S.N. CoppersmithUniversity of Wisconsin, Madison

• Building a Si quantum dot quantumcomputer

• Understanding valley splitting in Si/SiGequantum dots

Funding: ARO/NSA/ARDA, NSF

Building a silicon quantum-dotquantum computer: experiment,

simulation, and theory

UW QC collaboration (M. Eriksson, M. Friesen, et al.), andNEMO collaboration (P. von Allmen, T. Boykin, G. Klimeck, S. Lee)

• A computer obeying the laws of quantummechanics might solve problems a lot fasterthan one obeying laws of classical physics– Shor’s factoring algorithm (exponential (?))– Grover’s database search algorithm (√N vs N)– Quantum simulation (exponential)

Quantum versus classicalcomputing

Overview of our approach: Qubits are spins ofsingle electrons in quantum dots with variableelectron confinement

UW Si/SiGe quantumdot quantum computer

Modulation-dopedSi quantum well

Metal top gates

Graded SiGe

Si substrate

Back gate

Quantum dots

J ≅ 0

A

B

A

B

Control coupling between qubits via gate voltages

charge density maps at2 different gate voltages

swap

uncoupled

Silicon Advantages:• Coherence times much longer than GaAs

–Low spin-orbit coupling–Spin-zero nuclei (28Si)

Dot advantages:• Self-aligned to gates—no need to align to donors• Exchange coupling voltage-tunable• Fast operation• Scalable

Silicon Quantum Dot Disadvantages:• Need to work with relaxed SiGe heterostructures• Larger meff — smaller dots required• Heterostructure technology less mature

Advantages/limitations of Si Quantum Dots

Mark Eriksson (Physics)Mark Friesen (Physics)Robert Blick (ECE)Robert Joynt (Physics)Sue Coppersmith (Physics)Max Lagally (Materials Science)Dan van der Weide (ECE)Sucismita Chutia (Physics)Don Savage (Materials Science)Srijit Goswami (Physics)Levente Klein (Physics)Kristin Lewis (Physics)Lisa McGuire (Physics)Hui Qin (ECE)Nakul Shaji (Physics)Keith Slinker (Physics)Charles Tahan (Physics)

UW-Madison Solid-State QuantumComputing Team

NEMO collaboration:

Gerhard Klimeck (Purdue)Tim Boykin (UAH)Paul von Allmen (JPL)Fabiano Oyafuso (JPL)Seungwon Lee (JPL)

IBM Yorktown:

Pat MooneyJack Chu

Dartmouth:

Alex Rimberg

Progress in quantum dot fabrication

Schottky-gated device~1-3 electrons in dot

L.J. Klein et al., APL84, 4047 (2004).

L.J. Klein et al., APL90, 033103 (2007)

Source

Drain

1µm

G1

G2

Etched-gate device~100 electrons in dot

Device “Coulomb diamond”

0.5 µm

VoltsElectrostatics

100 nm50 nm

Rrms = 19 nm (~4.5 e)

groundstate

Simulations: how many electrons in the dot? (M. Friesen)

Theoretical issues

• Readout and initialization

• Sensitivity to imperfections

• Valley splitting(silicon is indirect-gap semiconductor)

Bulk Unstrained Silicon

Band Structure

Image : John H. Davies, Physics of Low Dimensional Semiconductors

Si band structure and conduction valley degeneracy

Bulk Unstrained Silicon

Band Structure

Image : John H. Davies, Physics of Low Dimensional Semiconductors

+X

+Y+Z

Si band structure and conduction valley degeneracy

conduction band minimum

Bulk Unstrained Silicon

Six Equivalent Valleys6-fold

degenerateEn

ergy

CB minimum

+X

+Y+Z

Strain and Valleys in Silicon

-Z

+ZSi1-xGex

6-fold degenerate

Ener

gy

CB minimum

4-fold degenerate

2-fold degenerate

Si

Bulk Strained Silicon

conduction through2 equivalent (+ Z) valleys

Strain and Valleys in Silicon

-Z

+ZSi1-xGex

6-fold degenerate

Ener

gy

CB minimum

4-fold degenerate

2-fold degenerate

Bulk Strained Silicon

+Z

conduction throughhybridized valleys

valley splittingΔv

Si1-xGexSi

Quantum confinement and electric fields lift the two-foldvalley degeneracy in Si/SiGe quantum wells

Simulation, theory, and experimentNEMO calculations with realistic atomicpotentials and smooth well surfacespredict large valley splitting

Analytic tight-binding calculations tohelp interpret NEMO results: suggestimportance of miscut quantum wells

Analytic effective mass theory tounderstand effect of steps in quantumwell

Experimentalmeasurements of

small valley splittingon Hall bars

NEMO calculations: small valleysplitting in miscut wells and identifyimportance of alloy disorder

Experimentalmeasurements of

large valley splittingin quantum point

contact

Valley splitting is ~1 meV: biggerthan spin splitting; big enough for QC

0.00

0.01

0.02

0.03

0.04

0.05

60 65 70 75 80 85 90 95 100

2-Band (Exact)

2-Band (Approx)

NEMO-spds

spds fit

E2

1 [m

eV]

Monolayers

10-4

10-3

10-2

10-1

100

101

10 100

0.0 mV/nm

1.04 mV/nm

2.08 mV/nm

3.12 mV/nm

4.16 mV/nm

E21 [

meV

]

Monolayers

Valley splitting in smooth quantum wells:

• Valley splitting even in zero electric field for thin quantumwells (splitting oscillates; envelope ∝ width-3)

• Magnitude for current quantum wells “should” be ~1 meV

•Simple, analytically solvable 2-band model agrees to within~20% with realistic NEMO calculations (40 orbitals/unit cell)

Width (Monolayers)

Spl

ittin

g (m

eV) NEMO

2-bandmodel

Width (monolayers)

Spl

ittin

g (m

eV)

electric fields:

T. Boykin, G. Klimeck, M. Eriksson, M. Friesen, SNC, P.von Allmen, F. Oyafuso, S. Lee, APL 84, 115-117 (2004).

0 1 2 3 40

200

400

600

800! =11 ! =7

Rxx

(")

B (T)

T=1.0 K

T=0.53 K

0 1 2 30

20

40

60

80

100

T = 0.53 K

T = 1.0 K

! v

(µeV

)

B (T)

Experimental data Linear Fit Estimate from SdH

S. Goswami, et al, Nature Physics 3, 41 (2007)

Val

ley

Spl

ittin

g (µ

eV)

R

• Valley splitting 10 to 100times smaller than predicted bytheory for smooth wells.• Many experiments all inagreement.

Microwave and SdH Measurements of Valley Splitting in Hall bars

Quantum well

SiGe

SiGe

z

M. Friesen, M.A.Eriksson, S.N. Coppersmith, Appl. Phys. Lett. 89, 202106 (2006).

Envelopes not shown

Hypothesis: atomic steps in the quantum wellreduce valley splitting

Wav

efun

ctio

nsin

a q

uant

um w

ell excited state

ground state

Interference from interfaces ontwo sides of a quantum well.

Friesen et al., cond-mat/0608229, PRB 75, 115318 (2007).

Valley splitting ina quantum well

effective masstight binding

Quantitative theory within a single plateau:

Quantum well

Barrier

Barrier

z z´

x´x

θ

s

Substrate 10 nm

8 T

B = 3 T

strong stepbunching

no stepbunching

weak disorder

Magnetic field (T)

Val

ley

split

ting

(µeV

)

Simulations of a2º miscut

Numerical simulation for random step profiles:

If magnetic confinement leads to larger valley splitting,then physical confinement should also increase the

valley splitting.

V. S

plitt

ing

(meV

)

Magnetic Field (T)Gate potential (V)

S. Goswami et al., Nature Physics 3, 41 (2007)

Experimentally measured valley splitting in aquantum point contact

Measured valley splitting is several tenths ofan meV, much bigger than the spin splitting

0.5 µm

100 nm

Electrostatics

50 nm

Rrms = 19 nm (~4.5 e)

groundstate

Predicted valleysplitting

= 90 µeV (2° miscut)= 360 µeV (1° miscut)~ 600 µeV (no miscut)

Predicted valley splitting in a quantum dot

Kharche et al., APL 90, 092109 (2007)

NEMO calculations of valley splitting in miscutquantum wells with atomistic alloy disorder

Kharche et al., APL 90, 092109 (2007)

Realistic alloy disorder increasesvalley splitting

Summary• Building a Si/SiGe quantum dot quantum computerinvolves solving problems amenable to concertedattack with coordinated experiment, simulation, andanalytic theory.• Experiments are key, since it matters whether thereare working devices at the end of the process.• Simulations and theory can make it more likely thatthe experimental devices work.

UW Si/SiGe QC groupNEMO collaboration

Support:

Acknowledgments

The end

Voltage on gate G1 (mV)40 30

Vol

tage

on

gate

G2

(mV

)

0.5

-0.5

0

50

Color Map of Source-DrainCurrent (red is more current)

Source

Drain

1µm

G1

G2

“Coulomb Diamond:” Coulomb Blockadein SiGe n-MODFET etched quantum dot

~100 electrons in dot

L.J. Klein et al., APL 84, 4047 (2004).

Source Drain

Gate

R R

CΣ

200 nm

TT

RRLL GG

CSCS

!"#$

!"#!

!"%$

!"%!

!"&$

!"&!

!"!$'())*+,!-+./

0%"!! 0&"1$ 0&"1! 0&"2$ 0&"2! 0&"3$ 0&"3!45!-4/

!"

!#

$"

$#

%"

%#

"

&'(()*+!,-./

0%123 0%124 0%12$ 0%12# 0%15567!,6/

Very few electron quantum dot

L. Klein et al., APL 90,033103 (2007)

Origin of valley splitting in squarewells

E

k

Two lowest energy eigenfunctions are each sums ofplane waves at 4 k-values ±k+, ±k-. Coefficientsdetermined by satisfying boundary conditions.

Values of k+, k- differ slightly for symmetric andantisymmetric states (can be found analytically).

-k+ -k- k- k+

T. Boykin, G. Klimeck, M. Eriksson, M. Friesen, SNC, P.von Allmen, F. Oyafuso, S. Lee, APL 84, 115-117 (2004).

T. Berer, D. Pachinger, G. Pillwein, M. Muhlberger, H. Lichtenberger, G.Brunthaler, F. Schaeffler: Lateral quantum dots in Si/SiGe realized by aSchottky split-gate technique, Appl. Phys. Lett. 88, 162112 (2006)

L.J. Klein, K.A. Slinker, J.L. Truitt, S. Goswami, K.L.M. Lewis, S.N.Coppersmith, D.W. van der Weide, M. Friesen, R.H. Blick, D.E. Savage,M.G. Lagally, C. Tahan, R. Joynt, M.A. Eriksson, J.O. Chu, J.A. Ott,P.M. Mooney: Coulomb blockade in a Si/SiGe two-dimensional electrongas quantum dot, Appl. Phys. Lett. 84, 4047–4049 (2004)

M.R. Sakr, H.W. Jiang, E. Yablonovitch, E.T. Croke: Fabrication andcharacterization of electrostatic Si/SiGe quantum dots with an integratedread-out channel, Appl. Phys. Lett. 87, 223104 (2005)

L.J. Klein, D.E. Savage, M.A. Eriksson: Coulomb blockade and Kondoeffect in a few-electron silicon/silicon-germanium quantum dot,Appl. Phys. Lett. 90, 033103 (2007)

K.A. Slinker, K.L.M. Lewis, C.C. Haselby, S. Goswami, L.J. Klein, J.O.Chu, S.N. Coppersmith, R. Joynt, R.H. Blick, M. Friesen, M.A. Eriksson:Quantum dots in Si/SiGe 2DEGs with Schottky top-gated leads,New Journ. Phys. 7, 246 (2005)

Recent Progress: Quantum Dots in Si/SiGe Quantum Wells

Importance of understanding valleysplitting in Si-based quantum computers

atomic-scale spatial oscillationsin charge density problemwith control of wavefunctionoverlap? (Koiller, Hu, DasSarma, PRL 88, 027903 (2002))

degeneracy decoherence?E

k

Our work:Heterostructure confinement lifts valley degeneracy

Strain + quantum confinement ⇒ wavefunction overlap varies smoothly

Si conduction band:

Spin and valley degeneracies havebeen lifted at finite field.

Tmix = 20mK

Each conductance step corresponds to a particular channel withunique orbital, valley, and spin combination.

Subband energy extraction: B. J. van Wees, et al., Phys. Rev. B 43, 12431 (1991).

Valley splitting: conductance steps at e2/h in QPC

Number of occupied channels:

Gate potential (V)

Mag

netic

fiel

d (T

)

N = 0 1

2

345

6 7>7

0

Quality of fit:

S. Goswami, et al, Nature Physics 3, 41 (2007)