r* log r* = log r/r B Figure 3.4: Comparison between ϕ (r ) [43] and suggested simple relation ϕ =12/r 2 for Yb. ϕ (r ) is approximated by 12/r 2 for r B /2 r r B. g μ, estimated for the rare-earth ions doped in the LaF 3 crystal. Figure 3.5 shows that the g μ values for a particular transition between similar set of initial and final states are close. Table 3.2 lists the values of g μ for the various rare-earth ions with different initial and final states. The magnitude of g μ for a transition between similar states is expected to be higher than for transitions between dissimilar states, due to a better overlap between the spherical harmonics of initial and final states in Eq. (3.12). For example, g μ for an 2 F 5/2 to 2 F 7/2 transition in Yb is expected to be higher than any other transitions in rare-earth ions. For Yb 3+ in CaF 2 crystal, e c ΔR e is estimated as C-m, which is nearly the same as the experimental value of C-m, which means an F F transition should also have a high value of g μ. 34

63 Pr 3+ Nd 3+ Ho 3+ Pm 3+ g μ Tm 3+ Pr 3+ Sm 3+ Dy 3+ Ho 3+ Er Tm 3+ D H P H F I G H F H S I Transitions Figure 3.5: The dipole factor g μ values for different rare-earth ions, with different initial and final electronic states. Transitions between similar sets of initial and final states have g μ values close to each other, showing dependence of g μ on the initial and final state wave functions. Table 3.2: Radial charge displacement ΔR e estimates and corresponding g μ values for some rare-earth elements doped in LaF 3 crystal, calculated using μ ph-e from Table 3.4 ż Element Transition ΔEe,g ΔR e e c ΔR e (C-m) g µ (ev) (Å) 59 Pr 3+ 1 D 2 3 H P 0 3 H Nd 3+ 4 F 3/2 4 I 9/ Pm 3+ 5 F 1 5 I Sm 3+ 4 G 5/2 6 H 5/ Dy 3+ 4 F 9/2 6 H 15/ Ho 3+ 5 S 2 5 I F 5 5 I Er 3+ 4 S 3/2 4 I 15/ Tm 3+ 1 D 2 3 H Tm 3+ 1 G 4 3 H

64 3.2.2 Judd-Ofelt Theory Estimation of Transition Dipole Moment Judd-Ofelt theory gives a accurate estimation of transition dipole moment with minimal experimental input, and μ ph-e is determine by the dipole approximation [4] μ 2 ph-e = 1 (2J + 1) [χ F 2 + n M 2 ], (3.15) where F 2 and M 2 represent the matrix elements of the electric and magnetic dipole operators respectively, joining an initial state J to the final state J, i.e., 2S+1 L J 2S +1 L J, χ =(n2 +2) 2 /9n, andn is the refractive index of the medium. The factor 2J+1 is added, since the matrix elements of μ e are summed over all components of the initial state i. Since the photon-induced electron transition is electric in nature, we will only discuss F 2 component. The transition dipole moment of induced electric dipole transition within the f n shell configuration is independently derived by Judd and Ofelt and is known as the Judd-Ofelt theory [76]. Judd-Ofelt theory uses semi-empirical data to find the various atomic parameters of the rare-earth (trivalent lanthandes) ions. F 2 matrix element is expanded as F 2 = e 2 c as i=2,4,6 B i ψj U (i) ψ J 2 to give the transition dipole moment μ 2 ph-e = 1 (2J + 1) (n 2 +2) 2 9n e 2 c i=2,4,6 B i ψj U (i) ψ J 2, (3.16) where B i is the crystal field parameter given in Table 3.3 and U (i) is a unit tensor 36

65 Table 3.3: Values of B i in the Judd-Ofelt transition dipole moment relation for some rare-earth ion in LaF 3 [15]. Ion B (cm 2 ) B (cm 2 ) Nd Eu Tb Ho Er Tm B (cm 2 ) Table 3.4: Calculated values of radiative life time and electric dipole moment for some rare-earth ions in LaF 3 crystal [15]. The index of refraction n is between 1.57 and Ion Transition ΔE e,g (cm 1 ) ψj U (2) ψ J 2 U (4) 2 U (6) 2 τ r (μs) μ e (C m) Nd 3+ 4 F 3/2 4 I 9/ Tb 3+ 5 D 3 7 F Ho 3+ 5 F 5 5 I Er 3+ 4 S 3/2 4 I 15/2 Tm 3+ 1 D 2 3 H Tm 3+ 1 G 4 3 H operator of rank i where i =2, 4, 6. The crystal parameter B i is defined by B i = (2i +1)Σ k D k 2 (2k +1) 1 I 2 (k, i), where D k (k odd) are the odd-parity terms in the static crystal field expansion and I 2 (k, i) contain integrals involving the radial parts of the 4f n wave functions, the excited opposite-parity electronic-state wave functions, and the energy separating these states [48]. The matrix element ψj U (i) ψ J 2 does not vary with the host, therefore, the value calculated in [15] can be used for most of the electronic transition of the rare-earth ions. Using the above relationship, the radiative lifetimes and the electric dipole moment are calculated for rare-earth ions in LaF 3 crystal as listed in Table

66 3.3 Electron-Lattice Interaction for Optical Phonon Electron-phonon interaction involves several physical parameters which characterize its strength. The electron-phonon coupling is defined using a simplified defect model at the ion site, and the available phonon density of states is estimated using the Debye-Gaussian model Electron-Phonon Coupling Electron-phonon coupling is analogous to the deformation potential theory in semi-conductors, however, the domain modification is necessary here, due to the discrete state nature of the bound electrons. For example, while semiconductor deformation potential is defined within the boundary of a unit cell, the electron-phonon coupling potential is here defined at the doped ion site. We adapt a defect model to estimate the electron-phonon coupling potential using electrons in an infinite square well. The solution is not exact, however, this interpretation provides the physical picture of the electron-phonon coupling for different host materials [42]. The doped ion is treated as a defect site surrounded by an infinite potential. Although the electron experiences potential by the nucleus of the doped ion, this is neglected when comparing electrons in different host materials thus, the interaction potential is not an absolute value, but a relative one which varies for different host constituents. Figure 3.6(a) shows the electron trapped in an infinite square well of width 2Q 1. The equilibrium bond lengths are determined by the intermolecular forces and the structure of the molecular complex, estimated from the 38

67 Table 3.5: Magnitude of structural parameter g s for some crystalline solids. Structure Host g s Structure Host g s Cubic CdF Trigonal CeF AlF InF LiF 0.23 LaF LiCl 0.36 FeCl Tetragonal MgF VF MnF Monoclinic SnF NiF ZrCl PbF IrCl TlF 0.48 Orthorhombic SbF Octahedra Fe 3 O VF structural metrics [46]. The peak stretching mode frequency is estimated using the combinative rule given as Γ AC = g s (Γ AA Γ CC ) 1/2, (3.17) where g s is a structural parameter which depends on the crystal structure, and Γ AC is the estimated force constant between anion and cation, using the monatomic force constants Γ AA and Γ CC. Figure 3.6(b) shows that electron in an infinite square well at the excited state. Figure 3.6 (c) shows the configuration coordinate diagram for the two states. Estimated magnitudes of g s, for some related crystals, are listed in Table 3.5. Satisfactory results have been reported in [26], using these estimates. The estimated peak phonon frequency, for some elements, is plotted in Fig. 3.7, for C-F crystal, where C is the cation element. As shown in the figure, the periodic-table first column 39

68 (a) Ligands Cation Anion Ion (b) C A I Г IAC Г IA Г m AC A m C Yb 3+ (c) -Q 1 Total Energy f i 2Q 1 Ground State 1 Δ ( h ) 2 p= 2m ω AC p Atomic Displacement Q 1 Square Well -Q 1 - Q f i 2Q 1 +2 Q Excited State 1 Δ ( h ) 2 p= 2m ω p AC Q Lattice Relaxation (under Energy Minimization) Q 1 + Q E f E i E 0 0 Q = 1 E e,f = Γ (2Q 1 +2 Q ) 2 2 IAC c i 1 E e,i = Γ (2Q 1 ) 2 2 IAC c f 3 2Q 1 Γ IAC Q c f + Δ 2 p + Δ 2 p Figure 3.6: Variation of electron energy with respect to the normal coordinate, at the doped-ion site, using an infinite square well model. (a) Ground state, and (b) excited state. (c) The configuration coordinate diagram for the process. 40

69 Al Ti Ni Zr W Ep,c =ħωp,c (ev) Na Mg V Fe Ca Mn K Rb Cd In La Pb Bi 0.03 C-F, C = Na, Al, Figure 3.7: (a) Peak stretching mode frequency using diatomic molecular data for cation-fluoride pair. z alkali-metals tend to have a lower peak energy. Generally, trend follows the periodic table row, indicating relation to the outer most electronic configuration of the cation. This relation is not pursued further here. For the remainder of the analysis, these phonon peaks are assumed as the most probable phonon energies in the coupling with electron of the ion, at moderate temperatures. Figure 3.8 shows the variation of g s with respect to the atomic number and some crystal structure groups are identified. No particular trend is found, however, all g s values are in the range. The ground-state energy of the electron in Fig. 3.6(a) is E i,1 = c i (2Q 1 ) Γ IACΔ 2 p, Δ p =( ) 1/2, (3.18) 2m AC ω p,p where c i is a constant that depends on the state quantum number and is independent of the bond, Γ IAC is the effective force constant of the ion-anion-cation set, Δ p is the displacement of the anion, due to a phonon with frequency ω p,p,andm AC is the 41

70 (a) Г AA (b) g s Г AA g s MgF 2 AlF 3 VF 3 VF 4 MnF 2 NiF2 InF 3 Orthorhombic TlF SbF3 SnF 2 Tetragonal 0.30 Cubic LaF CdF 2 PbF Figure 3.8: Variation of the structural parameter g s with respect to the atomic number, identifying some crystal structure grouping. The results are for C-F where C is the cation as shown. effective mass of the anion-cation pair. The excited state energy of the electron in Fig. 3.6(b) is z E f = c f (2Q 1 +2ΔQ) Γ IACΔ 2 p. (3.19) Since in general Q 1 ΔQ, weexpand(2q 1 +2ΔQ) 2 in terms of strain ΔQ/Q 1, i.e., E f = c f (2Q 1 ) Γ c f IAC(Δ p ) 2 ( c f ) 2 1, (3.20) 2Q 3 1Γ IAC 2Q 3 1 2Γ IAC where c f is again independent of the bond. In this treatment, since c i and c f are independent of ligands and are only a function of the ion, we compare the magnitude of change in electronic energy in response to ligand (lattice) vibration. In Eq. (3.20), the electron oscillates about the new equilibrium point c f /2Q 3 1Γ IAC and its energy is 42

71 reduced by (c f /2Q 3 1 )2 (1/2Γ IAC ). To analyze the electron energy change due to the displacement of the ligands (phonon), a simplified linear-chain model is considered. As discovered in [87], the most probable mode participating with electron of the ion is the breathing mode of the isolated ion-ligand complex. When considering the local available modes in the vicinity of the ion, stretching mode of the anion-cation pair is important [40]. Then from these structural metrics, the peak frequency ω p,p, and the cut-off frequency ω p,c, are estimated as ω p,p =( g sγ AC m AC ) 1/2, ω p,c =( 2g sγ AC m AC ) 1/2. (3.21) Due to the lattice vibration, the electron will oscillate as a harmonic oscillator which is displaced by Δ p.δ p is a function of m AC,andω p,p. Then using the following energy of the harmonic oscillator, the electron-phonon coupling (the rate of electronic energy change due to displacement of normal coordinates) is defined as ϕ e-p,o 1 2 Γ IACΔ 2 p Q 1, E e-p = 1 2 Γ IACΔ 2 p. (3.22) For simplicity, a linear derivative of the potential with respect to the normal coordinates is assumed. We have used the change in the energy of the ion electron as influenced by the anion-cation force constant. This is because the lattice is not composed of independent springs (as shown in the failure of the Einstein heat capacity model [3]). Figure 3.9 shows the calculated interaction potential using the square 43

72 3.0x x10-3 Na K Rb φ ' e-p,o = La 1 2 Γ IAC Δ p Q 1 φ ' e-p, O (ev/a) 2.0x10-3 Al Ca Mg Sr Sn Pb Tl 1.5x10-3 Cd V Zr 1.0x z Hf W C-F, C = Na, Mg,... Figure 3.9: Partial electron-optical phonon interaction potential using the infinitesquare-well model for cation-fluoride pair. well model, for various atoms in C-F, where C is the cation. The domain of the interaction is selected as two atomic spacings extending in the direction of the linear chain. Unlike semi-conductors, for doped ion the electrons of the rare-earth elements are highly localized within one or two atomic spacings [32]. Thus, it is reasonable to consider only the short-range interaction of the electron with the optical phonons. Figure 3.9 shows that the coupling is a slow decreasing function of the atomic number z. Preliminary ab initio calculations of the electron-phonon coupling in Yb 3+ :Cd-F using WIEN2K, to confirm the current hypothesis [10]. The discrete energy levels of the Yb 3+ : Cd-F was calculated using the electron band structure near the gamma point. The equilibrium structure energy level was first plotted and compared with the non-equilibrium (under longitudinal fluorine atom displacement) energy levels to estimate the electronic energy change due to longitudinal optical phonon mode (modeled as oscillation of fluorine atom). The results show that the electron-phonon 44

73 coupling is estimated to be slightly higher that the value predicted by the above analytical model. The difference is thought to be the result of contribution by adjacent anion and cation electrons which are shown to comprise a portion of the electron energy under oscillation near the Fermi level. Further ab initio studies are being conducted Phonon DOS Estimation Fernandez et al. [28] use the Debye-Gaussian model for the phonon density of states (DOS), which is close to the Debye model at low energies and has a Gaussian distribution at the center of the phonon spectrum, i.e., D p (E p )=c D E 2 p exp[ (E p E p,t ΔE p ) 2 ]. (3.23) Here, E p is the phonon energy, E p,t is the central frequency, ΔE p is the width of the phonon spectrum estimated to be approximately ev, and c D is a normalization constant. To find the normalization constant, we integrate the phonon DOS from zero to E p,c (the cut-off phonon energy). Such an approximation is useful for scaling D p in mixed glasses, where the structure is not known and the exact density of states is difficult to calculate. We use this approximation to estimate D p of the host material. The cut-off frequencies calculated in Section are used to estimate the total DOS of the bulk materials. Comparison among prediction by Eq. (3.23), prediction by MD simulation for Y 2 O 3 [87], and experimental result for Fe 3 O 4 [38] are shown in Figs (a) and (b). There are general agreement with both MD 45

74 predictions and the experimental results. However, since the frequency estimate uses the anion-cation pair, only the second peak is estimated and the first and third peaks which are likely due to cation-cation and anion-anion oscillations cannot be predicted using Eq. (3.23). In the interaction domain (in the vicinity of the rare-earth ion), only cation-anion pairs are relevant, due to absence of anion-anion and cation-cation oscillating pairs. The predicted peak and cut-off frequency are compared with the calculated values using the normal mode frequencies of the host complexes and these are listed in Table 3.6. The frequency estimation is compared with an analytic normal mode calculation using the interatomic potentials between the cation and anion, i.e., Γ Γ IA +4Γ IA for IA 6 type coordination Γ Γ IA +2Γ AA for IA 8 type coordination E p,p = ( Γ m A ) 1/2 for the breathing mode A 1g. (3.24) The peak phonon energy of the DOS, E p,p, is assumed to coincide with the breathing mode frequency and is related to the central phonon energy, E p,t by E p,p = E p,t ΔE2 E p,t. (3.25) Since the breathing mode frequency follows E p,a1g m 1/2 A, the population of the high-energy phonons decreases for the choice of a lighter anion. For CNBZn glass, E p,p,cdf ev and using E p,a1g m 1/2 A, E p,p,cdcl2 is estimated as

75 (a) 50 Y 2 O 3 MD Prediction (T = 300 K) Dp(Ep) (ev -1 ) Debye-Gaussian Model ΔE p 10 (b) 0 E p,p E p (ev) 50 Fe 3 O 4 Experimental Measurement (T = 296 K) 40 D p (E p ) (ev -1 ) Debye-Gaussian Model 10 E p,p E p (ev) Figure 3.10: (a) Comparison between the MD simulation phonon DOS [87] and the Debye Gaussian model, for Y 2 O 3. (b) Comparison between the experimental results [38] for phonon DOS and the Debye-Gaussian model, for Fe 3 O 4. 47

76 Table 3.6: Comparison of the phonon DOS peak energy E p,p values using the linear chain model, with the ion-ligand values, and the experimental values for some diatomic crystals. C-F Linear Chain E p,p (ev) Complex E p,p (ev) Experiment E p,p (ev) Cd-F Cd-Cl Zr-F In-F ev, which matches well with the Cd-F ( ev) and Cd-Cl ( ev) observed vibrations [1]. ΔE can be approximated as the difference in peaks of the Cd-F and Cd-Cl vibrations, therefore, ΔE ev, which is close to ΔE ev suggested by Fernandez et al. [28]. 3.4 Cooling Rate Optimal Photon Frequency As discussed in Section 3.1, the input off-resonance photon frequency determines the frequency of the phonon required to excite the electron from ground state. In turn, the input photon can be varied according to the distribution of the available phonon frequency in the lattice. By using the approximations made in Section 3.3 and using Eq. (2.17), we write the cooling rate as Ṡ ph-e-p Q ph,i = π 2ɛ o m AC μ ph-e 2 ϕ 2 e-p,o D p (E p )f o p (E p) E 3 p ω ph,i n d L (1 ω ph,e η e-ph). (3.26) u ph ω ph,i 48

77 S ph-e-p Q ph,i Heating Regime η p ph λph, λ i = λ ph,e ph,e (Resonance Emission and Phonon Side Bands) Yb 3+ : Zr-F φ' d,o = ev/å μ e = C m Experiment Gosnell [4] Yb 3+ : ZBLANP D p (E p ) D p (E p ) ( J -1 ) Cooling Regime E p (ev) Figure 3.11: (a) Idealized (resonance) and a realistic (side bands) normalized cooling rate as a function of phonon energy, for Yb 3+ : Zr-F. The cooling efficiencies are also plotted as solid line. The experimental result [34] is also shown. The cooling rate, is plotted in Fig. 3.11(a) by taking into account the multiple simultaneous transitions from the excited manifold of Yb 3+ to the four ground-state manifolds [64]. Then Eq. (3.26) becomes Ṡ ph-e-p Q ph,i = k=0,1,2,3 π 2ɛ o m AC μ ph-e 2 ϕ 2 e-p,o D p (E p )f o p (E p) E 3 p ω ph,i n d L (1 ω ph,e,k η e-ph), u ph ω ph,i (3.27) where j =0, 1, 2, 3 represents resonance, first, second and third phonon side bands, respectively. The results shows that the maximum cooling rate is to the left of the phonon peak. This is due to the phonon distribution function which suppresses phonons with higher energy and significantly influences the cooling rate at low temperatures. Figure 3.12 shows distribution of the maximum cooling rate as a function of the atomic number. The trend is fitted to a with 4th-order polynomial to guide the eye. 49

78 S ph-e-p Q ph,i T = 500 K Yb 3+ : C-F Ne Ca Zn Zr Sn Nd Yb Hg Th Fm Figure 3.12: Dimensionless cooling rate as a function of atomic number, for discrete values of temperature for Yb 3+ : C-F where C stands for cation. Some elements are added for reference. Also, a fourth-order polynomial fit is shown to guide the eye. Note that semiconductors and rare-earth materials have been omitted. The results show that there are two peaks, between atomic numbers 20 and 30 and 75 and 85. This trend supports the recent successes of blending of light and heavy cations as host materials for laser cooling of solids. However, as the temperature decreases and the available high energy phonons diminish rapidly (Bose-Einstein distribution fp o ), the cooling rate is quickly suppressed Optimal Host Pairs The above analysis provides a guide to the selection of ion-host materials for optimal performance. Here we compare various host materials, based on performance by atomic pairs, and we choose F as one of the atoms. Figure 3.13(a) shows variation of dimensionless cooling rate with respect to temperature, for some C-F pairs with Yb 3+ ion. The crystal structure assumed is FCC, which has C-F pairs as the ion immediate ligands. Here C is Tl, Zr, Hf, Nb Fe, Mg and Al. These structures may 50

79 .. (a) S ph-e-p Q ph,i (b) S ph-e-p Q ph,i x10-4 Tl Zr Hf Fe Al Nb Zr Hf T (K) Cs Rb K Lower Cooling Efficiency Tl Pb Mn Mg Na Al Nb Fe Mg Tl C-F, C = Tl, Zr,... Yb 3+ : C-F T = 300 K Lower Phonon Availability Zr Nb V Sc Al Experiment [4] Yb 3+ : ZBLANP E p (ev) Figure 3.13: (a) Dimensionless cooling rate as a function of temperature, for Yb 3+ : Tl-F, Zr-F, Hf-F, Nb-F Fe-F, Mg-F, and Al-F.. The results are for ideal conditions, i.e., quantum efficiency of one and identical FCC structures. (b) Variation of the maximum, normalized cooling rate for various cation-fluoride (C-F) pairs, as a function of the phonon energy for Yb 3+ : C-F, where C represents the various elements in the periodic table. Note that semiconductors and rare-earth materials have been omitted. The dashed line is only intended to guide the eye. 51

80 not be realized, for example, AlF 3 is an stable, existing compound. However, if a blend of different C-F pairs are made, the contributions of these pair ligands exist at the ion site. Figure 3.13(a) shows that Al which has relatively low phonon peak energy predicted by Fig. 3.11, exhibits high cooling rate over a wide range of temperatures, however, one can expect that the energy removed per transition is low (low capacity). On the contrary, one can expect the energy removed per transition is high (high capacity) for Zr (due to relatively high phonon peak energy), yet the performance decreases rapidly as the temperature decreases. The inset in Fig. 3.13(a) shows that at temperatures near 150 K, it is possible to reach even lower temperature with Tl compared to Zr. The maximum cooling rate, for some cation-fluoride pairs are given in Fig. 3.13(b). The results show that the first column alkali-metals from the periodic table are not good candidates for laser cooling at T = 300 K. However, due to the relatively lower phonon energy, these elements are expected to be more suitable at lower temperatures with the exception of Cs and Rb. The results are expected from Eq. (3.27), which shows that there are several competing processes in laser cooling of solids. These are, (a) higher phonon peak energy results in more energy removed per transition, (b) lower phonon peak energy results in higher phonon distribution values (c) low cut-off frequency results in higher phonon density of states, and (d) higher cut-off frequency results in higher a non-radiative decay. Using the above discussions, it is possible to quantitatively predict the cooling performance of a blended material. Figure 3.14 shows the cooling performance of an example blend of host materials. In practice, the composition discussed here may not 52

81 Yb 3+ : CF 4 or BF 4 S ph-e-p Q ph,i ZrF 4 ScF Zr 0.25 Al 0.25 Mn 0.25 Sc 0.25 F 4 Increased Abosorption Probability Using Blend MnF 4 AlF E p (ev) Figure 3.14: Dimensionless cooling rate as a function of phonon energy for diatomic host Yb 3+ : CF 4 or blend host, BF 4. The cooling rates have been calculated using ideal conditions, i.e., quantum efficiency of one and FCC structure, and are linearly superimposed. Although the magnitude of the cooling rate is moderated, the absorption probability increases as the phonon spectrum broadens. The cooling rates for diatomic hosts are shown in dashed lines and exhibit less broadening. be realized, however, the blend here provides an example providing a valuable general guide. The figure shows that by blending materials which have different phonon peak energies increase the half width of the transition. This, in turn increases the transition probability. One can expect that as the half width broadens, it increases the probability of various phonon modes available in the lattice coupling with the electron in oscillation. This blending strategy is expected to increase the absorption rate as much as factor of 2. Results of Figures 3.13(a) and 3.14 suggest using elements Al, Mn, Na and Mg in the host blend. Nevertheless, for optimized cooling performance, wide range of elements should be present in the blend for increased absorption probability (with the exception of Rb and Cs). 53

82 3.5 Discussion Transition Dipole Moment Optimization While selecting the dopant ion for a large μ ph-e, according to the charge displacement estimation model, the following general guidelines apply. First, a large energy gap ΔE e,g, would result in higher ΔR e,if, and therefore, larger μ ph-e. However, a large ΔE e,g also results in a lower cooling efficiency for the overall laser cooling process. A small energy gap, on the other hand, would lead to high non-radiative decay. Secondly, a transition between similar states, for example 2 F 5/2 2 F 7/2,leadsto a larger g μ, and therefore, a larger μ ph-e. The excited-state chosen for the transition should have the same total angular momentum as the ground state of the ion. Lastly, the energy levels for the transition should be selected in a manner such that there are no allowed energy levels in between the ground and the excited state manifolds, which can result in non radiative decay that severely affects the cooling efficiency. For example, the radiative lifetime of 4 I 9/2 state of Er 3+ is computed to be 20.7 ms, but its observed value is 0.15 ms [15], as a result of the non-radiative decay due to the presence of intermediate levels between 4 I 9/2 and the ground state 4 I 15/2. The calculated energy levels of various rare-earth ions are presented in [18]. Ce, Pr, Nd, Pm, Sm, Eu, Tb, and Dy, have energy gaps between the ground state and the lowest excited state, with magnitudes less than 2k B T (at room temperature). This would lead to high rates of non-radiative decay. Gd has a very large energy gap between the ground and the lowest excited state, which would result in a low laser cooling efficiency. Also, the ground and excited states have different L values, 54

83 (a). (b). S ph-e-p S ph-e-p Q ph,i Heating Cooling E p Zero Phonon Line Q ph,i Heating Phonon D p (E p ) Normal Modes E p,o. γ ph,a (Threshold E). S ph-e-p Q ph,i E p,c Multiple Phonon Peaks D p (E p ) D p (Ep) D p (Ep) Continuous Spectrum of Normal Mode E p Cooling. S ph-e-p Q ph,i E p,c Figure 3.15: (a) Variation of dimensionless cooling rate, absorption rate, and phonon DOS with respect to phonon energy. Maximum cooling rate can be achieved when the energy of maximum normal mode of the ion-ligand complex coincides with the cut-off phonon energy E p,c of the host and is the most available mode. (b) Multiple cooling peaks (increased cooling probabilities) are possible when multiple blends are present in the host and low symmetry is achieved at the ion site. therefore, a transition between these states would have a low value of g μ. Ho, Er, Tm, and Yb have ground and excited states with same total angular momentum L. Amongst these, Yb has the largest energy gap ΔE e,g between the ground and the lowest excited state, and should result in high ΔR e,if and low non-radiative decay. 55

84 3.5.2 Limits in Laser Cooling of Solids Figure 3.15(a) shows a qualitative prediction of the cooling rate when the phonon limit is removed. The results show that the maximum off-resonance absorption and the maximum anti-stokes cooling will occur when the maximum normal mode of the complex coincides with the peak of a single available mode, i.e., all of the available phonon modes are at the cut-off frequency. This is a hypothetical case since the integral of phonon DOS at the cut-off frequency has to be unity (a delta function) meaning there is no other phonon mode present (except for the phonon modes corresponding to the cut-off frequency). This will increase the phonon DOS by approximately a factor of 10. Note that when the limit from the structure (multiple phonon energy) is lifted, the absorption limit becomes only a function of the phonon distribution function fp o (T ). Figure 3.15(b) shows that when multiple-pair blends are used as hosts and low symmetry is achieved at the ion site, multiple cooling peaks are possible. Multiple-pair blends provided multiple phonon DOS peaks which can span across the phonon spectrum. When low symmetry is present at the ion site, the normal mode of the complex becomes broader, allowing multiple phonon mode couplings Off-Resonance Transition Dipole Moment The structural metrics which are developed above suggest multiple improvements and explain recent experimental success of laser cooling of blend solids. However, to scale various blended host material and doped ions, direct comparison amongst 56

85 various photon-electron-phonon interaction terms is necessary. We use Eq. (2.17) to define an a effective transition dipole moment which is analogous to the transition probability (Einstein coefficient A) when phonon participation is present, i.e., γ ph-e-p = 1 τ ph-e-p = ω 3 e,g 3π 2 2 u 3 ph ω ph,i ϕ e-p,o D p(e p )fp o (E p ) 3πɛ o u 3 ph 2m AC ωe,gv 3 ω 3 e,g E 3 p μ ph-e 2 μ 2 3πɛ o u 3 ph-e-p, (3.28) ph where μ 2 ph-e-p is the phonon-assisted transition dipole moment. This effective transition dipole moment gives the strength of phonon-assisted transition probability for material selection and is directly comparable to the resonance transition dipole moment μ ph-e Time Scales for Laser Cooling of Solids As mentioned in the Section 3.1, laser cooling has three processes and they are characterized by their time constants namely, τ ph-e, τ e-p and τ ph-e-p. Figure 3.16 shows the temperature dependence of these time constants. The radiative lifetime stays constant with respect to temperature, while the multi-phonon relaxation time is a strong function of temperature. The multi-phonon decy process is given by [86] γ e-p = γ ph-e[1 exp( ω p,c/k B T ) exp( ω p,c /k B T ) 1 ]Np, N p = E e,g ω p,c. (3.29) Then using the host selection metrics presented in the preceding sections, Eq. (3.29) is evaluated. The predicted photon-induced, phonon-assisted transition lifetime is 57

86 τ (ms) τ ph-e-p τ e-p (E p,c = ev) (E p,p = ev) Yb3+ : Zr-F Experiment Gosnell [5] τph-e T (K) Figure 3.16: Variation of time constants as a function of temperature. The radiative relaxation time is the shortest followed by the phonon-assisted, photon absorption at low temperatures. Multi-phonon relaxation exhibits strong temperature dependence and is dominant at high temperatures. compared with the experimental lifetime of [34] and is in good agreement. Then the dimensionless cooling rate is expressed as Ṡ ph-e-p Q ph,i = n d V τ ph,tr (1 ω ph,e η e-ph), (3.30) τ ph-e-p ω ph,i where τ ph,tr is the photon transit time which is defined by τ ph,tr = L/u ph,wherel is the sample length (along the beam). We have neglected the re-absorption of emitted photon since it is estimated to be only 0.005% of the total emission rate [87]. This indicates that unlike energy transport, the energy conversion process is a product of the time constants of the processes. The cooling rate is directly limited by the the phonon-assisted absorption process, which has a transition rate two orders of magnitude smaller than that of the purely radiative transition. Figure 3.16 also suggests that by optimizing the phonon-assisted transition rate, the cooling rate increase by a factor of 2, compared to experiment of [34], at room 58

87 8T Q ph,i Laser Input Power Q ph,b Ts -1 τ ph, tr ω ph, e τe ph S ph-e-p = nv d (1 ) Q τ ω τ -1 + τ -1 L ph ph e p ph, i e ph e p Q ph,e Luminescence Emission Power ph, i D Figure 3.17: Heat transfer mechanism of laser cooling of solids. The solid sample exchanges radiation heat with the surroundings through vacuum. The sample temperature will be determined by the balance between the external radiation (thermal load) and the cooling rate. temperature. The sample temperature depends on the balance between the thermal load ( radiative heat transfer from surroundings) and the laser cooling rate, shown in Fig The temperature of the sample is determined from the energy equation Q ph-b = Ṡph-e-p ( 1 2 πd2 + πdl)ɛ ph σ SB (Ts 4 T 4 ) = n dv τ ph,tr (1 ω ph,e τ 1 ph-e τ ph-e-p ω ph,i τ 1 ph-e + τ e-p 1 )Q ph,i, (3.31) where ɛ ph is the total emissivity of the sample, σ SB is Stefan-Boltzmann constant, T and T s are the sample and surroundings temperatures. This shows that as the cooling power increases, the thermal load increases rapidly restricting the maximum cooling temperature. With the factor of 2 increase in the cooling rate, a 30% decrease in the T s is expected, compared to the existing experiment [34]. Further improvement by photon trapping and possibility of using nanostructures to tailor D p for improved cooling rate have been discussed in [87]. 59

88 3.6 Summary Laser cooling of solids is interpreted as a phonon absorption followed by an offresonance photon absorption process. The transition dipole moment is expected to be the highest for Yb 3+, due to high energy separation and no intermediate available states between the ground and excited states. The anti-stokes process is expected to have the highest probability when the normal modes of the ion-ligand complex coincide with the maximum available phonon modes predicted by the diatomic Debye- Gaussian phonon DOS model. The model predicts that the ideal laser cooling host material vary with target temperature and the model successfully predicts the recent success in using blends of elements for laser cooling of solids. Materials which were never been used such as Li, Al, Mn, Nb, Hf and Mg are suggested here for improved cooling at low temperatures. The time constants for the individual transition processes are evaluated and the anti-stokes process is found to be limited by the phonon-assisted absorption time. The models (materials metrics) developed here suggest improvements (over current record) in the cooling rate (and lowered target temperature) are possible using the identified optimal ion-host materials. 60

89 Chapter 4 Phonon recycling in rare-earth ion-doped lasers 4.1 Introduction Energy recycling is promising in photonic devices, such as high-power lasers, where laser inefficiency is critical [47]. Laser inefficiency can be viewed as quantum defect (emitted photon energy is lower than that absorbed, ω ph,p > ω ph,l ) and emitted phonons carry away excess entropy produced in the absorption of multimode pumping photons. The second law requirement leads that emitted phonons have higher entropy compared to photons. Thus, in high power lasers, elaborate cooling systems are unavoidable and hinders development of efficient portable units. Recent efforts on radiation balanced laser system have provided alternative to mechanical cooling systems. However, this is challenging due to limited photon wavelength contrast (between lasing sites and cooling sites) when single crystal laser medium is used 61

90 Energy flow diagram Cooling layer (C) (Phonon absorption ) Phonon recycling Lasing layer (L) (phonon emission) Diode laser pump photons Incoherent fluorescent emission Cooling cycle Phonon upconversion (3-phonon process) Phonon Lasing cycle Rare-earth ion (e.g., Yb 3+ ) Phonon emission (heat generation) Coherent laser emission Low cut-off frequency phonon lasing crystal (e.g., YAG) High cut-off frequency phonon cooling crystal (e.g., ZrF 2 ) Excess pump photons Figure 4.1: Phonon recycling in ion-doped lasing layer and its adjacent cooling layer. The energy-flow diagram, interactions among ions, photons and phonons. Photon and phonon energy flows are indicated with clear and grey arrows, respectively. The phonons emitted during lasing cycle are transmitted through the interface by acoustic and optical-phonon transmission and three-phonon upconversion due to an energy gradient. [12, 13, 14]. In this letter, we present detailed prediction of phonon recycling in multilayer ion-doped lasers using the anti-stokes luminescence and the FGR (Fermi Golden Rule) treatment of the carrier transition processes. Anti-Stokes luminescence has cooled ion-doped solids from room temperature to 208 K and its theory has been advanced recently [107, 25, 34, 27, 87, 53, 96]. Anti- Stokes luminescence is reverse of quantum defect (i.e., ω ph,sp > ω ph,p ) and the excess entropy is removed by multimode incoherent spontaneous photons. The phononassisted photon absorption rate strongly depends on high energy phonon occupation, making it inefficient at low temperatures. Then theoretical and experimental improved laser cooling of solids face significant challenges [40, 53]. Conversely, at moderate to high temperatures, the high energy phonons are more accessible. Phonon 62

91 recycling in ion-doped lasers use this by absorbing the high energy phonons created in the lasing layer to drive anti-stokes process in the adjacent cooling layer as shown in the energy flow diagram in Fig The excess photons subsequent to the laser cycle is redirected to the adjacent cooling layer and used in the cooling cycle. Thus, phonon recycling involves dynamics of the three energy carriers (photon, electron, and phonon) interactions within and across the two heterogeneous layers. 4.2 Ion-doped Lasers and Anti-Stokes Cooling Kinetics E e Excited state (L) Absorption ħω ph,p. γ ph,a ħω ph,l γ e-p,s γ ph,a. γ ph-e,l. ħω p,o. e γ e-p,m (T) 4 ħω p,o Excited state (C) ħω ph,l 5 ħω p,o. γ ph-e-p 6 e. γ ph-e,sp ħω ph,sp Ground state (L) Lasing layer (L) 1 Absorption of pump photons 2 Rapid single-phonon decay 3 Stimulated emission (laser) 4 Multiphonon emission e 5 6 Ground state (C) e Cooling layer (C) Single-photon phonon (O) absorption (anti-stokes) Spontaneous emission Figure 4.2: Electronic energy levels of laser and cooling layer dopants and interactions with energy carriers. The corresponding transition rates are indicated with numbered boxes. The single-phonon rapid decay follows the absorption rate, while the mutiphonon decay is independent of the photon transitions. In high power lasers, rare-earth (Lanthanides) ions are used for their limited interaction with ligands yielding high quantum efficiency [32]. The photon, electron, and phonon interaction kinetics of such lasers are shown in Fig 4.2. The laser diode- 63

92 pumped photons excite the electrons in the ground state to the excited state which in turn decay rapidly to the metastable upper level of the ion. As the photons are amplified in the laser cavity, stimulated emission begins to dominate beyond a threshold intensity producing coherent photons as predicted by the Einstein B coefficient [44, 50]. The stimulated emission competes with multiphonon decay at moderate to high temperatures which becomes the dominate source of heat. Figure 4.2 also shows the anti-stokes cooling kinetics where its resonance transition frequency designed approximately 2k B T higher than that of the lasing layer. The off-resonance photons give rise to phonon assisted photon absorption as phonon population increases during lasing. This off-resonance transition process is treated using FGR as γ ph-e-p = 2π M ph-e-p 2 δ D (E e,f E e,i ω p,o ), (4.1) where γ ph-e-p is the phonon assisted photon transition rate, photon (ph), electron (e) and phonon (p), M ph-e-p is the matrix element of this interaction, δ D is the Dirac delta function, E e,f and E e,i are the final and initial electronic energies, and ω p,o is the optical phonon frequency involved. Here, M ph-e-p, is from a second-order perturbation theory and the final form of the transition rate is γ ph-e-p = π 2ɛ o m (s ph,i μ ph-e) 2 ϕ 2 e-p,o D p(ω p,o ) f p(ω p,o ) E 3 p,o L Q ph,p Q ph,a, (4.2) u ph n d,c V where m is the reduced mass of host atom pair, s ph,i is the the polarization vector, μ ph-e is the electronic transition dipole moment vector, ϕ e-p,o is the optical-phonon 64

93 deformation potential, D p (ω p,o ) is the phonon density of states of phonon having frequency ω p,o, f p is the phonon distribution function, L is the photon transit length, u ph is the speed of light, and (Q ph,p Q ph,a )/n d,c V represents the photon intensity per dopant ion. As can be seen in equation (2), the phonon assisted photon transition rate is a function of atomic properties of the dopant ion and the host which are optimized for desired performance [53]. 4.3 Phonon Transmission and Up Conversion In ion-doped lasers, phonons are emitted in the lasing layer by temperatureindependent rapid single-phonon decay of excited electrons to the metastable upper levels and by temperature-dependent multiphonon decay from the metastable levels to the ground state. This sudden nonequilibrium phonon distribution drives phonon propagation to the adjacent cooling layer. The propagation process involves acoustic-optical phonon up and down conversion (three-phonon processes) [117, 103] and optical and acoustic phonon transmission [106] which are influenced by phonon spectrum of both layers [50, 106]. Figure 4.3(a) shows the Debye-Gaussian model of phonon DOS [density of state, D p (E p )] with low cut-off energy (E p,c,l ) phonon host in the lasing and high cut-off (E p,c,c ) in cooling layer. The three phonon processes involved in heat transfer across layers is shown in Fig. 4.3(b). Note that the figures are not to scale, and the phonon DOS integrates to 3N, wheren is the number of atoms, so depending on the cut-off phonon energies, the curve moves up and down. The shaded area under the curve in Fig. 4.3(a) represents occupied phonon levels at 65

94 Dp(Ep) Low cut-off frequency phonon crystal (a) Occupied levels E p,c,l High cut-off frequency phonon crystal E p,c,c E p. A γ A+A O ħω p,a Boundary (~ps) ħω p,a ħω p,o Acoustic-optical upconversion (three-phonon process). B γ tr, O (~ns) ħω p,o Optical phonon transmission. C γ tr, A (~ps) ħω p,a Acoustic phonon transmission (b) Figure 4.3: (a) Overlap of phonon density of states (Debye-Gaussian model) for low (red) and high (blue) cut-off frequency crystals (not to scale). Filled areas under the curve are the occupied phonon states at a given temperature. Saturation of low frequency phonons in the cooling layer makes upconversion favorable when transmission of phonons are needed from the lasing layer. (b) The three phonon transfer mechanism are shown as letters. a given temperature and is the product of phonon DOS and the Bose-Einstein distribution function. When E p,c,l <E p,c,c, the average value of phonon DOS is higher in the lasing layer, and for a given energy, phonons occupy lower energy states because of their higher availability. In contrast, for the cooling layer, the value of phonon DOS is lower, but extends further into the high-energy regime. Thus, equilibrium phonons occupy higher energy states in the cooling layer. During laser operation, generated phonons begin to occupy higher energy states in the lasing laser and creates energy nonequilibrium between the lasing and cooling layer. Then an energy gradient forms which promotes propagation of phonons to the adjacent cooling layer. In crossing the interface, due to the argument above, most low-energy phonon states are already occupied in the cooling layer. Thus, for phonons to propagate into the cooling layer, among the three-phonon processes the acoustic-optical phonon up conversion process becomes favorable. The added high-energy phonons in the cooling layer are used in 66

95 the phonon-assisted absorption (anti-stokes absorption). 4.4 Nonequilibrium Phonon Distribution and Photon Re-absorption Because of phonon transmission and upconversion processes, equilibrium phonon treatment (Bose-Einstein distribution, fp o ) used in [87, 53] is no longer valid in calculating the transition rate involved in phonon recycling. Calculation of the exact nonequilibrium distribution is very challenging. However, we calculate the time scale for each process to identify the bottleneck mechanism. It is then reasonable to assume the non-bottleneck processes as equilibrium. For example, acoustic phonon thermalization has time scale of the order of ps, the upconversion of two acoustic phonons to one optical phonon is also of the order of ps [50, 117, 103]. However, due to their low speed, the optical phonon transmission is expected to have time scale of the order of 10 ns [79] for sub mm length scale. Thus, the bottleneck process is identified as optical phonon transmission. Therefore, in nonequlilibrium phonon populations, deviation from equilibrium distribution for transmitted optical phonons is accounted for by introducing a transmission coefficient, τ p,tr,o while equilibrium distribution is used for three-phonon upconversion process. These treatments of nonequlibrium phonon populations give f p = f o p (ω p,o )+τ p,tr,o f o p (ω p,o )+f o p (ω p,a1 )f o p (ω p,a2 )[f o p (ω p,o )+1]. (4.3) 67

96 fp o(ω p,a 1 )andfp o(ω p,a 1 ) are distributions of any two acoustic phonon frequency which can be annihilated to create one optical phonon of frequency ω p,o and must be allowed by phonon dispersion relation and momentum conservation [117]. Note that as the time-scale contrast decreases, the population deviate from equilibrium, resulting in lower occupation number for the three-phonon process. 4.5 Improved Laser Efficiency with Phonon Recycling The phonon recycling transition rates appear in macroscopic energy equation, i.e., Q ph-e-p = n d,c γ ph-e-pv, energy conversion (Ṡ) energy transfer (Q) and used for steady state behavior of ion-doped laser with adjacent anti-stokes cooling. Figure 4.4(a) shows cooling efficiency η C (heat removed by anti-stokes cooling/total heat generated by lasing) as function of temperature, for three different cooling crystals with dopant Yb 3+ and Yb 3+ :YAG lasing crystal (1 cm in diameter and 1.5 mm in thickness). The pumping, laser and spontaneous emission wavelengths are at λ ph,p = 998 nm, λ ph,l = 1030 nm and λ ph,sp = 975 nm respectively. The most probable optical and cut-off frequency of hosts are found by material metrics given in [53]. The results show ZrF 2, optical phonon energy of ev, phonon recycling rate peaking at the conventional laser operation temperatures T L = 300 to 325 K with approximately 30% of phonons recycled. At low temperatures, the anti-stokes cooling rate rapidly diminishes, while the temperature-independent, rapid single-phonon decay 68

97 Q ph-e-p η C = Q e-p,s + Qe-p,m Cooling layer : 2nd Law Limit Yb 3+ :ZrF 2, (ħω p,o = ev) Yb 3+ :CdF 2, (ħω p,o = ev) Yb 3+ :WF 4, (ħω p,o = 0.053eV) η L = Q e-ph,l Q ph,a Q e-ph,l = W Q ph-e-p = 11.6 Q e-ph,sp = Q = 26.2 n d,l : n d,c = 1:5 Cooling layer : Yb 3+ :ZrF 2 Q P = 750 W Q = 40.3 Cooling Layer Included. 1 Q e-ph,l = W Q = T (K) (a) T (K) (b) Figure 4.4: (a) Phonon cooling performance as a function of temperature for phonon recycled ion-doped laser using three different cooling crystals. 1 cm in diameter and 1.5 mm thick Yb 3+ :YAG disk is used for the laser layer, and emission spectra of Yb 3+ :ZBLANP is used to calculate the spontaneous emission peak [99, 34, 63]. The second law limit of anti-stokes cooling is also shown. (b) Laser operation conditions as a function of temperature with (1 ) and without (1) phonon recycling. For 1, 12.8 W of heat is removed by phonon recycling, while producing W of spontaneous emission. When the same external cooling (40.3 W) is used, this reduced cooling load corresponds to a reduced laser crystal temperature and moves the operation condition to 2. remains constant. At high temperatures, the anti-stokes cooling rate increases, and temperature-dependent multiphonon decay dominates over all other mechanisms thus decreasing the phonon recycling. For crystals with low optical phonon energy, e.g., CdF 2, phonon recycling peaks at lower temperatures and decreases rapidly at high temperatures (due to low optical phonon transmission rate). In contrast, high phonon energy crystals such as WF 4 result in much lower phonon recycling rate (due to their rather large reduced mass and lower phonon occupancy number). However, experiments use combination of different materials to achieve maximum cooling because of statistical distribution of spontaneous emission spectrum resulting from various defects and impurities [40, 64]. In Fig. 4.4(b), variation of laser efficiency η L (laser 69

98 emission/total absorption) is shown for a laser system as a function of temperature. The laser with no phonon recycling is pumped by Q ph,p = 750 W diode laser, while producing Q ph,l = W laser emission and Q e-p, = 40.3 W of heat [99]. With an added cooling layer of the same volume and having 5 times the dopant concentration, we predict Q ph-e-p = 12.8 W of heat is removed by phonon recycling producing Q ph,sp = W of spontaneous emission. When same external cooling unit is used, for example, Q e-p, = 40.3 W, the reduced cooling load translates to 12 K lower crystal operating temperature, moving the operating conditions to higher laser efficiency. Alternatively, for the same operating temperature, the phonon-recycled laser requires 30% less cooling resulting in significant decrease in the cooling unit, which may allow for switching to coolant from water to air and allow for portability. The thermodynamic limit is found by examining the second of the entire phonon recycled laser as shown in Fig.4.5. The incoming and outgoing entropies are evaluated using Q/T of the energy conversions. To find the equilibrium temperature of the photon source, the thermodynamics of radiation is used [60, 61]. The flux temperatures (defined by Q/Ṡentropy) of 11,000, 44,000, and 4,500 K are used for the diode laser T P,laserT L and the spontaneous emission T ph,sp, respectively [73, 90] while, entropy of the phonons are evaluated at the laser crystal temperature T L. In order to satisfy the second law limit, the summation of entropy fluxes need to be positive but zero, for reversible process (limit). The second law limit for cooling layer alone is at 60% at 300 K as shown in Fig. 4.4(a), while the limit for the lasing and cooling is at 70% (not shown). This shows that phonon recycling satisfies the second law and is open to future improvement (removing inefficiencies in carrier interactions). 70

99 Incoherent photon Q ph-e,sp T ph-e,sp Qe-p, T Q P T P Cooling layer (C). S ph-e-p T Q e-p T+ T Q ph-e-p T P Pump photon Q L T L Pump photon 1 Lasing layer (L) Thermal system Coherent photon T P Q e-p Q P Q L Lasing layer : + 0 T P T L T+ T. Q ph-e-p S ph-e-p Q ph-e,sp Q e-p, Cooling layer : + 0 T P T T ph-e,sp + T Q e-p Q e-p Layer boundary : T+ T + T 0 Q Entire system : P Q ph-e-p Q L Q ph-e,sp Q e-p, + TP + T L + T ph-e,sp + T 0 Figure 4.5: Entropy flow diagram for phonon recycling in ion-doped laser system. Second law requires every component in the system to satisfy the inequality. Individual components of the laser system are marked by numbers and corresponding inequalities using the heat transfer rates at source or sink temperatures. 4.6 Summary Synergy effect of introducing the anti-stokes cooling in ion-doped lasing through phonon recycling is proposed and the predicted performances presented. The lasing and cooling kinetics are governed by atomic quantities comprising the carrier interaction rates which then are introduced in the macroscopic energy equation. Phonon transmission between the adjacent lasing and upconversion cooling layer are discussed and included in the nonequilibrium phonon occupancy. For the example considered, the increased phonon population enhances the anti-stokes cooling rate resulting in 71

100 30% phonon recycling and improved laser efficiency is predicted when using the same cooling unit. The phonon recycling is well within the second law limit. This letter present theoretical advancements in phonon recycling by applying atomic carrier interactions to thermodynamics and heat transfer of photonic devices and with potential in application to other areas of photonics. 72

101 Chapter 5 Phonon-assisted enhanced absorption of alloyed amorphous silicon for solar photovoltaics 5.1 Introduction a-si: H is a common solar photovoltaic (SPV) material with low energy conversion efficiency compared to c-si [66]. Alloyed a-si: H has not been among the next generation SPV materials because a-si technology has been considered to be relatively mature and the abstruse carrier transport physics of random structures pose as obstacle [105]. Earlier studies explored this complex carrier transport kinetics and recent ab-initio calculations have dealt with the origin of the Urbach tails [78]. First-principles calculations on a-si is challenging due to absence of long-range order. However studies have used periodic disordered cells with good results [22, 20, 21]. 73

102 Despite the challenges, amorphous solids provide a unique opportunity for theoretical prediction of phonon properties because of absence of the selection rule, especially momentum conservation, due to loss of translational symmetry [105]. In addressing SPV efficiency improvements, the role of phonon has not been properly explored than in other photonic devices [55]. Using X alloying of a-si with a phonon-participation perspective on solar energy conversion, we show by alloying, the electronic density of states D e of a-si x X 1 x is minimally altered, and that there is enhancement in the second-order, phononcoupled photon absorption due to lower phonon energy and enhanced phonon density of states D p. We select group IV element to avoid doping effects due to extra electrons or holes, focusing on elements known to form similar tetragonal structure as Si (e.g., Ge and Sn). Especially, for Sn forming soft bonds, it is predicted to significantly lower the phonon energy in the alloy and reduce the electronic bandgap [58, 71]. Also, as the mass mismatch increases the desired separation of the phonon peaks (similar to phonon bandgaps) resembles highly mismatched crystals [70]. While Si- Sn crystal faces phase separation [58], introduction of Sn into an amorphous phase Si can circumvent this problem. Here, a-si is simulated with disordered 64-atom cells generated by the Wooten-Winer-Weaire method [113]. To simulate an a-si-x alloy we replace the Si atoms with X atoms. We average over 6 different configurations of X atoms in a-si structure. We use the Fermi golden rule, the harmonic vibration, and the ground-state electronic and phonon calculations for this preliminary atomistic model which allows for the optimization of the alloy. 74

103 10 4 E e,g Conduction Band Phonon Assisted Energy States 298 K (cm -1 ) 10 3 ~ħω p,o Valence Band T = 12.7 K σ ph,a 10 2 Phonon-Assisted Transition (2nd Order Transition) 1.2 Phonon Enhanced Absorption 298 K 3 2 Regime 1 Direct Transition ħω ph (ev) Figure 5.1: Absorption coefficient for a-si: H at 12.7 and 298 K [81]. At low temperatures, single photon absorption is dominant but significant temperature dependence is shown near room temperature. The dotted line shows qualitative enhancement of absorption when the phonon coupling is enhanced with no temperature change. The inset shows the energy processes involved in the transitions. 5.2 Absorption Coefficient of a-si: H We begin by examining the absorption coefficient of a-si: H, as shown asa function of photon energy in Fig. 5.1 [81]. There exists a significant temperature dependence of the absorption spectrum near the optical bandedge and the spectrum shows directband-like behavior. In semiconductors, the majority of this temperature dependence is known to be the result of electron-lattice interaction [111] which makes the role of phonon important at finite temperatures. Still, the optical transitions of a-si: H and the effect of disorder, thermal vibration, and hydrogen content have not yet been clarified [72, 17]. However, strong experimental data and theoretical analysis show the bandgap is strongly dependent on the hydrogen content and the band-to-band optical transition is dominated by indirect (phonon-coupled) transition similar to c-si [33, 30]. So, the direct-band like absorption behavior is explained by phonon 75

104 coupling in the absence of the momentum conservation rule. We decompose the absorption coefficient into three distinct regimes, i.e., 1) direct absorption where only one photon is absorbed in the transition process, 2) phonon-assisted absorption where one phonon and one photon (with energy higher than the bandgap) are absorbed, and 3) the same but for photon energy slightly lower than the bandgap. When phononassisted absorption is enhanced, the absorption coefficient will increase at constant temperature, as shown by the dotted line. The electronic energy changes due to these processes are shown as an inset in Fig. 5.1 where phonon assisted energy states appear due to electron-phonon interaction. 5.3 Photon-Electron-Phonon Interactions and Absorption Figure 5.2. renders conceptual photon-electron-phonon interaction in a-si 1 x X x. The initial i and final f are the semiconductor valence and conduction electron states. At T>0phonons propagate through the lattice causing continuous excitation and de-excitation of electrons in the ground state. This interaction results in oscillation of the electrons through the acoustic and optical electron-phonon coupling ϕ e-p,a and ϕ e-p,o and create the phonon-coupled energy states l [50]. These energy states contrast the strong-ion coupled phonon states in isolated ions [89], since the electrons are not tightly bound to any ionized dopant. The a-phase allows for an isolated electron wave function ψ e representation, due to broken symmetry. Therefore, the interaction 76

105 Amorphous Si Isolated Electron Wavefunction Representation ψ e Alloy X Final State f Positive r Negative Charge Charge Electron Overlap μ e = ψ e,f e c r ψ e,i ψ e,f - Γ Si-Si Initial State i Γ Si-X ψ e,i + ψ e 2 m Si m X ψ e 2 Vibration Energy ħω p (Phonon) Ground State Oscillation (Phonon) Coupling ψ e,i ψ e,l ψ e,f φ e-p,a and Phonon- Coupled State l φ' e-p,o r Phonon-Coupling Effective Overlap Photon Energy ħω ph μ ph-e-p = ψ e,f e c r ψ e,l Figure 5.2: Conceptual rendering of photon-electron-phonon coupling in a-si-x alloy using isolated electron representation. The electron distributions for initial (i), final (f ), and phonon coupled (l) states are shown along with the photon and phonon energies. matrix can be expanded using the weak-phonon-coupling Hamiltonian, where the phonon-coupled states are treated as intermediate states during the transition. Then the electronic overlap (μ e ) turns into a phonon-coupling effective overlap (μ ph-e-p) which can be enhanced by alloy optimization [52]. When an electron is excited by absorption of a photon ( ω ph ), it leaves behind a positive charge (hole). This three carrier interaction can be represented by weak-phonon-coupled photon absorption (2nd-order transition) using Fermi golden rule and is written as [28] γ ph-e-p = 2π M ph-e-p 2 δ D (E e,f E e,i ω ph ω p ), (5.1) f where δ D is the Dirac delta, E e,i and E e,f are, respectively, the initial and final energies of the electron, and ω ph = E ph and ω p = E p are photon and phonon energies. M ph-e-p is the interaction matrix from a second-order perturbation theory and total 77

106 Hamiltonian of the system is H=H e +H p +H ph +H ph-e +H e-p, (5.2) where H e,h p,h ph,h ph-e, andh e-p are electron, phonon, photon, electron-photon interaction, and electron-phonon interaction Hamiltonians respectively. The acoustic and optical electron-phonon interaction can be written as H e-p,a = ϕ e-p,aɛ, H e-p,o = ϕ e-p,oq κp, (5.3) where ɛ and Q κp are the strain and the normal coordinate respectively. Then the second order interaction matrix becomes M ph-e-p = l l f H int l l H int i E e,i E e,l ψ f,fph o,fo p H ph-e ψ l,fph o +1,fo p ψ l,fph o +1,fo p H e-p ψ i,fph o +1,fo p +1, E e,i (E e,i ω p ) (5.4) where ψ is the isolated electron wave function, state l is phonon assisted state, H int = H ph-e +H e-p is the interaction Hamiltonian, fph o and f p o are boson distribution functions. When the above matrix is expanded and plugged into Eq. (1), the transition rates for acoustic and optical phonon-coupled photon absorption become 78

107 [87, 52] γ ph-e-p = π 2 ɛ o ( ϕ2 e-p,a mu 2 p or ϕ 2 e-p,o mωp 2 )(s ph,i μ ph-e) 2 D p (E p ) f o p (E p ) E p e ph,i, (5.5) where m is the reduced mass of oscillating atom pair, u p is the speed of sound, ɛ o is the permittivity, s ph,i is the polarization vector, μ ph-e is the electronic transition dipole moment vector, D p (E p ) is the phonon density of states of phonon having energy E p, and e ph,i is the energy density of the incoming photon. This transition rate is directly proportional to the absorption coefficient through σ ph-e-p,a = ω phn e γ ph-e-p u ph e ph,i = πω phn e 2 u ph ɛ o ( ϕ2 e-p,a mu 2 p or ϕ 2 e-p,o mωp 2 )(s ph,i μ ph-e) 2 D p (E p ) f o p (E p) E p, (5.6) where n e is the number density of absorption sites and u ph is the speed of light) and the coefficients are theoretically evaluated by ab-initio calculations or MD (Molecular Dynamics) [50]. 5.4 Electronic Structure of a-si x X 1 x First, we examine the electronic structure of a-si 1 x Sn x alloy by calculating the electronic density of states D e of a-si with and without X (group IV elements) using VASP [59], within the frame work of DFT (Density Functional Theory) and PAW (Projector Augmented Wave) potentials generated with GGA-PW91 pseudopotential Monkhorst-Pack k-point grid was used with energy cut-off of 370 ev. 79

108 50 40 a-si 0.5 X 0.5 Si X D e (1/eV-cell) a-si a-si 0.5 Ge a-si 0.5 Sn 0.5 Ee,g Urbach Tail E e (ev) Figure 5.3: Calculated electron density of states of a-si, a-si 0.5 Ge 0.5,anda-Si 0.5 Sn 0.5 with disordered 64-atom cells generated by the Wooten-Winer-Weaire method [113]. The bandgap E e,g and the Urbach tail are shown. A sample structure of a-si 0.5 X 0.5 is shown as an inset. The results are shown in Fig. 5.3 for a-si, a-si 0.5 Ge 0.5,anda-Si 0.5 Sn 0.5 and one of Si 0.5 X 0.5 structure is shown as an inset as an example. Although DFT cannot accurately calculate the excited states and predicts smaller bandgaps for insulators and semiconductors [35, 5], we can still examine the effect of X near the bandgap within the same structure. The results clearly show a bandgap E e,g of approximately 0.58 ev (which is similar to that of c-si predicted by DFT and smaller compared to a-si: H due to no H content) and the Urbach tail states indicating mostly 4-coordination. For such small cell sizes, defect states due to dangling bonds are not evident. However, the defect states are not relevant for the scope of this study. The comparison of the D e show that even with high X content, the overall density distribution near the bandedge is virtually unchanged. This is because most of the electronic density is known to be composed of s and p orbitals near the bandgap [9] which is retained in the a-si 1 x X x alloy (X being a group IV element). The results indicate that D e is 80

109 0.030 Dp (1/meV-cell) Si-X (A) Si-Si (A) Si-Si (O) Si-X (O) X-X (O) a-si a-si 0.5 Ge 0.5 a-si 0.5 Sn a-si (Experiment) E p (mev) Figure 5.4: Ab-initio calculated phonon density of states of a-si, a-si 0.5 Ge 0.5,and a-si 0.5 Sn 0.5 with disordered 64-atom cells generated by the Wooten-Winer-Weaire method [113]. The experimental D p of a-si using neutron scattering is also shown [11]. not sensitive to the type of the element but on the coordination of the atoms. However, for Sn, the results indicate a smaller bandgap which may be advantageous in increasing the energy conversion efficiency where the ideal single junction bandgap is found to be approximately 1.1 ev [66] whereas bandgap for a-si: H is approximately 1.7 ev. 5.5 Phonon Density of States of a-si x X 1 x We perform ab-initio phonon calculation of this 64 atom a-si cell using CASTEP [93]. Finite displacement method with GGA-PW91 pseudopotential are used along with k-point grid and energy cut-off of 200 ev. Figure 5.4 shows calculated D p of a-si, a-si 0.5 Ge 0.5, and a-si 0.5 Sn 0.5 using 0.5 THz smearing, along with the experimental neutron scattering result of a-si. The ab-initio phonon peaks of the 81

110 a-si is in good agreement with the experiment near 25 (A) and 60 mev (O). When element X is added, the lower phonon energy peaks (due to Si-X and X-X) appear. For a-si 0.5 Ge 0.5, there are no distinguished peaks near 15 and 55 mev, indicating mixed phonon states between the Si and Ge acoustic and optical oscillating modes and this trend is also found in the Raman experiment and MD (Molecular Dynamic) simulation results [11]. However, a strong Ge-Ge optical mode peak is found near 37 mev. For a-si 0.5 Sn 0.5, due to significant mismatch between the two elements, large segregation of the phonon peaks are found. The Si-Sn and Sn-Sn optical phonon peaks are near 45 and 30 mev. As the phonon peak energy decreases, the occupation probability of these modes increase (due to boson function) which predict enhancement of the phonon-coupled photon absorption and the overall absorption coefficient. Unlike D e, D p shows significant dependence on the alloying element, indicating that the phonon spectrum is readily altered by composition. Experiments taking advantage of these trends are found in laser cooling of glasses (amorphous), varying the composition to enhance the cooling performance [107, 96]. 5.6 Phonon-Coupling Enhanced Absorption and Current Density We now assess the increased current generation due to increased absorption using the following, ideal current density generation equation under illumination within the 82

111 limits of the hemispherical solid angle [66] j e = e c [1 ρ r (E ph )]{1 exp[ σ ph-e-p,a(e ph )L]} πi ph(e ph,t)de ph, (5.7) E ph >E e,g ω ph where e c is the electron charge, ρ r (E ph ) is the spectral reflectivity, σ ph-e-p,a(e ph )is the absorption coefficient due to phonon-coupled photon absorption, L is the optical length, and I ph (E ph,t) is the spectral intensity of the sun (approximated as blackbody). The optically thin limit gives 1 exp [ σ ph-e-p,a(e ph )L] σ ph,a (E ph )L [50, 23]. Now, to assess the current density generation improvement using enhanced phonon-coupled process using alloy element X, we use a scaled total current density generation equation j e = j e,n = j e,n (a-si 1 x X x ) j e,n (a-si) = E ph >E e,g σ ph-e-p,a(a-si 1 x X x )I ph (E ph,t)de ph E ph>e e,g σ ph-e-p,a(a-si)i ph (E ph,t)de ph, (5.8) where je,n is the normalized current for peak phonon energies, j e,n(a-si) is the current generated by acoustic and optical phonon peaks of a-si. We assume that ρ r and L remain the same over the entire solar spectrum. Then, this normalized current density depends only on the absorption coefficient and the spectral intensity. Now, using Eq. (6) and the validated assumption that the electronic structure near the bandgap does not change with alloyed X, all the quantum quantities related to electron-photon 83

112 interaction cancels and the normalized current becomes je = E ph >E e,g [( ϕ2 e-p,a mu 2 p [( ϕ2 e-p,a E ph>e e,g mu 2 p or ϕ 2 e-p,o mωp 2 or ϕ 2 e-p,o mωp 2 ) Dp(Ep)f p o(ep) E p ] a-si 1 x X x I ph (E ph,t)de ph ) Dp(Ep)f p o(ep) E p ] a-sii ph (E ph,t)de ph, (5.9) which is only dependent on the acoustic and optical phonon-coupling related quantities integrated over the solar spectrum. The electron-phonon couplings ϕ e-p,a and j e,n * Net Enhancement a-si 0.5 Sn 0.5 a-si 0.5 Ge 0.5 Optical E p (mev) a-si 0.5 Sn 0.5 a-si 0.5 Ge 0.5 Si-Si Optical 0.7 Acoustic Figure 5.5: Normalized phonon-coupled current density using the peak phonon energies of a-si 0.5 Ge 0.5 and a-si 0.5 Sn 0.5. The scaled total phonon-coupled current density is shown as net enhancement using broken lines. ϕ e-p,o are the electronic energy changes caused by the bond displacements of phonon modes and here we use the theory of harmonic oscillators. The stretching mode frequency is ω p =(Γ/m) 1/2, where Γ is the equivalent force constant found using the combinative rule and monatomic bond properties [46, 52, 42] and m is the reduced mass. Then the energy change by displacement Δ p =( /2mω p ) 1/2 becomes ϕ e p,a 0.5ΓΔ 2 p. For the optical phonons the energy change is defined over the bond length as ϕ e p,o 0.5ΓΔ2 p /r e,wherer e is equilibrium bond length (i.e., Si-Si, Si-X, 84

113 and X-X). Then Eq. (9) becomes je = E ph >E e,g [( 2 Γ 1/2 or 2 )D u 2 pm 3/2 m 1/2 reγ 2 1/2 p (E p )fp o (E p )] a-si 1 x X x I ph (E ph,t)de ph [( 2 Γ 1/2 E ph>e e,g or 2 )D u 2 p m3/2 m 1/2 re 2 p (E p )f o Γ1/2 p (E.(5.10) p)] a-sii ph (E ph,t)de ph The normalized, peak generated current density showing the phonon-coupling enhanced absorption, for a-si 0.5 Ge 0.5 and a-si 0.5 Sn 0.5 as a function of the phonon energy, are presented in Fig The peak phonon energies used are those in Fig The net current which is the summation over all peak phonon energies, is also shown (and indicated as net enhancement). The results show that the current generation is significantly increased due to the Si-X and X-X optical phonon-coupled absorption (high absorption rate at lower phonon energies and larger D p ). However, for the acoustic and Si-Si optical phonon-coupled absorption, there is a reduction in the current. a-si 0.5 Ge 0.5 shows a larger degradation for the acoustic modes, compared to that of a-si 0.5 Sn 0.5 (because of the small shift in the peak due to mixed phonon states). Degradation at the Si-Si optical mode is alleviated due to higher electron-phonon coupling. Although a-si 0.5 Ge 0.5 shows larger enhancement of the optical phonon-coupled absorption, when the reduced acoustic and Si-Si optical coupled absorptions are included, a-si 0.5 Sn 0.5 shows a higher net current j e enhancement of approximately 11%. The trend for the optical phonon-coupled current generation shows the combined effect of Γ 1/2 and f o p dependence in Eq. (10), favoring low energy optical phonon modes. The higher enhancement associated with a-si 0.5 Ge 0.5 in this regime is due to high ϕ e-p,o and D p. The current generation due to the acoustic modes show similar behavior, but the f o p dependence is reduced (due to Γ 1/2 Dependence). Therefore, 85

114 alloys with low energy optical phonons enhance the absorption and efficiency of solar energy conversion. 5.7 EffectsonCarrierTransport We now address the carrier transport properties which is related to the collection efficiency. Unlike c-si, transport kinetics in a-si are more complex and several theories have been proposed [105] and here we address them qualitatively and assess the dominant scattering mechanism. The total electron scattering rate, τ 1 e is expressed as a combination of various scattering mechanisms through Matthiessen rule as 1 = , (5.11) τ e τ i,im τ a τ n,im τ c-c τ p,a τ p,o where τ 1 i,im, τ a 1, τ 1 1 n,im,andτc-c are ionized impurity, alloy, neutral impurity, and carrier-carrier scattering respectively. τ 1 p,a 1 and τp,o are the acoustic and optical phonon scattering. The ionized impurity scattering rate shows temperature dependence of T 3/2 so, any increase ionized impurity due to Sn alloying is expected to be negligible at operating temperatures of SPV. Alloy scattering rate is highly dependent on cluster size and show T 1/2 dependence [100]. For a-si 1 x Sn x, the cluster size dependence is not relevant when uniform distribution of the alloy is assumed. Carriercarrier scattering rate is also negligible for carrier concentrations of less than cm 3 and the effects decreases as film thickness is reduced [105]. Similar to phonon-coupled absorption, acoustic phonon scattering is expected to be lower for Ge and Sn alloyed 86

115 a-si, however, higher optical phonon scattering is expected due to enhancement of Si- X and X-X oscillation modes and increased defect density is predicted due to higher defect concentration. This may result in degradation of the electron mobility of the cells. Nevertheless, in a-si, further hydrogen passivation can be used to significantly reduce the defect density (as much as 4 orders of magnitude), and the presence of the hydrogen atoms may shift the phonon peaks to higher frequencies thus reducing the optical phonon participation during collection. Still phonon scattering can be alleviated by predicted increased absorption that allows for thinner films which in turn reduce the amount of materials used. 5.8 Summary We have expanded the second-order interaction matrix using the weak-phonon coupled state approximation and the isolated electron representation (noting the covalent nature and the disorder phase of the amorphous semiconductors). We predict that alloying with Ge or Sn increases phonon-coupled photon absorption and enhances the current generation near the optical bandedge for a-si. The D e calculations show that the electronic bandgap is not significantly altered for up to 50% alloying. However, due to the lower force constant and higher mass mismatch, the phonon spectrum shows significant red-shifting of peaks resulting from Si-Sn (or Ge) and Sn-Sn (or Ge-Ge) vibrations. These shifted optical-phonon energies result in higher phonon occupation, thus enhancing the phonon-coupled absorption. Therefore, this soft-bond alloying of a-si results in net phonon-coupling enhanced absorption suit- 87

116 able for solar photovoltaics. Synthesis by nonequilibrium processes such as ultra-fast laser melting of laminates, makes this finding promising. Addition of Sn may also help with long-term stabilization of the amorphous Si SPV films, which has been a concern [104]. 88

117 Chapter 6 Phonon tuned thermionic cooling in light emitting diodes 6.1 Introduction LED lighting currently has up to 35% plug-in-efficiency (3-fold larger than incandescent lighting) [91], however its concentrated heat loss requires elaborate heat sinks [2]. In incandescent lamp the heat loss is due to the nonvisibile photon emission, and this radiation is spread over the large surface area of the lamp [49]. However, in LEDs the loss is through phonon emission (by nonradiative decay and other processes) which is transported through the solid structure (chip and its substrate) [41]. Moreover, the current LED thermal-damage threshold is about 100 o C, thus making its thermal management challenging [91]. Heterogenous thermionic cooling is ballistic transport of nonequilibrium electrons over a potential barrier created by a heterogenous layer with the energy supplied by absorption of phonon [67, 68, 95, 69]. Various 89

118 theoretical models and predictions have shown that its coefficient of performance is much larger than that of the thermoelectric cooling [65, 37, 95, 116, 36]. Such models focus mainly on the electron transport across potential barriers created by the work function differences and current density j e predicted by the Richardson equation which does not address the electron-phonon kinetics at the atomic level (done for various photonic systems [52, 55, 56]). In this letter, we consider recycling a part of the emitted phonons through the use of a barrier transition cooling (BTC) layer (or BTCL) at the LED chip level. We address various energy conversion and transport processes and rates (including multiphonon absorption) in BTCL-LED, identify the bottleneck process. Then, we tune the BTCL barrier height for maxium energy conversion efficiency. 6.2 Integrated LED and BTC Layers BTCL, not requiring a separate circuit, suits well in a series arrangement in the LED chip, as shown in Fig. 6.1(a). We consider a heterogenous GaN/InGaN based LED chip with an adjacent metal/algaas/gaas/metal BTCL. External potential drives the electrons over the potential barrier Δϕ b (by absorption of phonons) created by a heterogeneous layer. Then the electrons are injected into the LED chip for radiative transition and nonradiative decays. The phonon generated by the nonradiative transition in the LED chip is transferred by various phonon transport mechanism such as optical/acoustic phonon transmission and phonon up/down conversion at the interface. The conceptual energy diagram and energy conversion is shown in Fig. 90

119 InGaN/GaN Quantum-Well LED AlGaAs/GaAs Potential Barrier cooling Layer (BTCL) - LED Layers (a) BTC Layers n-type - p-type Multiphonon Absorption E e ħω ph ħω p,o φ b - - Photon Emission x Photon Energy. S e-ph Radiative Emission Recombination Phonon (Heat) Energy Q e-p = Nonradiative Emission Nonradiative Emission. S e-p.. S e-p + S p-e,c Phonon Up/Down Conversion Potential Barrier Phonon Transmission (b) LED. S p-e,c BTCL Phonon Recycling (c) Multiphonon Absorption Phonon Transmission Figure 6.1: (a) Integrated GaN/InGaN LED layer and metal/algaas/gaas/metal BTCL. (b) The conduction electron energy diagram and related carrier transitions. (c) The energy flow diagram of such BTCL integrated LED chip. - J e φ e 6.1(b) and the energy flow diagram of input power J e Δϕ e (product of current and electric potential across LED), barrier transition cooling Ṡp-e,C, nonradiative emission Ṡ e-p, radiative emission Ṡe-ph, and net phonon produced Q e-p are shown in Fig. 6.1(c). By recycling phonons emitted by nonradiative emission, larger electrical energy is not necessary to drive the cooling system, as long as the phonon absorption rate is not the bottleneck process. We now analyze these related interactions to identify the bottleneck process under appropriate assumptions. 91

120 - -. γ e-ph Recombination. γ e-p,srh -. γ e-p,auger ħω ph ħω p,o - ħω p,o Radiative Shockley-Read-Hall Auger (a).. γ p-p γ tr, O ħω p,o ħω p,a Optical Phonon Transmission ħω p,o. γ tr, A Acoustic-Optical Upconversion ħω p,a (c) Acoustic Phonon Transmission - Multiphonon ħω p,o ħω Absorption p,o ħω p,a Interfacial Optical-Acoustic Downconversion (b) Figure 6.2: Detailed carrier transitions are shown for (a) radiative and nonradiative transitions in LED, (b) phonon transmission in LED/BTCL boundary, and (c) phonon upconversion (d) multiphonon absorption in BTCL. 6.3 Carrier kinetics in LED and BTCL (d) - The relevant carrier transitions in LED and BTCL are shown in Figs. 6.2(a) to (d). Figure 6.2(a) shows the radiative transition and the nonradiative transitions in LEDs producing photons and phonons respectively. Figure 6.2(b) shows the phonon transmission processes between LED and BTCL, i.e., the optical and acoustic phonon transmissions and the optical phonon decay. Figure 6.2(c) shows the phonon-phonon upconversion and Fig. 6.2(d) electron-phonon transition processes in BTCL. We only consider the optical phonon absorption, because this is found to be an order of magnitude larger than acoustic phonon absorption rate [100] (which is extremely low multiphonon absorption due to relatively low energies). The radiative transition rate γ ph in LED is linearly dependent on the carrier concentration and is simply given by 92

121 [75] γ e-p = A e-phn e, (6.1) where A e-ph is the radiative decay constant and n e is the carrier concentration. Note that we use units of s 1 for transition rates. The dominant nonradiative decays are the Shockley-Read-Hall (SHR) and Auger recombinations [97]. The SHR recombination is due direct multiphonon decay at the defect sites, while the Auger recombination is due to transfer of energy to a conduction electron followed by a multiphonon decay. These nonradiative recombinations are commonly presented with simple expressions (e.g., for GaN [75]). The total nonradiative transition rate γ e-p depend on the carrier concentration (SHR is given as a constant coefficient, while the Auger has a quadratic dependence) as γ e-p = A SHR + A Auger n 2 e, (6.2) where A SHR and A Auger are the Shockly-Read-Hall and Auger recombination coefficients respectively. The phonon frequencies generated by these nonradiative transitions depend on the highest available phonon frequency in the lattice and decay to the lower frequencies via phonon-phonon interaction processes [57]. The phonon generated by the nonradiative transition in LED is transmitted to the BTCL by various phonon-phonon transitions and transmission processes. Significant optical phonon population is available in GaN, due to the large optical phonon life- 93

122 times {when cation and anion mass ratio m C /m A > 4, LO (longitudinal optical) 2TA (transverse acoustic) or 2LA (longitudinal acoustic) decay channels are unavailable and the dominate decay channel is found to be LO LO+LA(TA) [109]}. These optical phonon modes can be directly transmitted to the adjacent BTCL layer despite low optical phonon velocities and decay rapidly as soon as they enter AlGaAs/GaAs BTCL. Acoustic phonon transmission rate can be estimated by dividing transmission length (L) by acoustic velocity and the transmission coefficient is predicted by diffusion mismatch model (DMM) using the difference in the mode velocities across the interface [106, 50]. Considering transverse acoustic waves have velocities of 3960 m/s [114] and 2800 m/s [112] respectively in zinc-blend GaN and GaAs in [111] direction (we take a conservative approach and take the smaller values). then the inverse of the acoustic mode propagation time through GaN and GaAs can be estimated to be of the order of s 1 up to 1 μm in combined length which is extremely fast. Although we have transmission coefficient predicted by the DMM model to be approximately 0.67, availability of the acoustic modes in BTCL layer not expected to limit the kinetics of the system. Acoustic modes available in the BTCL can be upconverted to form higher energy phonons to be used in the multiphonon absorption process across the potential barrier and this rate can be calculated by three phonon interaction model given as [103] γ p-p = M p-p 2 R ω 8πρ p,aω 2 p,of 3 o 3 p (ω p,a )fp o (ω p,a )[fp o (ω p,o )+1], u 7 p,a u2 p,o M p-p 2 =4ρ 2 γ 2 Gu 2 p,au 2 p,o, R = 2 3 1/2 Γ AA Γ CC Γ AA +Γ CC, (6.3) 94

123 where ρ is the density, u p,a and u p,o are the acoustic and optical phonon velocities respectively, ω p,a and ω p,o are the acoustic and optical phonon frequencies respectively, fp o is the phonon distribution function, γ G is the Grüneisen parameter, and Γ AA and Γ CC are the force constants which can be estimated by material metrics [46]. We have used the Debye-Einstiein phonon density of states model and assumed that the two acoustic phonons are identical. We formulate the electron transition over the potential barrier Δϕ b as an optical phonon absorption process in the presence of a bias voltage. The relation for single optical phonon absorption rate in semiconductors is derived using the Fermi golden rule and is [100] γ p-e = ϕ 2 e-p,o m3/2 e,e Ee 1/2 fp o(e p), (6.4) 2 1/2 π 3 ρω p,o where ϕ e-p,o is the optical phonon deformation potential, m e,e is the reduced electron mass, and E e is the electron energy. We have used the parabolic band assumption for the electronic density of states near the bandedge. Table 6.1: Various transition rates (s 1 ) for LED and BTCL using properties found in ref. [97, 112, 46, 102]. n e = cm 3 and T = 323 K are used throughout. γ a e-ph γ b e-p,shr γ c e-p,auger γ d p-p γ e p-e a. A e-ph = cm 3 s 1 γ G =0.8 b. A SHR = s 1 R = c. A Auger = cm 6 -s 1 e. ϕ e-p,o =9.0 ev/å d. ρ = 5317 kg/m 3 m e,e =0.0819m e u p,a = 2800 m/s E e =0.05 ev ω p,o =8.6 Trad/s ω p,o =0.035 ev ω p,a =4.3 Trad/s 95

124 From Table 6.1, the single-phonon absorption rate is much faster than other processes. This suggests multiphonon absorption can be accommodated in the TCL, i.e., we can use a barrier height exceeding the available single phonon energy. We use the commonly used exponential dependence (of the phonon distribution function) for the multiphonon transition processes [62, 85] given as γ p-e,c = γ p-ef onp 1 p, (6.5) where N p is the number of absorbed phonons. The variation of the multiphonon transition rate with respect to N p is shown in Fig. 6.3 along with the LED total transition rate γ LED. Form the results, the multiphonon transition rate becomes the bottleneck process at N p =11for ω p,o =0.035 ev. So, when the barrier height exceeds approximately 0.4 ev, the electron transition over the potential barrier becomes the limiting process, thus reducing the electron injection into the LED layer. This will then significantly reduce the LED performance and is not desirable. 6.4 Phonon recycling in LED We now calculate the performance of LED with a BTCL by using the transition rates under constant current assumption. When BTCL is added to the LED, the barrier acts as an extra resistance and has adverse effect on the current. This adverse effect will reduce the total radiative transition rate i.e., γ e-ph e (Ee,c Np ωp)/k BT γ LED where E e,c is the electron energy over the conduction bandedge at the bottom of the 96

125 10 12 γ i-j (s -1 ) γ p-p. γ LED =. γ e-ph +. γ p-e,c.. γ e-p,srh + γ e-p,au Barrier Bottleneck Regime N p Figure 6.3: Variation of multiphonon transition rate with respect to number of phonons absorbed N p. The total LED transition rate γ LED = γ e-ph+ γ e-p,srh+ γ e-p,auger and the acoustic phonon upconversion rates are also shown. barrier potential and results in additional nonradiative emission i.e., γ e-p+e (Ee,c Np ωp)/k BT γ LED atthebarrier. HerewetakeE e,c as 0.45 ev to ensure a constant current up to the barrier bottleneck (i.e., N p = 11). The energy conversion rate is given by Ṡ i-j = ω i,j n e V γ i-j where V is the volume. We define the overall quantum efficiency as η e-p = Ṡe-ph/(Ṡe-ph +Ṡe-p +Ṡp-e,C), which is the ratio of the photon emission rate to the total input power (including the power required for cooling). The calculated results are shown in Fig. 6.4 for V =10 12 m 3 the sample LED produces 1.5 and 1.4 W of phonon and photon energy, respectively. When the BTCL of 100 nm is added (shown with N p > 0), the adverse effect on the current increases the phonon emission and decreases the photon emission as the barrier height is increased. Note that BTCL thickness should be large enough to avoid any quantum well effect [77]. The barrier transition cooling rate Ṡp-e,C increases with N p due to higher energy absorbed per transition, while the kinetics is limited by LED total transition (i.e., γ e-ph + γ e-p,srh + γ e-p,auger). Above N p = 11, performance is limited by the barrier 97

126 Increase in Phonon. Emission S e-p due to Barrier 1.5. Decrease in Photon. S e-ph Emission due to Barrier 1.0 η e-ph Si-j (W) 0.5. S p-e,c Barrier Bottleneck Regime η e-ph Peak N p Efficiency Figure 6.4: Variation of the energy conversion rate with respect to the number of phonons required to over come the potential barrier. The overall quantum efficiency is also shown. bottleneck in Fig We find that η e-p peaks at N p = 8 which corresponds to a barrier height of ev which is a 17% improvement over the LED with no such BTCL. This recycles up to 30% of heat (phonon) generated by the nonradiative decay reducing the required heat removal load in operating LED (then requiring a smaller heat sink). Note that the above calculations are for the cooling rate Ṡp-e,C based on the equilibrium phonon population at T = 323 K. However, when phonon transport path short (as is here) and for materials (such as GaN and InN) containing significant mass miss match (between cations and anions), the multiphonon transition rate is influenced by nonequilibrium optical phonons due to long lifetime of optical phonon modes. Such nonequilibrium contribution also encountered and discussed in other photonic systems [55], but is rather difficult to quantify the LED system. 98

127 6.5 Summary In summary, the carrier kinetics of the BTCL-integrated LED are discussed and evaluated for GaN/InGaN based LED and AlGaAS/GaAs based barrier system, showing the barrier height can be up to ev without significantly altering the LED performance. 99

128 Chapter 7 Summary and Future work 7.1 Contributions This study examines the fundamentals pertaining to the atomic-level carrier kinetics and the structural metrics of phonon recycling in photonics. The significance of this work is highlighted by the initiation and attempt to shift the focus on the role of thermal energy in energy conversion to the phonons. The work starts from an interesting and intriguing physical phenomena of the anti-stokes laser cooling of solids and expands the carriers (phonon, electron and photon) interactions kinetics to other photonic systems such as ion-doped lasers, amorphous silicon solar photovoltaics, and potential barrier integrated light emitting diodes. Common ground is established by looking at the influence of the phonons (equilibrium and nonequilibrium) in energy conversion systems, establishing the materials metrics that can be used for optimal phonon-assisted transitions, and ultimately predicting the performance (e.g., efficiency) improvements for a few photonic systems. In the process of 100

129 doing so, atomic-level kinetics described by the FGR is discussed and the relevant atomic parameters are examined (e.g., carriers couplings, densities of states, equilibrium and nonequilibrium populations). Simple harmonic oscillator based analysis is used along with ab-initio calculations to predict the optimized phonon frequencies. Electronic bandstructure and density of states are also examined using ab-initio calculations to predict phonon-electron couplings and the effect of alloying in energy conversion materials. The significant contributions of this work are summarized below: Development of a second-order interaction matrix using optical phonon-assisted photon transition The optical-phonon deformation potential interaction is used to derive the second-order phonon-assisted photon transition using the FGR. We complement the existing acoustic-phonon based interaction matrix [28, 87] by realizing the high electric field and energy associated with the optical phonons are a more likely candidates for the optical interaction. Optical phonon interaction is through the derivative of the electronic energy with respect to the displacement of the nearest neighboring ligand stretching mode and we optimize this through material selection [52]. Application of the material metrics to prediction of the phonon frequency and the electron-phonon couplings in disordered structures. In disordered atomic structures such as laser host glasses and amorphous semiconductors, the phonon frequency is found to be highly dependent on the composition. The dominant vibration frequency is predicted using the homonuclear force constant data and 101

130 the harmonic oscillator assumption [46]. The peak and the cut-off phonon frequency determines the height of the density of states (when the Debye-Gaussian DOS model is used) and determines the availability of the phonons when combined with the Bose-Einstein distribution function. This model is similar to the Einstein phonon DOS model where a single non-interacting harmonic oscillator is assumed for the optical phonon modes, but broadening is taken into account for a more realistic description of the phonon spectrum. Also, the electron-phonon coupling is calculated using the average displacement and the electronic energy change predicted by the harmonic oscillation of pair atoms [52]. Development of a predictive method for phonon recycling in photonic systems such as laser cooling of solids, ion-doped lasers, a-si solar photovoltaic, and potential-barrier intergraded LEDs. Based on the optical-phonon assisted photon absorption model and the phonon frequencies and couplings predicted using the harmonic oscillator model, the phonon-assisted transition kinetics of various photonic systems are calculated. In laser cooling of solids, Li, Al, and Na bond with F are found to be optimal for improved cooling rate (up to 200 % at room temperature, compared to currently used blends). However, at lower temperatures, low-phonon materials such as Cs, Rb, and K bond with F are optimal. So, blending is suggested to achieve optimal cooling rate over a wide range of temperatures. The lowest phonon frequency of the blend is expected to determine the temperature limit of the laser cooling of solids. In ion-doped 102

131 lasers, high phonon cut-off energy material are proposed which use the excess pump photons and drive the anti-stokes cycle. This cycle can be used to remove the nonequilibrium phonons emitted during the laser cycle [55] and we predict recycling up to 35 % of emitted phonons. Although adding the recycling layers adds complexity and requires multiple mirrors to redirect the pumping photons, the reduced cooling load is expected to significantly increase the portability of the laser unit which is desired for some applications. In a-si solar photovoltaic, phonons influence approximately 75 % of the total absorption near the bandedge. The low phonon energy Sn alloying enhances the phonon-assisted transition and shifts the absorption coefficient towards the lower photon energy. This may increase the current generation rate up to 11 % when the possible negative effects on the electron mobility are neglected. We also use, ab-initio calculations to predict changes to the electronic band structure and the phonon spectrum peak shifts [56]. The alloying effect on the mobility can be reduced by further passivation of hydrogen which lowers the defect density by up to 4 orders of magnitude [92, 105]. In the potential-barrier integrated LED, the phonons emitted in the nonradiative processes are used to promote the conduction electrons over a heterogenous potential barrier. This reduces the required external cooling by 30 % which is significant in the high-light intensity LEDs. Verification of material metrics in predicting the electron-phonon coupling in isolated ions using ab-initio electronic bandstructure calculation. Ab-inito calculations of the isolated clusters can be used to calculate the electron-phonon 103

132 coupling constant based on the harmonic oscillator model. Unlike the deformation potential calculation in the semiconductors which deals with the entire electronic bands, only the optically active bands (f orbitals for rare-earth ions) are used in the coupling calculation. In our simulations, atoms are explicitly displaced to simulate the oscillation of ligands in the phonon propagation. Then the initial and final electronic energies are compared to the estimates of the electronic energy change for a given displacement [54]. 7.2 Proposed Future Work There are three extensions of this work which should be pursued: Full ab-inito Molecular dynamics (MD) simulations should be carried out to complement the predictions made by the material metrics and to investigate any temperature induced structural instabilities. To accurately model and predict disordered structures, number of atoms of the order of thousands are required and MD calculations should be done in conjunction with the zero-temperature ab-inito calculation. However, this is very challenging with the current hardware and software capabilities. Nevertheless, as the computational resources become more abundant, the full ab-initio MD calculations should become routine in examining the properties of new materials. Include detailed charge carrier transport for the compound semiconductors. This study mainly focuses on realization of the phonon recycling in photonic systems. However, the electron transport is an important aspect, especially in 104

133 the compound semiconductors when alloying is introduced for improved phonon participation. Therefore, further studies should be done on the electron mobility which influences the electron collection efficiency in the photovoltaic and the electron injection into the quantum well in the heterogenous LED systems. Experimental investigation should be carried out on the phonon recycling examples to identify technological and economic challenges. This theoretical study uses simplified theories and sound assumptions to predict the benefit of phonon recycling in example photonic systems. However, there may be numerous technical and economic issues to be considered before the actual realization of such proposed systems. Therefore extensive experimental investigations are recommended. 7.3 Outlook Although this study predicts modest efficiency improvements using the concept of phonon recycling (these modest improvements are in part bound by the second law of thermodynamics), it shows that improvements are possible by carefully engineering atomic structures influencing the phonons in the system. This study may be a step towards the atomic-level -based approach research as evolutionary extension of the continuum-based research in mechanical engineering. Such future scientist-engineers are essential in realizing new and better materials for improved energy conversion. 105

134 Appendix A Ab-initio calculations of f -orbital electron phonon interaction in laser cooling A.1 Introduction In laser cooling of solid materials, many experimental and theoretical investigations have been carried out in the past decade to enhance the cooling performance. Experiments with rare-earth doped crystals have been successful in cooling to 208 K, from the room temperature [34, 27, 107]. Theoretical investigations focus on individual transition rates under assumed quantum models and have made significant progresses [87, 40, 24, 53, 6]. Still, very few experimental and theoretical studies have attempted to characterize the fundamental material properties involved in the interaction between the three carriers (photon, electron, and phonon) in laser cooling 106

135 of solids [?]. The challenge in the study of interaction carriers is that the existing theoretical investigations have been exclusively for semiconductors with band treatment of conduction electrons and they do not fully describe the carrier interaction for isolated ions. Especially the electron optical phonon interaction is described by a normalization constant times deformation potential over the lattice constant. These values range up to tens of ev/å for various semiconductor elements [82]. However, in isolated rare-earth atoms, these definitions need to be modified and recent study shows that for rare-earth ions with insulator host, electron-phonon interaction potential was found to be significantly lower [53]. In this study, we review the analytical harmonic approximation of the electron phonon interaction potential made in Ref. [53], and compare it with ab-initio calculations using isolated rare-earth host cluster and point out the validity of both methods and their limitations. In anti-stokes laser cooling of solids, the second-order process, where electron interacts with photon and phonon, for the cooling rate and the transition rate (using Fermi golden rule) is [50, 83] γ ph-e-p = 2π M ph-e-p 2 δ D (E e,f E e,i ω p,o ), (A.1) where M ph-e-p is the interaction matrix element of the photon, electron and phonon, δ D is the Dirac delta function, E e,f and E e,i are the final and initial electronic energies, and ω p,o is the phonon frequency involved in the process. Here M ph-e-p is expanded 107

136 using second-order perturbation theory and is given as M ph-e-p = m m f H int m m H int i E e,i E e,m ψ f,f ph,f p H ph-e ψ m,f ph +1,f p ψ m,f ph +1,f p H e-p ψ i,f ph +1,f p +1, E i (E m ω p ) (A.2) where subscript m designates intermediate states subsequent to phonon absorption but prior to photon absorption. With the interaction Hamiltonian for photon-electron and optical phonon-electron, the transition rate becomes γ ph-e-p = π 2m AC (s ph,i μ ph-e) 2 ɛ o ϕ 2 e-p,o D p (E p )f o p (E p ) E 3 p ω ph,i D ph, (A.3) V where m AC is the reduced mass of the oscillating pair, s ph,i is photon polarization vector, μ ph-e is transition dipole moment vector, ϕ e-p,o is electron-phonon interaction potential when optical phonon interaction is assumed, D p (E p ) is density of states of phonons having energy E p, f p is Bose-Einstein distribution function, ω ph,i is incoming photon frequency, and D ph is the photon density of states which integrates to unity, since the incoming laser light is assumed to be monochromatic (and one-photon interaction is assumed). In rare-earth ions, the f electron shell is optically active and when placed in a crystal field environment, the electron energy levels split. Particularly for Yb 3+, due to the large spin-orbit coupling, 2 F 5/2 is approximately 10,000 cm 1 ( 1.2 ev) above ground state 2 F 7/2. The crystal field interaction further splits the excited and 108

137 the ground state electron energy levels, depending on the ligand environment. This energy splitting is within a few k B T (comparable to phonon energy) and makes the anti-stokes cooling possible. The electronic energy levels of the shell move up and down at finite temperature, because of the continuous excitation and de-excitation of phonon. Under the adiabatic approximation, the electron-phonon interaction is represented by 2 Ψ(Q) Φ Q (r) m e Q Q =H e-pψ(q)φ Q (r), (A.4) wherem e is the electron mass, Ψ(Q) is the lattice wave function which depends on the normal coordinate Q, Φ Q (r) is electronic wave function which depends on r which is the coordinate of the ion, and H e-p is interaction Hamiltonian. We use an simple model developed in [117] to model the interaction. As mentioned above, the ion continuously moves due to lattice vibration and this movement changes the potential in the lattice. This change in the potential is experienced as a perturbation for the electron and scatters it to new electronic states and the energy change can be written as ΔE e-p i Δ p,i ϕ e-p,o Δ p, (A.5) where ΔE e-p is the energy change of the electron due to phonon (lattice vibration), Δ p is the displacement of the atom, and ϕ e-p,o is the potential change due to the perturbation which is called the electron-phonon coupling. 109

138 Ligands Cation Anion Harmonic Potential Interaction Model Ion Yb Г IA 1 E e-p = 2 Г IAC Г AC Γ IAC Δ 2 p Δ ( ħ ) p= 2m ω p Q 1 AC 1/2 Band Energy at Nonequilibrium, E e Band Energy at Equilibrium, E o e Partial Band Structure of Yb Γ 2ϖ λ X Figure A.1: A conceptual rendering of the ion-ligand complex under atomic displacement due to phonon excitation. The energy change of the electron using harmonic potential can be represented by the equivalent potential energy change of the complex. Ab-initio calculations show the change in the partial electronic bandstructure of the ion at the Γ point due to atomic displacement. The equilibrium and displaced bandstructures are indicated by solid and broken lines, respectively, and the corresponding electronic energies are marked on y axis (E o e is for the equilibrium and E e is for the displaced). In Yb 3+ the electronic energy level oscillations by phonon takes place within the energy level splitting due to crystal field interaction at ground or excited energy levels. In laser cooling of solid using rare-earth ions, optically active 4f electronic shell (partial electronic energy change) are significant. The energy change is found by comparing the electronic bandstructure of the equilibrium cluster with nonequilibrium (displaced). Here we do this for ground state 4f orbital of Yb at the Γ point. Figure A.1 shows a conceptual (model) rendering of the electronic energy change of ionligand complex under ligand displacement. The partial electronic bandstructure of Yb in host (Yb electron interaction with potential created by the neighboring ligands) 110

139 is also shown. In Ref. [53], a simplified harmonic potential is used between atoms and estimates the electron-phonon interaction potential for rare-earth materials are made. It uses experiential results for force constants for mononuclear pair atoms and the combinative rule to extrapolate the force constants between different atoms. This gives Γ IAC =[(Γ AA Γ CC ) 1/2 Γ II ] 1/2, (A.6) where Γ AA, Γ CC,andΓ II are the force constants of the anion-anion, cation-cation, and ion-ion respectively. The atomic displacement is found by using the stretching-mode amplitude of the ligand pair (cation and anion) by a quantum of the most probable phonon energy and this is given by Δ p =( ) 1/2, 2m AC ω p (A.7) where Δ p is the amplitude of displacement, m AC is the reduced mass of the anion and cation and ω p is the angular frequency of the phonon. Then assuming that the optically active electron of the ion is in an infinite square-well, we estimate the energy change of the electron using appropriate boundary conditions. The potential change between the atoms under ligand displacement and the corresponding interaction po- 111

140 tential are given by (also shown in Figure A.1) ΔE e-p = 1 2 Γ IACΔ 2 p, ϕ e-p,o ΔE e-p Q 1, (A.8) where Q 1 is the distance between the ion and closest cation. Despite its simplicity, the harmonic potential estimate of the electron-phonon interaction gives reasonable estimates and is used in laser cooling of solids [53]. To further verify the validity of this model, below we perform ab-initio calculations using Yb ion in various hosts to determine the electron-phonon interaction potential. A.2 Calculation The calculation of the electronic energy change due to ligand displacement is performed using WIEN2K which is based on the full potential (linearized) augmented plane-wave (L)APW and local orbital method. (L)APW method is considered as one among the most accurate schemes for bandstructure calculations. A 2 x 2 x 2 supercell is first created then the ligands are removed until irreducible unit cell is created. The supercell and the irreducible unit cell is shown in Figure A.2. Then the central cation atom is replaced by and Yb ion to simulate doping. Then using WIEN2K, the whole structure is relaxed until, equilibrium is reached. Table 1. shows the structure, symmetry, Q o 1 of bulk, Q 1 of relaxed structure and corresponding ligand displacements for the compounds used in the ab-initio calculations. At this equilibrium geometry, the partial bandstructure of the Yb atom is calculated and compared 112

141 with an displaced structure, where ligand displacement is simulated by displacement of the immediate neighboring atoms. The bandstructure calculation is carried out until energy convergence of Ry is reached. 113

142 (a) Cd F Yb F Cd (b) Figure A.2: (a) shows the 2 x 2 x 2 supercell created for doping of Yb atom in CdF 2 host. (b) shows the Yb: CdF 2 complex constructed by removing outer ligands which is not expected to influence 4f electron shell of Yb. The simulated ligand movement is also shown by arrows pointing away from the central ion. 114