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. 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

4. 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?

5. 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)

6. 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

7. 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)

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

9. 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

10. 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)

11. 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

12. 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.

13. 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

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

15. 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.

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

17. 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

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. 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)

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

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

22. 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 , depending on the B-field or lateral confinement potential.
F(x)
For shallow donors, the Stark effect causes spectrum narrowing.
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