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 StructuresSanremo, March 6-8, 2006
“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)
(Fowler, et al., 1966)
(Nicholas, von Klitzing, & Englert, et al., 1980)
Valley Splitting: An Old Problem
New Methodology, New Directions
Different Tools:• New tight binding
tools• New effective mass
theory
Different Materials:• Si/SiGe
heterostructures
500
nmSi substrate
Si95Ge05
Si90Ge10
Si85Ge15
Si80Ge20SiSi80Ge20
Different Knobs:• Microwaves• QD and QPC
spectroscopy• (No MOSFET gate)
200 nm
Different Motivation:• Qubits• Single electron limit• Small B fields
J 0
Uncoupled
J > 0
Swap
Quantum Computing with Spins
Open questions:• Well defined qubits?• Wave function oscillations?
Electron density for P:Si(Koiller, et al., 2004)
Orb
ital s
tate
s
B f
ield
Con
finem
ent
Ener
gy
Zeeman Splitting
Valley Splitting
Ener
gy
qubit
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
Ener
gy [m
eV]
Theory Li P
P:Si
Electron Valley Resonance (EVR)
Motivation for an Effective Mass Approach
• Valley states have same envelope• Valley splitting small, compared to orbital• Suggests perturbation theory
|(z)
|2
Si (5.43 nm)
Si 0.
7Ge 0
.3
(160
meV
)
Si 0.
7Ge 0
.3
2-2+1-1+
Effective Mass Theory in Silicon
kx
ky
kz
bulk siliconvalleymixing
incommensurateoscillations (fast)
Bloch fn.(fast)
envelopefn. (slow)
Ec
kz
Fz(k) • Kohn-Luttinger effective mass theory relies on separation of fast and slow length scales. (1955)
• Assume no valley coupling.
Effect of Strain
kx
ky
kz
strained silicon• Envelope equation contains
an effective mass, but no crystal potential.
• Potentials assumed to be slowly varying.
Valley Coupling
V(r)
F(r)
central cell
Ec
kz
interaction
F(k)
• 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
Valley Splitting in a Quantum Well
Si (5.43 nm)
Si 0.
7Ge 0
.3
(160
meV
)
Si 0.
7Ge 0
.3
|(z)
|2
cos(kmz)sin(kmz)
Two -functionsInterference
Interference between interfaces causes oscillations in Ev(L)
Tight Binding Approach
dispersion relation
Boykin et al., 2004
Si (5.43 nm)
Si 0.
7Ge 0
.3
(160
meV
)
Si 0.
7Ge 0
.3
|(z)
|2
confinement
Two-band TB model captures
1)Valley center, km
2)Effective mass, m*3)Finite barriers, Ec
Calculating Input Parameters
L
Ec
• Excellent agreement between EM and TB theories.
• Only one input parameter for EM• Sophisticated atomistic calculations
give small quantitative improvements. Boykin et al., 2004
2-band TBmany-band theory
Val
ley
split
ting
[μeV
]
E
Quantum Well in an Electric Field
Tight Binding
Effective Mass
Boykin et al., 2004
Single- electron
Self-consistent 2DEG from Hartree theory:
asymmetric quantum well
Miscut Substrate
• Valley splitting varies from sample to sample.
• Crystallographic misorientation? (Ando, 1979)
Quantum well
Barrier
Barrier
z z'
x'x
θ
B
s
Substrate
Magnetic Confinement
Large B field
Small B field
F(x)-fn. at
each stepinterference
experiment
uniform steps
Val
ley
Spl
ittin
g, E
v
Magnetic Field, B
• Valley splitting vanishes when B → 0.
• Doesn’t agree with experiments for uniform steps.
Step Disorder
10 nm
a/4
step bunching
[100]
(Swartzentruber, 1990)
Vicinal Silicon - STMSimulationGeometry
Simulations of Disordered Steps
Correct magnitude for valley splitting over a wide range of disorder models.
strong step bunching
no step bunching
weak disorder10 nm
• Color scale: local valley splitting for 2° miscut at B = 8 T
• Wide steps or “plateaus” have largest valley splitting.
8 T confinement3 T confinement
Plateau Model
“plateau”
• Linear dependence of Ev(B) depends on the disorder model
• “Plateau” model scaling:
• Scaling factor (C) can be determined from EVR
Ev ~ C/R2θ2
Confinement models: R ~ LB (magnetic) R ~ Lφ (dots)
Valley Splitting in a Quantum Dot
0.5 μm
100 nm
VoltsElectrostatics
50 nm
Rrms = 19 nm (~4.5 e)
groundstatePredicted valley
splitting = 90 μeV (2° miscut)= 360 μeV (1° miscut)~ 600 μeV (no miscut)~ 400 μeV (1e)
Stark Effect in P:Si – Valley Mixing
• 3 input parameters are required from spectroscopy.
• Only envelope functions depend on electric field.
Ener
gy [m
eV]
Stark Shift
spectrumnarrowing
• Electric field reduces occupation of the central cell.
• Ionization re-establishes 6-fold degeneracy.
0
Conclusions
F(x)
spectrum narrowing
5. For shallow donors, the Stark effect causes spectrum narrowing.
1. Valleys are coupled by sharp confinement potentials.2. Valley coupling potentials are -functions, with few input parameter.3. Bare valley splitting is of order of 1 meV. (Quantum well)4. Steps suppress valley splitting by a factor of 1-1000, depending on the
B-field or lateral confinement potential.
AcknowledgementsTheory (UW-Madison):Prof. Susan CoppersmithProf. Robert JoyntCharles TahanSuchi Chutia
Experiment (UW-Madison):Prof. Mark ErikssonSrijit Goswami
Atomistic Simulations:Prof. Gerhard Klimeck (Purdue)Prof. Timothy Boykin (Alabama) Paul von Allmen (JPL)Fabiano OyafusoSeungwon Lee
