Valley Splitting Theory for Quantum Wells and Shallow Donors in Silicon

1. 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

2. 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)

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

4. Confinement B field Orbital states Energy Energy Zeeman Splitting Valley Splitting Quantum Computing with Spins qubit Electron density for P:Si (Koiller, et al., 2004) • Open questions: • Well defined qubits? • Wave function oscillations?

5. Theory Li P Energy [meV] P:Si 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 Electron Valley Resonance (EVR)

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

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

8. ky kx kz strained silicon Effect of Strain • Envelope equation contains an effective mass, but no crystal potential. • Potentials assumed to be slowly varying.

9. Ec kz Valley Coupling V(r) 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

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

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

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

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

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

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

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

17. 8 T confinement 3 T confinement strong step bunching no step bunching weak disorder 10 nm • Color scale: local valley splitting for 2° miscut at B = 8 T • Wide steps or “plateaus” have largest valley splitting. Correct magnitude for valley splitting over a wide range of disorder models. Simulations of Disordered Steps

18. 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)

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

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

21. 0 Stark Shift spectrum narrowing • Electric field reduces occupation of the central cell. • Ionization re-establishes 6-fold degeneracy.

22. F(x) spectrum narrowing 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. • For shallow donors, the Stark effect causes spectrum narrowing.

23. 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