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Doctorate Slides, title: "Modeling charge transport-induced phenomena in colloidal double quantum dots and developing new probes for inter-dot interactions"

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- 1. Modeling charge transport induced phenomena in colloidal double quantum dots and developing new probes for inter-dot interactions Roni Pozner, Efrat Lifshitz and Uri Peskin
- 2. Typical Atom Quantum Dot What Are Quantum Dots? Size Alloying: PbS…..PbSexS1-x…..PbSe “Artificial Atom” Semiconductor Bulk Discrete Energy Levels Energy Bands Quantum Size Effect
- 3. How a random experiment led us to a theoretical work which ended with a novel idea for a memory device Scanning Tunneling Microscopy
- 4. Solar Cells Silicon Solar Cell 1st Generation Colloidal QDs Solar Cell 3rd Generation Quantum Dot Arrays Quantum Dot Gratzel Cells
- 5. Quantum Dots Assembly How do various transport properties of quantum dots with different sizes, structures, coupling and compositions vary by shifting from a Single Quantum Dot to a Quantum Dots Array? Lithography QDs Electron Coupling Mechanical Coupling Electron Coupling Discrete energy levels Electron-Phonon coupling Dark States, Coloumb Blockade and more Disorder effects Macroscopic transport models The focus of today’s talk: Colloidal QDs where is the experiment picture? The Ligands: ➡ Preventing agglomeration between QDs in the solution ➡ Passivating the QDs surface ➡ Controlling interaction strength between QDs Ligands
- 6. x Solving the generalized eigenvalue problem: Single electron Hamiltonian matrix Overlap matrix Based on STM experiments: ACS NANO Vol 3. No. 2. Daniël Vanmaekelbergh, Karin Overgaag Coulomb Blockade Nature, Vol. 400, 1999, Uri Banin, Oded Milo Double Quantum Dot Model Overlap between the two localized orbitals Inter-dot distance Electronic coupling 3D localized non-orthogonal Gaussian orbitals Quantum Dot Ligands The orthonormal DQD orbitals Single particle hamiltonian terms two particles terms Small q: Strong dependence on S,t Large q: Weak dependence on S,t
- 7. I erential Resistance Probe for Inter-dot Interactions in a Double Quan R. Pozner, E. Lifshitz and U. Peskin, J. Phys. Chem. Lett, 6, 1521−1528 (2015) Negative Resistance
- 8. = = Surface STM Tip Reservoirs of non-interacting electrons Single electron hopping between localized orbitals & electrode states Full Hamiltonian: Projection of DQD Hamiltonian onto the orthonormal DQD orbitals: Weak Weak Weak STM Tip-DQD-Surface Rigid Coupling term depends on the inter-dot distance
- 9. Model Calculation Steady State Response Transition Rates States Populations Liouville-von Neumann Equation under the Quantum Master Equation Redfield approach - electrons number in the (mn)th DQD eigenstate
- 10. Negative Differential Resistance Negative Resistance: Increase in voltage results in a decrease in the current
- 11. E1-E2 (eV) Current Rigid STM Tip-DQD-Surface
- 12. Various Reasons for NDR in Double Quantum Dot • Lead-QD bias coupling dependence A. Nauen, F. Hohls, N. Maire, K. Pierz, and R. J. Haug, Phys. Rev. B 70, 033305 (2004) • Level renormalization with respect to coupling to the leads B. Wunsch, M. Braun, J. König, and D. Pfannkuche, Phys. Rev. B 72, 205319 (2005) • Ratio between lead-DQD and inter-dot coupling I. Djuric, B. Dong, and H. L. Cui, J. Appl. Phys. 99, 063710 (2005) • Series and parallel configurations of asymmetric DQD Physical Review B 70, 085301 (2004) • Stronger coupling to an excited state then to a ground state J. Weis, R. J. Haug, K. v. Klitzing, and K. Ploog, Phys. Rev. Lett. 71, 4019 (1993) NDR Theories What is the NDR mechanism in our STM-DQD model?
- 13. The non-zero transition rates (at zero T) are: Rates and Probability equations: Model Analysis
- 14. Symmetric Case T DQD S Transition Rates The NDR reveals the molecule-like nature of the DQD due to destructive interference in the coherent coupling to the shared surface. : :
- 15. A Probe For Inter-Dot Interactions
- 16. Orthodox Theory Coulomb Blockade = But what about: • Tunnelling • Interferences • Co-tunnelling • Coherences/Noncoherences • Adiabatic/Nonadiabatic • Level Shifting • Level Broadening • Kondo effect… ? “Negative Resistance stands in contrast to a simple ohmic resistor” Double Quantum Dots
- 17. Charge Transport Induced Recoil and Dissociation in Double Quantum Dots R. Pozner, E. Lifshitz and U. Peskin, Nano Lett., 14, 6244−6249 (2014) I II Dissociation Negative Resistance
- 18. The inter-dot interaction potential V(q) should reflect universal: A. Short-range repulsion B. Long-range attraction x “Weak” “Strong” “Intermediate” STM Tip-DQD-Surface with Mechanical Coupling Surface STM Tip Full Hamiltonian: Projection of DQD Hamiltonian onto the orthonormal DQD orbitals: Weak Weak Weak Bound Floating
- 19. The states of the mechanical coupling are quasi- continuous, and therefore can be treated classically. Model Calculation Steady State Response Why do we treat the mechanical coupling classically? Transition Rates States Populations QDs are heavy in comparison with atoms Back on the envelope calculation leads to: Liouville-von Neumann Equation under the Quantum Master Equation Redfield approach Electro-mechanical Classical dynamical equations Time-Dependent Momentum and Position operators + + ★ We find the steady-state distance (q) between the dots for every bias potential ★ Finally, we calculate the steady state current - electrons number in the (mn)th DQD eigenstate
- 20. Strong Mechanical Coupling Current & Mechanical Displacement
- 21. “Intermediate” “Strong” Strong Mechanical Coupling Weak Mechanical Coupling Surface Recoil Dissociation Current & Mechanical Displacement Transitions at: After dissociation the current is measured through a single quantum dot “Weak”
- 22. Nonadiabatic force terms can be neglected Density-matrix is diagonal at steady- state Assuming coherences can be neglected Adiabatic Force Terms Nonadiabatic Force Terms Born-Oppenheimer Picture Har̈tle,R.;Millis,A. J.Phys.Rev.B 2013, 88, 235426.
- 23. Born-Oppenheimer Picture Strong Coupling Potential Energy Curves Weak Coupling doubly occupied state “antibonding" orbital state “bonding” orbital state empty state leads to dissociation The net force is a weighted average over the different electronics states
- 24. Dissociation Recoil Born-Oppenheimer Picture Population of the Anti-Bonding state
- 25. Triple Quantum Dots NEMS Memory Device Negative Resistance I II Dissociation III NEMS Memory Device Negative Resistance R. Pozner, E. Lifshitz and U. Peskin, Appl. Phys. Lett. 107, 113109 (2015)
- 26. STM Tip L Insulating layer Surface R Insulating layer Surface -5 0 5 ‘OFF’ State ‘ON’ State TQD NEMS Memory Device Nanoelectromechni cal
- 27. L Insulating layer Surface R Insulating layer Surface TQD NEMS Memory Device Probing Current Inducing Motion ״ 0 ״ ״ 1 ״ bit DQD SQD Writing Region Reading Region
- 28. Asymmetry Interdot Distance Performance Analysis
- 29. T DQD S Transition Rates : : Performance Analysis “Strong”
- 30. NEMS Floating Beam Solid State NAND Flash NEMS Anchor-less Shuttle Triple Quantum Dot NEMS Non-Volatile Memory Devices Potential for: High frequency operation No hystersis Zero current leakage Zero steady-state power consumption Very very small
- 31. NEMS Floating Beam Solid State NAND Flash NEMS Anchor-less Shuttle Triple Quantum Dot NEMS Non-Volatile Memory Devices Potential for: High frequency operation No hystersis Zero current leakage Zero steady-state power consumption Very very small Poisson’s equation Poisson’s equation & Navier’s equation Poisson’s equation & Navier’s equation Quantum Master Equation
- 32. What did we learn along the way? Inducing mechanical motion in nanoscale systems. Evaluation of mechanical forces in nanoscale systems. 1. Dissociation - R. Pozner, E. Lifshitz and U. Peskin, Nano Lett., 14, 6244−6249 ( A probe for inter-dot interactions 2. NDR - R. Pozner, E. Lifshitz and U. Peskin, J. Phys. Chem. Lett., 6, 1521−1528 ( 3.There is maybe an untapped world of NEMS devices that can exploit the mobility properties of Colloidal Quantum Dots.
- 33. Quantum Dissociation Intuition STM Tip Mass & Momentum Effect Orbitals Effect Classical Intuition Quantum Intuition STM Tip
- 35. 1 Spins 2 Spins
- 36. 40 30 20 10 0 4 3 2 1 0 0 0.5 1 1.5 2 2.5 Current Population Multiple NDRs
- 37. Apply Reach S.S Apply A B C D E Measure F 2 Spins - 3 Bits Memory Device
- 38. 20 0 15 0 10 0 50 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 50 0 40 0 30 0 20 0 10 0 1 Spin
- 39. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 80 0 70 0 60 0 50 0 40 0 30 0 20 0 10 60 0 50 0 40 0 30 0 20 0 10 0 0 0 0.4 0.8 1.2 1.6 2 2 Spins
- 41. PARALLEL STM • Coherences • I(t) What about Dark States here? What about the NDR here?
- 42. 3 Sites 4 Sites
- 43. Oscillator + GLE De-phasing Surface STM Tip Full Hamiltonian: DQD Hamiltonian: Generalized Langevin Equation: Morse potential + GLE Oscillator + GLE
- 44. Macroscopic Model Interpolation Mott - Varying Range Hopping: Efros-Shklovskii - Variable Range Hopping: Temporal Energy Fluctuations: Interpolation of our kinetic quantum-classical model to macroscopic arrays models
- 45. Beyond weak coupling Using the Hierarchical Quantum Master Equation approach to explore the Recoil / Dissociation effect for strong Electrode-Dots coupling
- 46. Future Work (Experimental) Random Embedded Trenches
- 47. More accurate treatment of the QDs and Ligands structure should yield better predictions… Improved Orbital Treatment Gaussian Orbitals 1st Improvement 2nd Improvement Orbial and LUMO dependency More Realistic Orbitals Calculations New more accurate orbital Sum of Gaussians
- 48. Material Force constant (K) which leads to dissociation Gaussian QD PbSe QD CdSe QD PbS QD Core/Shell Improved Orbital Treatment CdSe Core CdSe/ZnS Core/Shell ? ? ? ?
- 49. Acknowledgements Efrat Lifshitz Uri Peskin all the group members my girlfriend and my family
- 50. Thank You
- 51. OTHERS
- 52. Electron Correlations Pumps Populations Time [fsec]
- 53. No tunneling Just electron correlations! R R Relaxation Satellite D A B 3 Sites donor population oscillates periodically to the acceptor
- 54. Weak Correlation Effects Strong Correlation Effects Quasi-particle Approximation Hole Mixing Correlation Satellite Relaxation Satellite Ground State Correlation Satellites Breakdown of the molecular-orbital-picture Charge Migration Mechanisms inner-valence shells (2nd charge migration) (1st charge migration) outer-valence shells φi φj beatings (3rd charge migration) Above Below
- 55. NEMS
- 56. L Insulating layer Surface R Insulating layer Surface TQD NEMS Device Probing Current Inducing Motion ״ 0 ״ ״ 1 ״ bit
- 60. Implementation A short break for a science-fiction fabrication before we finish
- 61. Implementation A short break for a science-fiction fabrication process before we finish
- 63. DISSOCIATION
- 64. The inter-dot interaction potential V(q) should reflect universal: A. Short-range repulsion B. Long-range attraction x “Weak” “Strong” “Intermediate” = = STM Tip-DQD-Surface with Mechanical Coupling Surface STM Tip Reservoirs of non-interacting electrons Single electron hopping between localized orbitals & electrode states Full Hamiltonian: Coupling term depends on the inter-dot distance Projection of DQD Hamiltonian onto the orthonormal DQD orbitals: Weak Weak Weak Under the weak coupling approximation, the DQD is treated under the Reduced Density Matrix method. Each eigenstate is associated with an occupation number of the DQD orbitals. Bound Floating
- 65. Model Analysis Nonadiabatic force terms can be neglected Adiabatic Force Terms Nonadiabatic Force Terms Density-matrix is diagonal at steady- state Assuming coherences can be neglected populations potential energy surfaces (PES) or BO surfaces The effective force is a weighted average of the populations doubly occupied state “antibonding" orbital state “bonding” orbital state empty state potential energy curves
- 66. Adiabatic Force Terms Non-adiabatic Force Terms
- 67. Assuming coherences can be neglected Density-matrix is diagonal at steady- state Non-adiabatic force terms can be neglected
- 68. Dissociation Recoil Born-Oppenheimer Picture Weak Coupling Potential energy curves:
- 69. RC Model
- 70. D is the inter-dot binding energy and alpha is the interaction range param
- 71. r is the electronic coordinate, and each dot has a different center of mass. The dots dimensions are characterized by an effective size parameter, sigma (the standard deviation of the probability distribution, |xhi| square). xhi, can be refined to include details for any specific CQDs of specific shape and chemical compositions.
- 72. - Spikes: Small deviations in the threshold energies due to changes in the steady state inter-dot distance lead to the sharp picks. Notice that the sharp steps are due to the weak coupling between the DQD and the leads and the low temperature. - We note that the reduced density matrix approximation invoked in this work underestimates the step widths under these conditions. Sharp spikes in the current are due to minor displacement of the transport channels thresholds. - A detailed discussion of the dependence of the steady state populations on the DQD structure is beyond our scope here and will be given elsewhere
- 73. atom like behaviour maximum localization bulk like behaviour maximum delocalization Quantum Dots Arrays How do various electronic transport properties of quantum dots with different sizes, structures, coupling and compositions change when moving from a Single Quantum Dot to a Quantum Dots Film? tradeoff between maximum localization and maximum delocalization ideal photogeneration ideal current flow.
- 74. Recoil Dissociation Dissociation empty state bonding orbital antibonding orbital electron interactions Born-Oppenheimer Picture Intermediate Weak Strong
- 75. NDR
- 76. A Probe for Inter-Dot Interactions
- 77. T DQD S 00 10 01 11 0 0 00 10 01 11 0 0 0 0 1 0 0 1 1 1 0 00 10 01 11 Surfac e Symmetric Case Transition Rates Coupling Matrix Elements Destructive Interference
- 78. 1 2 = Count the number of particles at state ‘1’ (occupation number) 2nd Quantization lets us deal with multi-electrons systems without taking care of the underlying complex multi-electrons orbitals 2nd Quantization ‘in a nutshell’ let’s go a few steps back…
- 79. DQD Hamiltonian: Rigid Double Quantum Dot Model Transitions: 00 10 01 11 doubly occupied state “antibonding" orbital state “bonding” orbital state empty state = T S T S
- 80. Solving the generalized eigenvalue problem (Sites Representation): Single electron Hamiltonian matrix Overlap matrix Rigid Double Quantum Dot Model Surface STM Tip 1 2 Orthonormal superpositions of local dot orbitals (Eigenstates Representation): 1 2 Orbitals coefficients matrix
- 82. 00 10 01 11 T S Surface STM Tip 1/2= =X Matrix elements:
- 83. Solving the generalized eigenvalue problem (Sites Representation): Single electron Hamiltonian matrix Overlap matrix Double Quantum Dot Model 1 2 Orbitals coefficients matrix
- 84. Symmetric case 2nd Point of View E1-E2 E1-E2 E1-E2 E1-E2 q (nm) q (nm) q (nm) q (nm)
- 85. Negative Differential Resistance in asymmetric 2 terminal device with a shared contact. Rigid STM Tip-DQD-Surface
- 86. MODEL
- 87. Operators in the interaction frame: Rewritten Liouville–von Neumann equation: third (non-Markovian) term Approximated second-order equation of motion : Assumption: weak coupling Reduced molecular density Tracing over the bath degrees of freedom - obtain the reduced equation in subspace Redfield approach
- 88. Leads to a closed equation:
- 90. Model Calculation Full Hamiltonian Reduced Density Matrix Approach Reduced Density Matrix in the basis of the DQD Hamiltonian eigenstates Surface [Energy (eV] Tip Rate of single electron transition induced by the leads: Spectral function: Quantum Master Equation Approach Fermi distribution functions: Solving Liouville-von Neumann equation - Kappa is the rate for transitions induced by the Kth lead. - ehta_n_k equals one or zero if the Kth lead is coupled or uncoupled to the nth dot.
- 91. A tight-binding model of a linear conductor single-particle electrode’s eigenstates single-particle energy in the band semi-infinite linear chains Restricting the coupling between the electrode and the subspace to a specific molecular site Subspace- electrode coupling has a product form
- 92. Model Calculation Substrate [Energy (eV] Tip population potential energy curve Steady state response
- 93. Dependence of the electronic parameters on the inter-dot distance x Orthonormal superpositions of local dot orbitals: Double Quantum Dot Hamiltonian Two electrons interaction terms: Explicit derivation of Interaction terms:
- 94. three-dimensional localized Gaussian orbitals: Single electron Hamiltonian matrix: Overlap matrix: Solving the generalized eigenvalue problem: Double Quantum Dot Model Single particle terms two particles term E1 and E2 are assumed to be independent of the inter-dot distance Our local basis is not orthonormal. But we can construct an orthonormal basis by using a super Using this method we move from the local orbital representation to the DQD eigenstate represe
- 95. Solving
- 97. Secular
- 98. Weak Coupling
- 99. DERIVATIONS
- 104. For N=0, the electronic space is spanned by the vacuum, For N=1, the electronic space spanned by the single particle state, For N=2, the electronic space spanned by two particle determinants: In general Projection: N - Partcie M - Orbitals
- 105. where are the N particle determinants that span the M particle space. The respective force matrix for N=1, M=2, in the basis reads, And for M=2, N=2: N - Partcie M - Orbitals
- 106. Let us define a density operator in the N=0,1,2 space, Let us define a density operator in the N=0,1,2 space, N - Partcie M - Orbitals
- 107. The many-electron Hamiltonian The full (non-relativistic) Hamiltonian for a many electron system in the basis of N spin-orbitals reads: and create and annihilte an electron in the ith spin-orbital, and the cofficients are defined as Two 3D localized non-orthogonal Gaussian orbitals In the case of two (real) spatial orbutal, i,j (שייך1,2) (no spin), the hamiltonian is:
- 108. Without considering the details of the associated single particle Hamiltonian, , we assume that two of its eigenstates are orthonormal super positions of the two Gaussians, Orthogonality => e1 and e2 are the eigenvalues and t=0. Notice that our signle particle state phi1(r) and phi2(r) de Using:
- 109. Defining Gaussian product terms:
- 110. The Gaussian integrals are written as Auxiliary functions:
- 111. Denoting the inter-dot distance as , we obtain:
- 113. We can now obtain explicity expression for the Columb integrals as function of R.
- 114. dependence of the electronic parameters on the inter-dot distance 1 2 3 4 5 6

- Discrete Energy Bands Also Discrete -> Quantum Size Effect Artificial Atoms -> Change Size -> Reduce Bandgap -> Change Materials
- Emphasise the fact that the QDs sit on top of a macroscopic surface Does DQD important for the operation of a solar cell Most focus today is on excition splitting and no one talk about the transport through the QDs Relvent to order as well as order system (coherence effects, defects, dephasing…) And then small clusters of DQD are also important and maybe influcence how much the interaction bettwen dots is important in a big array
- Shifting Question Single -> STM -> Results Large -> Transistor -> Results Our Focus -> Lithogrpahy -> Rigid -> Electron Coupling Collodial -> Solution -> Ligands -> Ligands Roles -> Where is the picture? -> Electron Coupling -> Mechanical Coupling
- Present 1 Dot -> 2 Dots -> Coloumbe Blockade -> Gaussian -> Parameters -> E1 & E2 -> q -> S -> t -> STM t experiment Matrices -> Eigenvalue Problem -> DQD eigenenergives or the single electron terms as a function of q -> Double electron terms Q large -> Q small Summary
- Incease morse to infinity until the DQD becomes rigid moving to measure electronic interactions, we need to assume to the DQD is rigid. there are many relevent cases. Under the weak coupling approximation, the DQD is treated under the Reduced Density Matrix method -> Each eigenstate is associated with an occupation number of the DQD orbitals. Projection -> Add a surface (2 connections) -> Add a tip (left connection) -> Model descirption Full Hamiltonian -> H_DQD -> H_Leads -> H_coupling Instead of Rigid -> Non Rigid -> Represent Effective Mechanical Interaction -> Wall -> Full Picture Complete H_DQD -> Explain new terms -> Electrons using full QM - > Mehnical Interaction using CM Just a few studies treated the ligands at the atomistic level -> and the mechanical force between DQD is far from understood -> but the ligands should probably affect V(q) and reflect… Morse
- Solve Classic Equation -> Trace -> p’ -> q’ -> equal 0 Solve Liouville Equation -> QME -> Rate -> equal 0 Solve coupled -> Until convergence Why Treat Classically -> Morse -> Black Dot -> Heavy -> Induce Change -> Potential Energy Change -> 10^3 states -> States are continuities inside deltaE -> Can be treated classically - The reduced DQD density states evolves according to the second-order time-dependent Liouville equation in respect to the coupling to the leads.
- - Coupling to surface is the new science. Molecule like behaviour. We are the first to talk about interference effect due to a coherent surface Destructive Interference to the surface
- Subtracting the threshold energies (both measurable in this case) Subtracting the threshold energies (both measurable in this case) the value of the interaction can be revealed, provided that the overlap between the two localized orbitals, s, is sufficiently small. - the value of the interaction can be revealed, provided that the overlap between the two localized orbitals, s, is sufficiently small.
- Under the weak coupling approximation, the DQD is treated under the Reduced Density Matrix method -> Each eigenstate is associated with an occupation number of the DQD orbitals. Projection -> Add a surface (2 connections) -> Add a tip (left connection) -> Model descirption Full Hamiltonian -> H_DQD -> H_Leads -> H_coupling Instead of Rigid -> Non Rigid -> Represent Effective Mechanical Interaction -> Wall -> Full Picture Complete H_DQD -> Explain new terms -> Electrons using full QM - > Mehnical Interaction using CM Just a few studies treated the ligands at the atomistic level -> and the mechanical force between DQD is far from understood -> but the ligands should probably affect V(q) and reflect… Morse
- Solve Classic Equation -> Trace -> p’ -> q’ -> equal 0 Solve Liouville Equation -> QME -> Rate -> equal 0 Solve coupled -> Until convergence Why Treat Classically -> Morse -> Black Dot -> Heavy -> Induce Change -> Potential Energy Change -> 10^3 states -> States are continuities inside deltaE -> Can be treated classically - The reduced DQD density states evolves according to the second-order time-dependent Liouville equation in respect to the coupling to the leads.
- Strong Mechanical Coupling -> Explain Figure & Axes -> Annotations -> deltaI & deltaQ, deviations from the rigid case -> Tip on top of the little dot -> Show Results -> Throw Away Weak Coupling -> Tip still on top of the little dot -> Please Ignore the specific transitions -> Relatively strong deviations from rigid -> 50% movement -> Recoil Tip on top of the big dot -> Strong derivation -> more then 100% movement -> Dissociation
- p’ equal trace over multiplication of density matrix and derivative of H -> Again, we are dealing with a 2 orbitals basis set -> dH/dq in Fock Space Diagnoal are Adiabatic Forces Terms -> Represent Force due to the population on each DQD eigenstate NonDiagnoal are NonAdiabatic Forces Terms -> Represent Force due to population transfer between different DQD eigenstate Because the relatively large DQD eigenvalues separation -> and because electronic interactions exceed levels broadening induced by the DQD-leads coupling and by the temperature -> Rapid Dephasing -> Coherences -> D.M. is diagnoal as S.S. -> NonAdiabatic Terms can be ignored upon taking the trace over D.M. & dH/dQ Left with Diagonal Contriubations -> Example of Potential Energy Curve -> Empty -> Bonding -> AntiBonding -> Double Occupied 10 is a “bonding” orbital and is associated with a single electron tunnelling between the dots. 01 state is due to quantum de-localization of single electrons between the dots in an “anti-bonding” orbital (having a nodal plane between the dots centers) 11 gets populated at higher bias voltages, and includes additional repulsion due to two-body interactions. The Final Equation Effective Force -> Multiplication We can learn from this equation that the effect of the leads amount to inducing particle transfer between the DQD eigenstates, and there is no charge transfer due to kinetic energy of the dots themselves.
- The derivative of the curves it the force. The effective force weights the probabilities 10 is a “bonding” orbital and is associated with a single electron tunnelling between the dots. 01 state is due to quantum de-localization of single electrons between the dots in an “anti-bonding” orbital (having a nodal plane between the dots centers) 11 gets populated at higher bias voltages, and includes additional repulsion due to two-body interactions. Strong -> Q>Q0 -> All Curves Attractive -> Because Attractive, there is no large shift from rigid case Our model calculations predict that the DQD should dissociate when the Force constant of the Morse potential is smaller then 10^-2 Weak -> Two Repulsive (does not have a minimum for finite q) -> Two Attractive Explain Population Map -> We see Population for Each State -> Blue means little probability for population -> Red mean high probability for population Like in di-atomic molecule, also here we have repulsive states
- This is for Recoil Case -> Let’s see the Dissociation Case -> Look at P01, the replusive -> Recoil there is a shift to Attractive states -> Significant population at the replusive states leads to Dissocation. Derivation of K -> More accurate treatment -> Better predictions
- Induce -> Probe -> NEMS Storing -> Zero power for storing the data
- - Coupling to surface is the new science. Molecule like behavior. We are the first to talk about interference effect due to a coherent surface Destructive Interference to the surface The NDR reveals the molecule-like nature of the DQD due to destructive interference in the coherent coupling to the shared surface.
- In physics, the Navier–Stokes equations describe the motion of viscous fluid substances. These balance equations arise from applying Newton's second law to fluid motion.
- Elaborate about dephasing. It will eliminate the NDR due to double/single substrate in the rigid case. The surface is not 100% shared anymore Will lead to different behavior of the dissociation effect. “REAL” dissociation Describe expected result?
- - Percolation - Non Arrhenius behavior
- Induce -> Probe -> NEMS Storing -> Zero power for storing the data
- NOR Flash cell size NAND Flash cell size is much smaller than NOR Flash cell size—4F2 compared to 10F2—because NOR Flash cells require a separate metal contact for each cell.
- Projection -> Add a surface (2 connections) -> Add a tip (left connection) -> Model description Full Hamiltonian -> H_DQD -> H_Leads -> H_coupling Instead of Rigid -> Non Rigid -> Represent Effective Mechanical Interaction -> Wall -> Full Picture Complete H_DQD -> Explain new terms -> Electrons using full QM - > Mechanical Interaction using CM Just a few studies treated the ligands at the atomistic level -> and the mechanical force between DQD is far from understood -> but the ligands should probably affect V(q) and reflect… Morse
- p’ equal trace over multiplication of density matrix and derivative of H -> Again, we are dealing with a 2 orbitals basis set -> dH/dq in Fock Space Diagonal are Adiabatic Forces Terms -> Represent Force due to the population on each DQD eigenstate NonDiagnoal are NonAdiabatic Forces Terms -> Represent Force due to population transfer between different DQD eigenstate Because the relatively large DQD eigenvalues separation -> and because electronic interactions exceed levels broadening induced by the DQD-leads coupling and by the temperature -> Rapid Dephasing -> Coherences -> D.M. is diagnoal as S.S. -> NonAdiabatic Terms can be ignored upon taking the trace over D.M. & dH/dQ Left with Diagonal Contriubations -> Example of Potential Energy Curve -> Empty -> Bonding -> AntiBonding -> Double Occupied 10 is a “bonding” orbital and is associated with a single electron tunneling between the dots. 01 state is due to quantum de-localization of single electrons between the dots in an “anti-bonding” orbital (having a nodal plane between the dots centers) 11 gets populated at higher bias voltages, and includes additional repulsion due to two-body interactions. The Final Equation Effective Force -> Multiplication We can learn from this equation that the effect of the leads amount to inducing particle transfer between the DQD eigenstates, and there is no charge transfer due to kinetic energy of the dots themselves.
- p’ equal trace over multiplication of density matrix and derivative of H -> Again, we are dealing with a 2 orbitals basis set -> dH/dq in Fock Space Diagonal are Adiabatic Forces Terms -> Represent Force due to the population on each DQD eigenstate NonDiagnoal are NonAdiabatic Forces Terms -> Represent Force due to population transfer between different DQD eigenstate Because the relatively large DQD eigenvalues separation -> and because electronic interactions exceed levels broadening induced by the DQD-leads coupling and by the temperature -> Rapid Dephasing -> Coherences -> D.M. is diagnoal as S.S. -> NonAdiabatic Terms can be ignored upon taking the trace over D.M. & dH/dQ Left with Diagonal Contriubations -> Example of Potential Energy Curve -> Empty -> Bonding -> AntiBonding -> Double Occupied 10 is a “bonding” orbital and is associated with a single electron tunneling between the dots. 01 state is due to quantum de-localization of single electrons between the dots in an “anti-bonding” orbital (having a nodal plane between the dots centers) 11 gets populated at higher bias voltages, and includes additional repulsion due to two-body interactions. The Final Equation Effective Force -> Multiplication We can learn from this equation that the effect of the leads amount to inducing particle transfer between the DQD eigenstates, and there is no charge transfer due to kinetic energy of the dots themselves.
- p’ equal trace over multiplication of density matrix and derivative of H -> Again, we are dealing with a 2 orbitals basis set -> dH/dq in Fock Space Diagonal are Adiabatic Forces Terms -> Represent Force due to the population on each DQD eigenstate NonDiagnoal are NonAdiabatic Forces Terms -> Represent Force due to population transfer between different DQD eigenstate Because the relatively large DQD eigenvalues separation -> and because electronic interactions exceed levels broadening induced by the DQD-leads coupling and by the temperature -> Rapid Dephasing -> Coherences -> D.M. is diagnoal as S.S. -> NonAdiabatic Terms can be ignored upon taking the trace over D.M. & dH/dQ Left with Diagonal Contriubations -> Example of Potential Energy Curve -> Empty -> Bonding -> AntiBonding -> Double Occupied 10 is a “bonding” orbital and is associated with a single electron tunneling between the dots. 01 state is due to quantum de-localization of single electrons between the dots in an “anti-bonding” orbital (having a nodal plane between the dots centers) 11 gets populated at higher bias voltages, and includes additional repulsion due to two-body interactions. The Final Equation Effective Force -> Multiplication We can learn from this equation that the effect of the leads amount to inducing particle transfer between the DQD eigenstates, and there is no charge transfer due to kinetic energy of the dots themselves.
- Weak -> Two Repulsive (does not have a minimum for finite q) -> Two Attractive Explain Population Map -> We see Population for Each State -> Blue means little probability for population -> Red mean high probability for population This is for Recoil Case -> Let’s see the Dissociation Case -> Look at P01, the repulsive -> Recoil there is a shift to Attractive states -> Significant population at the repulsive states leads to Dissociation. Derivation of K -> More accurate treatment -> Better predictions
- De-Localization: The first, associated with the “01” state (see in Fig. 2) is due to the quantum de-localization of single electrons (tunneling) between the dots in an “anti-bonding” DQD orbital (having a nodal plane between the dots centers). A second contribution is associated with the state ”11”, which gets populated at higher bias voltages, and includes additional repulsion due to two-body (electrostatic) interactions. Averaged Force: Mechanical recoil between the two dots becomes apparent when the averaged force is repulsive, due to population transfer into repulsive states (“01” and/or “11”) at steady state. Comparing the three different DQD structures, the strong mechanical response observed for weak binding (up to dissociation) in the cases E1>E2 and E1=E2 is indeed correlated with a relatively significant population of the repulsive electronic states, as demonstrated in the middle and right columns of Fig.5. For , the states “00” and “10” dominate the steady state for the accessible inter-dot distances, and the recoil is insignificant. A detailed discussion of the dependence of the steady state populations on the DQD structure is beyond our scope here and will be given elsewhere. ——— H_DQD eigenvalues (BO surfaces) as functions of the inter-dot distance, q, in steady state conditions (p=0). The left, middle and right plots correspond to decreasing binding interactions, i.e., V(q0+2sigma)=0.25,0.025,0.0025 respectively. Each many-body eigenstate is marked according to the electronic occupation of the two DQD orbitals ‘00’,’01’,’10’,11 correspond respectively to the electronic energies: 0,e1,e2,e1+e2+U.
- - Subtracting the threshold energies (both measurable in this case) - the value of the interaction can be revealed, provided that the overlap between the two localized orbitals, s, is sufficiently small.
- “Crush course” on 2nd quantization, Density matrix (before coherences)
- Basis: The reduced DQD density is represented in the basis of the DQD Hamiltonian eigenstates, defined by H_DQD|mn>=E_mn|mn> Since H_D commutes with the electronic number operator, each eigenstate can be identified by the occupation numbers of the two DQD orbitals (m,nE(0,1)), where the respective orbitals and orbital energies depend on the classical variables, p and q. Reduced DM approach: Considering the typical electronic tunneling barriers between the DQD and the STM tip, and between the DQD and the surface, the dynamics of the electronic density can be approximated in the limit of weak DQD-leads electronic coupling, using the reduced density matrix approach. Liouville: The leads are assumed to maintain a quasi-equilibrium density, while the reduced DQD state evolves according to the time-dependent Liouville equation to second order in the coupling to the leads. Considering rapid de-phasing (decay of coherences) between DQD eigenstates, the effect of the leads amounts to inducing transitions between the DQD eigenstates via electron hopping events, according to the kinetic equations dPmn=K*Pmn. Pmn is the transient population of the respective eigenstate of the DQD Hamiltonian, Kappa is the rate for transitions induced by the Kth lead, where Gamma is the equation and f is the Fermi distribuation function. Notice that the transition rate constants depend explicitly on the mechanical degree of freedom via Uij(q) and Emn(p,q).
- Steady state response of the system to applied bias voltage between the tip and the substrate is associated with a self-consistent solution of the coupled electro-mechanical dynamical equations under the constraint, q’=p’=P00’=P10’=P01’=P11’=0 . These equations define the inter-dot distance and the steady state current from the tip to the surface, I_t->s, where Nmn is the electrons number in the Nmn_th DQD eigenstate. Notice that in principle there are multiple steady state solutions. In particular, there is always a solution with p=0 and q->infinity which corresponds to transport through a single dot. The results below relate to the steady state values of q, which are nearest to the equilibrium distance, q0. BO: The differences in the mechanical response of different DQDs to a given bias voltage can be rationalized using a Born-Oppenheimer(BO) picture. PES: The eigenvalues of H_dqd for p=0 (steady state) define potential energy curves for the inter-dot mechanical motion, from which the mechanical force is derived. Taking the quantum mechanical trace in the classical force equation in the basis of the (many body) electronic eigenstates gives, . As one can see the averaged force depends on the electronic state populations, . Attractive/Repulsive: For strong binding all four curves are attractive. For intermediate binding the potential curve associated with the “11” state becomes repulsive (i.e., does not have a minimum for finite q), and for weak binding both the “01” and the “11” curves are repulsive. Notice that the model accounts for two different contributions to the repulsion.
- phi 1,2: The respective orthonormal DQD orbitals, , and , define an electronic Fock space. DQD H: the DQD Hamiltonian reads. a is the creation operator for an electron in the mth orbital, muo is the reduced mass of the two dots. The electronic coupling to the mechanical degree of freedom is inherent via the explicit dependence of the single electron energies and the electronic interaction term on q (see supplementary material). U(q): Considering the explicit form of the local orbitals and the expansion coefficient, these integrals can be calculated as follows,. The dependence of U12(q) and Uex(q) on the inter-dot distance is therefore uniquely determined by the specific single particle model invoked in the text. U(q): the electronic interaction term on (see supplementary material). Add: Notice that at q->infinity, e(1), e(2) approximate E1,E2, but at the beginning there is a strong dependence on S & t. Add: Very consistent model - just 2 parameters from which we derive all the parameters.
- 2 Dots: A simple analytic model for the two dot orbitals is invoked, representing them as three-dimensional Gaussians. where and are the dots center of mass coordinates and is the electronic coordinate. The dots dimensions are characterized by an effective size parameter, (the standard deviation of the respective probability distribution, ). The detailed structure of the electronic envelope function, as well as the rapid oscillations associated with the semiconductor periodicity, are excluded here from , but these functions can be refined to include these details for any specific CQDs of specific shape and chemical compositions. Overlap: Setting,, and , the inter-dot distance,, defines the overlap between these two localized orbitals, S. H,S matrices: The matrix representation of the effective single electron Hamiltonian in the basis of the two localized orbitals and the respective overlap matrix take the following generic form, H,S. Thereby, the on-site energies denoted here as , are assumed to be independent on the inter-dot distance, and their values are taken as the LUMO energies of the two separated dots at . The choice accounts for differences between the dots in terms of their LUMO energy due to internal structure, disorder, local external fields, etc. The electronic coupling matrix element is assigned a typical exponential decay form, where the coupling matrix element at the reference distance is , and is the decay length of the effective single electron hopping interaction which depends on the nanoparticle boundaries and on the surrounding ligands. Variation principle, e1(q), e2(q): Each dot is represented by a single localized spin orbital. The single particle energies of the DQD structure at a fixed inter-dot distance, , can be calculated using the standard linear variation principle. A typical distance dependence of the electronic model parameters e1(q),e2(q),U(q). The equilibrium inter-dot distance was set to q0=5sigma, which corresponds to an overlap integral, S(q0)=0.044. The Hamiltonian parameters were set to , E1=E2=0.25eV, t0=-0.01eV and, gamma=0.4nm-1, which correspond to an energy splitting, e1(q0)-e2(q0)=2(SE1-t0)/(1-S^2)~0.042eV between the two DQD orbitals at the equilibrium geometry. Bottom: Illustrative plots for the ligands-mediated electronic interactions.
- BO: The differences in the mechanical response of different DQDs to a given bias voltage can be rationalized using a Born-Oppenheimer(BO) picture. PES: The eigenvalues of H_dqd for p=0 (steady state) define potential energy curves for the inter-dot mechanical motion, from which the mechanical force is derived. Taking the quantum mechanical trace in the classical force equation in the basis of the (many body) electronic eigenstates gives, . As one can see the averaged force depends on the electronic state populations, . Attractive/Repulsive: For strong binding all four curves are attractive. For intermediate binding the potential curve associated with the “11” state becomes repulsive (i.e., does not have a minimum for finite q), and for weak binding both the “01” and the “11” curves are repulsive. Notice that the model accounts for two different contributions to the repulsion.
- 1. We start by associating the eigenstates of the electronic single particle dimer Hamiltonian with orthonormal super positions of local dot orbitals. 2. Xhi_A,B are non-orthogonal, normalized, three-dimensional Gaussians, with an overlap integral S, where q=|Ra-Rb| is the inter-dot distance. Using the normalization of phi_1,2 we rewrite, and the orthogonality of pho_1,2 implies 3. The coefficients a1,b1,a2,b2 and the respective single particle energies e1,e2, are uniquely defined by the generalized eigenvalue problem according to the standard linear variation principle. C is the coefficients matrix. Notice that the matrices H and S depend explicitly on the inter-dot distance, q, and therefore the solutions to the generalized eigenvalue problem, i.e. the respective single particle energies (e1,e2) and the expansion coefficients (a1,a2,b1,b2) also depend on the inter-dot distance. 4. Within the Fock space spanned by the two orthonormal single particle states, the electronic many-body dimer Hamiltonian reads. U(q) is the two electron interaction term. where U12(q) and Uex(q) are respectively the coulomb and exchange integrals. 5. Finally, we consider the dependence of the dimer-lead coupling on q. The coupling between the DQD and the STM tip(T) and the surface(S), corresponds to electron hopping between the localized orbitals (Xhi_a,b) and the lead states, H_dqd-leads, where d_n is the annihilation operator of an electron at the nth dot orbital. 6. Expanding the dot orbital in terms of the orthonormal DQD orbitals d_n,the coupling term becomes dependent on the inter-dot distance. Using Eq.(A1), it follows that 6(2), and therefore 6(3)