#### Transcript Mark Friesen "Valley Splitting Theory for Quantum - MDM-CNR

Valley Splitting Theory for Quantum Wells and Shallow Donors in Silicon Mark Friesen, University of Wisconsin-Madison International Workshop on ESR and Related Phenomena in Low-D Structures Sanremo, March 6-8, 2006 Valley Splitting: An Old Problem (Fowler, et al., 1966) “It has long been known that this [two-fold] valley degeneracy predicted in the effective mass approximation is lifted in actual inversion layers…. Usually the valley splitting is observed in … strong magnetic fields and relatively low electron concentrations. Only relatively recently have extensive investigations been performed on these interesting old phenomenon.” (Ando, Fowler, and Stern, RMP, 1982) (Nicholas, von Klitzing, & Englert, et al., 1980) New Methodology, New Directions Different Materials: • Si/SiGe heterostructures Different Knobs: • Microwaves • QD and QPC spectroscopy • (No MOSFET gate) Different Motivation: • Qubits • Single electron limit • Small B fields SiSi80Ge20 Si80Ge20 Si85Ge15 Si90Ge10 200 nm Si95Ge05 Uncoupled 500 nm Si substrate Different Tools: • New tight binding tools • New effective mass theory J0 Swap J>0 Energy Energy Confinement B field Orbital states Quantum Computing with Spins qubit Zeeman Valley Splitting Splitting Electron density for P:Si (Koiller, et al., 2004) Open questions: • Well defined qubits? • Wave function oscillations? Outline Develop a valley coupling theory for single electrons: 1. Effective mass theory (and tight binding) 2. Effect steps and magnetic fields in a QW 3. Stark effect for P:Si donors Li P Energy [meV] Theory P:Si Electron Valley Resonance (EVR) Si (5.43 nm) 22+ 11+ Si0.7Ge0.3 Si0.7Ge0.3 (160 meV) |(z)|2 Motivation for an Effective Mass Approach • Valley states have same envelope • Valley splitting small, compared to orbital • Suggests perturbation theory Effective Mass Theory in Silicon ky incommensurate oscillations (fast) kx envelope fn. (slow) kz valley mixing bulk silicon Bloch fn. (fast) Ec Fz(k) kz • Kohn-Luttinger effective mass theory relies on separation of fast and slow length scales. (1955) • Assume no valley coupling. Effect of Strain ky kx kz strained silicon • Envelope equation contains an effective mass, but no crystal potential. • Potentials assumed to be slowly varying. Valley Coupling Ec V(r) kz F(k) central cell interaction F(r) shallow donor • Interaction in k-space is due to sharp confinement in real space. • Effective mass theory still valid, away from confinement singularity. • On EM length scales, singularity appears as a delta function: Vvalley(r) ≈ vv (r) • Valley coupling involves wavefunctions evaluated at the singularity site: F(0) Si (5.43 nm) Si0.7Ge0.3 Si0.7Ge0.3 (160 meV) |(z)|2 Valley Splitting in a Quantum Well cos(kmz) sin(kmz) Interference Two -functions Interference between interfaces causes oscillations in Ev(L) Tight Binding Approach dispersion relation Boykin et al., 2004 |(z)|2 Si (5.43 nm) Si0.7Ge0.3 Two-band TB model captures 1) Valley center, km 2) Effective mass, m* 3) Finite barriers, Ec Si0.7Ge0.3 (160 meV) confinement Calculating Input Parameters 2-band TB many-band theory • Excellent agreement between EM and TB theories. • Only one input parameter for EM • Sophisticated atomistic calculations give small quantitative improvements. Valley splitting [μeV] Ec L Boykin et al., 2004 Quantum Well in an Electric Field Effective Mass E Singleelectron Tight Binding asymmetric quantum well Self-consistent 2DEG from Hartree theory: Boykin et al., 2004 Miscut Substrate z z' • Valley splitting varies from sample to sample. • Crystallographic misorientation? (Ando, 1979) B x x' θ Barrier Quantum well Substrate Barrier s Magnetic Confinement Large B field Small B field interference • Valley splitting vanishes when B → 0. • Doesn’t agree with experiments for uniform steps. Valley Splitting, Ev F(x) -fn. at each step uniform steps experiment Magnetic Field, B Step Disorder Vicinal Silicon - STM Simulation Geometry a/4 [100] (Swartzentruber, 1990) step bunching 10 nm Simulations of Disordered Steps 8 T confinement 3 T confinement strong step bunching no step bunching 10 nm • Color scale: local valley splitting for 2° miscut at B=8T • Wide steps or “plateaus” have largest valley splitting. weak disorder Correct magnitude for valley splitting over a wide range of disorder models. Plateau Model • Linear dependence of Ev(B) depends on the disorder model • “Plateau” model scaling: “plateau” Ev ~ C/R2θ2 • Scaling factor (C) can be determined from EVR Confinement models: R ~ LB (magnetic) R ~ Lφ (dots) Valley Splitting in a Quantum Dot Volts Electrostatics 0.5 μm Predicted valley splitting = 90 μeV (2° miscut) = 360 μeV (1° miscut) ~ 600 μeV (no miscut) ~ 400 μeV (1e) ground state 50 nm Rrms = 19 nm (~4.5 e) 100 nm Energy [meV] Stark Effect in P:Si – Valley Mixing • 3 input parameters are required from spectroscopy. • Only envelope functions depend on electric field. Stark Shift spectrum narrowing 0 • Electric field reduces occupation of the central cell. • Ionization re-establishes 6-fold degeneracy. Conclusions 1. 2. 3. 4. Valleys are coupled by sharp confinement potentials. Valley coupling potentials are -functions, with few input parameter. Bare valley splitting is of order of 1 meV. (Quantum well) Steps suppress valley splitting by a factor of 1-1000, depending on the B-field or lateral confinement potential. F(x) 5. For shallow donors, the Stark effect causes spectrum narrowing. spectrum narrowing Acknowledgements Theory (UW-Madison): Prof. Susan Coppersmith Prof. Robert Joynt Charles Tahan Suchi Chutia Experiment (UW-Madison): Prof. Mark Eriksson Srijit Goswami Atomistic Simulations: Prof. Gerhard Klimeck (Purdue) Prof. Timothy Boykin (Alabama) Paul von Allmen (JPL) Fabiano Oyafuso Seungwon Lee