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Physics of Confined Electrons

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  1. Physics of Confined Electrons D. G. Kanhere, Bhalchandra Pujari and Kavita Joshi Department of Physics and Centre for Modeling and Simulation University of Pune, Pune -411007 Kolkata Jan 2007 ACL, Physics Dept, Pune University

  2. Physics of Confined Electrons • Confined Electron systems are realized in Quantum dots • Fabrication: Molecular Beam Epitaxy method, Electron Beam Lithography, Self assembly via Electrochemical means etc. Molecular beam epitaxy  ACL, Physics Dept, Pune University

  3. Properties: Good Ref: Reimann and Manninen: Rev Mod Phys 74 ( 2002) . • Quantum Dots exhibit magic numbers Like atoms or clusters Quantum Dots also show the stability peaks in the addition energy spectrum ACL, Physics Dept, Pune University

  4. Properties • Coulomb Blockade • Transport through QD is in Coulomb Blockade regime • Conductance is Quantized ACL, Physics Dept, Pune University

  5. Issues • Wigner Crystallization In bulk solid, at low densities, Coulomb energy dominates and Kinetic energy approaches zero resulting in ‘crystallization’ of electrons to their positions Transition from Fermi liquid to Wigner crystal is the transition from metal to insulator and of immense importance in electronics Do we obtain it in confined system? Wigner Molecule ?! Wigner, E., Phys Rev, 46, 1002 [1934] ACL, Physics Dept, Pune University

  6. Issues • Singlet – Triplet transition • Singlet-triplet transition is extremely important to tune magnetic and electrical properties. • Usually achieved by means of application of magnetic field (2 stage Kondo effect) Can we obtain it without Magnetic filed ? W.G. van der Wiel et al, PRL 88, 12 [2002] ACL, Physics Dept, Pune University

  7. Issues • What are the effects of impurities or disorder ? Electronic structure, Charge density, phases, magnetic structures – spin order • What about quantum dot molecules ? Vertically / Laterally coupled quantum dots… • Transport, optical properties ? ACL, Physics Dept, Pune University

  8. Computational Model QDs are modeled as 2D electron gas confined by external potential. External potential could be: Square well Parabolic well OR any arbitrary We model our system in square well potential. ACL, Physics Dept, Pune University

  9. Computational Details Exact Hamiltonian SDFT Hamiltonian V effective Exchange Correlation* Computational units ħ = e2/ε= m* = 1 Exact Hamiltonian : Configuration Interaction SDFT : Spin Density functional theory *Attaccalite et al;PRL,88,256601 [2002] Tanater & Ceperley, PRB,39,5005-5016 [1989] ACL, Physics Dept, Pune University

  10. Real Space Method • System being non-periodic (finite) real space technique has the edge • 2D Cartesian grid • High-order finite difference (5 Point formula) • Davidson algorithm for diagonalization • 2D Poisson equation solved by discrete convolution method ACL, Physics Dept, Pune University

  11. Results • DFT 2-20 electrons • Homogeneous • Attractive Impurity • Configuration interaction • 2-6 electrons • g(r,r’) spin and charge correlations • Impurity Induced effect ACL, Physics Dept, Pune University

  12. Impurity • Impurity model Impurity is modeled by Gaussian function G (x,y) = A exp(B(x2 + y2)) • Impurity is insignificant with respect to the height of the rigid walls • Can be attractive or repulsive • Width is fixed to ~ 30nm and strength is 1 Ha* ACL, Physics Dept, Pune University

  13. Non – interacting picture Non-interacting eigenfunctions N = 4, Size of dot ~ 295nm , rs ~ 8 N=4 Non interacting Charge density ACL, Physics Dept, Pune University

  14. N=4Non interacting Vs DFT densities Non interacting Rs ~ 2 Rs ~ 8 DFT ACL, Physics Dept, Pune University

  15. Known Problems with DFT • Symmetry breaking Although the total charge density is symmetric, the spin-polarized charge density given by DFT may not retain the symmetry of the potential ↑ ↓ Total Charge density for N = 4, size ~ 175 nm; rs ~ 5; Sz =1 There are several low-lying near equilibrium states which make self consistency difficult (especially in Wigner regime) ACL, Physics Dept, Pune University

  16. Configuration Interaction • Construct Ψ as a linear combination of all possible configurations. ΨCI = Σi CiΨiSlater ΨiSlater is the Slater determinant constructed for ith configuration • Construct H matrix. Diagonalize the N-electron Hamiltonian in a basis of N-electron functions i.e. Slater determinants to get lowest few eigenvalues and eigenvectors.  ΨCI  Ĥ exact ΨCI • If the basis is complete, we can obtain the exact energies not only of the ground state, but also for all excited states of the system. • Full CI : The basis set is complete. The answer is exact !!! • Exact CI : Exact within the given basis set but the basis set is truncated. Conceptually straightforward but not computationally!! ACL, Physics Dept, Pune University

  17. Configuration Interaction • Memory requirement (HUGE) !! Table for N= 4 In practice Full CI is possible only for few electron systems. ACL, Physics Dept, Pune University

  18. Configuration Interaction • Energies against configuration N=2 ACL, Physics Dept, Pune University

  19. The Charge densities N=2-20 • Total charge density as a function of number of electrons. • Fixed BOX size : increase the number of electrons: • Watch the density grow. ACL, Physics Dept, Pune University

  20. Charge densities (DFT) N=2 to 20; Rs = 8.5 to2.7; Box length=250nm IMPURE PURE ACL, Physics Dept, Pune University

  21. N=2 ↓ ↓ ↑ ↑ Results N = 2 DFT • Ground state is Singlet Sz=0 Total charge density DFT results for N= 2, rs ~ 3 to rs ~ 12 Wigner crystal is observed at rs ~ 9 (size ~ 235 nm) 4 peaks in the density reflects the symmetry of the potential well Wigner, E., Phys Rev, 46, 1002 [1934] ACL, Physics Dept, Pune University

  22. CI :Densities and Correlations : N=2 Density High Density Low density Up-Dn Correlation ACL, Physics Dept, Pune University

  23. Densities and Correlations :N=2 Attractive Impurity CI Results Density Low density High density Correlations Magnetic Non-magnetic ACL, Physics Dept, Pune University

  24. ↑ ↑ ↓ DFT Results : Density • Wigner Crystal for N = 4 Total charge density DFT results for N= 4, rs ~ 1.5 to rs ~ 11 (size ~60 nm to ~295 nm) Wigner crystal is observed at rs ~ 8 (size ~260nm) ACL, Physics Dept, Pune University

  25. DFT energy components N = 4 RS ACL, Physics Dept, Pune University

  26. DFT Results N=4: Repulsive Impurity : DFT Results S=0 • Repulsive Impurity enhances the Localization Without Impurity N=4, ~60nm rs ~ 1.5 Total charge densities N=4, ~295nm, rs ~ 11 With Impurity ACL, Physics Dept, Pune University

  27. Densities and Correlations : N=4 CI Results : Up-Up Up -Dn N=4, S=1, Rs ~ 1 Inhomogeneity in correlations * See A. Ghosal et al, Nat Phys Lett. 2006 ACL, Physics Dept, Pune University

  28. Densities and Correlations : N=4 CI Results : Up-Up Up-Dn N=4 S=1, Rs~ 10 ACL, Physics Dept, Pune University

  29. Effect of Correlation on density • 4 electron system Rs ~ 1.5 Rs ~ 4 Rs ~ 6.5 Charge density along diagonal Dramatic difference in correlated charge densities! ACL, Physics Dept, Pune University

  30. UP - DN DN - UP UP - UP CI result :N=4 Effect of impurity on spin correlations! Pure N=4, Rs = 4, Sz=1 with and without impurity Impure ACL, Physics Dept, Pune University

  31. UP - DN DN - UP UP - UP UP - DN DN - UP UP - UP CI result :N=5 (Sz=1.5) Effect of impurity on spin correlations! PURE N=5, Rs = 4, Sz=1.5 with and without impurity IMPURE ACL, Physics Dept, Pune University

  32. N=4 ,3-Up,1- Dn, Rs=3.5 (DFT and CI) Charge Density • Exciting phase for ‘Spintronics’! ↓ ↑ size ~ 120 nm Down electrons are completely delocalized (conducting) while Up electrons are showing ‘incipient’ Wigner crystal phase (insulating). ACL, Physics Dept, Pune University

  33. Single electron trap! • N=10 Rs=4.0 Sz=0 UP DOWN Down Density Down Density Impure Pure UP Density UP Density ACL, Physics Dept, Pune University

  34. Addition energy spectra Rs=1.5 Rs=4 A(n) = E(n+1) -2E(n) + E(n-1) No qualitative change with the addition of impurity! ACL, Physics Dept, Pune University

  35. DFT : Sz=1/2 ‘Spin density wave’ • A typical SDW observed in SDFT (N=7, Rs~4) UP DOWN Spin density With Impurity ACL, Physics Dept, Pune University

  36. Attractive impurity enhances delocalization!! DFT Result Down UP Pure N=7 Rs~1.5 UP Down Impure ACL, Physics Dept, Pune University

  37. N=2 CI RESULT :Impurity effects on charge density N=2 Pure Rs ~ 1.5 Rs ~ 4 N=2 Impure Sz=0 ACL, Physics Dept, Pune University

  38. N=4 CI RESULT :Impurity effects on charge density N=4 Pure Rs ~ 1.5 Rs ~ 4 Impure Sz=1 ACL, Physics Dept, Pune University

  39. N=5 N=5 CI RESULT :Impurity effects on charge density Pure Rs ~ 1.5 Rs ~ 4 Impure Sz=0.5 Sz=1.5 ACL, Physics Dept, Pune University

  40. N=6 CI RESULT :Impurity effects on charge density N=6 Pure Rs ~ 1.5 Rs ~ 4 Impure Sz=0 ACL, Physics Dept, Pune University

  41. N=7 N=7 CI RESULT :Impurity effects on charge density Pure Rs ~ 1.5 Rs ~ 4 Impure ACL, Physics Dept, Pune University

  42. Conclusion • Confinement produces dramatics effects on the nature of the electronic states. DFT captures many of these. • DFT gives qualitatively good results (in agreement with CI). • Localization, spin order, correlation induced inhomogeneity…. • Singlet – Triplet transition can be achieved by b) introducing the impurity • Introduction of impurity enhances the localization , in some cases does induce delocalization. ACL, Physics Dept, Pune University

  43. Thank you! ACL, Physics Dept, Pune University