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

nm

Si substrate

Si95Ge05

Si90Ge10

Si85Ge15

Si80Ge20

SiSi80Ge20

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

sta

tes

B

fiel

d

Co

nfi

nem

ent

En

erg

y

Zeeman Splitting

Valley Splitting

En

erg

y

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

Ene

rgy

[meV

]

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

[μ

eV]

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'

xθ

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,

Ev

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 confinement

3 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.

Ene

rgy

[meV

]

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.

Acknowledgements

Theory (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