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 500 nm Si substrate Si95Ge05 Si90Ge10 Si85Ge15 Si80Ge20 Si Different Knobs: Microwaves QD and QPC spectroscopy (No MOSFET gate) 200 nm Different Tools: New tight binding tools New effective mass theory Different Motivation: Qubits Single electron limit Small B fields J  0 Uncoupled J > 0 Swap

Electron density for P:Si Quantum Computing with Spins Orbital states B field Confinement Energy Zeeman Splitting Valley Splitting Energy qubit 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: Effective mass theory (and tight binding) Effect steps and magnetic fields in a QW Stark effect for P:Si donors Energy [meV] Theory Li P P:Si Electron Valley Resonance (EVR)

Motivation for an Effective Mass Approach |(z)|2 Si (5.43 nm) Si0.7Ge0.3 (160 meV) 2- 2+ 1- 1+ Valley states have same envelope Valley splitting small, compared to orbital Suggests perturbation theory

Effective Mass Theory in Silicon incommensurate oscillations (fast) envelope fn. (slow) kx ky kz bulk silicon valley mixing Bloch fn. (fast) Ec kz Kohn-Luttinger effective mass theory relies on separation of fast and slow length scales. (1955) Assume no valley coupling. Fz(k)

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) 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) shallow donor

Interference between interfaces causes oscillations in Ev(L) Valley Splitting in a Quantum Well Si (5.43 nm) Si0.7Ge0.3 (160 meV) |(z)|2 cos(kmz) sin(kmz) Two -functions Interference Interference between interfaces causes oscillations in Ev(L)

Tight Binding Approach dispersion relation Boykin et al., 2004 Si (5.43 nm) Si0.7Ge0.3 (160 meV) |(z)|2 confinement Two-band TB model captures Valley center, km Effective mass, m* Finite barriers, Ec

Calculating Input Parameters Ec Boykin et al., 2004 2-band TB many-band theory Valley splitting [μeV] Excellent agreement between EM and TB theories. Only one input parameter for EM Sophisticated atomistic calculations give small quantitative improvements.

Quantum Well in an Electric Field Effective Mass Self-consistent 2DEG from Hartree theory: asymmetric quantum well E Single- electron Tight Binding Boykin et al., 2004

Miscut Substrate Quantum well Barrier z z' x' x θ B s Substrate Valley splitting varies from sample to sample. Crystallographic misorientation? (Ando, 1979)

Magnetic Confinement Large B field Small B field -fn. at each step interference Large B field Small B field F(x) experiment uniform steps Valley Splitting, Ev Magnetic Field, B Valley splitting vanishes when B → 0. Doesn’t agree with experiments for uniform steps.

Step Disorder Simulation Vicinal Silicon - STM Geometry a/4 [100] (Swartzentruber, 1990) Vicinal Silicon - STM 10 nm a/4 step bunching

Simulations of Disordered Steps 10 nm Color scale: local valley splitting for 2° miscut at B = 8 T Wide steps or “plateaus” have largest valley splitting. 8 T confinement 3 T confinement Correct magnitude for valley splitting over a wide range of disorder models. strong step bunching no step bunching weak disorder

Plateau Model Linear dependence of Ev(B) depends on the disorder model “Plateau” model scaling: Scaling factor (C) can be determined from EVR Ev ~ C/R2θ2 “plateau” Confinement models: R ~ LB (magnetic) R ~ Lφ (dots)

Predicted valley splitting Valley Splitting in a Quantum Dot 0.5 μm Volts Electrostatics 100 nm 50 nm Rrms = 19 nm (~4.5 e) ground state Predicted valley splitting = 90 μeV (2° miscut) = 360 μeV (1° miscut) ~ 600 μeV (no miscut) ~ 400 μeV (1e)

Stark Effect in P:Si – Valley Mixing Energy [meV] 3 input parameters are required from spectroscopy. Only envelope functions depend on electric field.

Stark Shift spectrum narrowing Electric field reduces occupation of the central cell. Ionization re-establishes 6-fold degeneracy.

Conclusions 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) 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