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
J0
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