Rajib Rahman

Orbital Stark Shift of donor-interface states

Lansbergen, Rahman, GK, LH, SR, Nature Physics, 4, 656 (2008)

ε

Oxide-Si-impurityOxide-Si-impurity

ε=0

Donor-interface system

Smit et al. PRB 68 (2003)Martins et al. PRB 69 (2004)Calderon et al. PRL 96 (2006)

Rajib Rahman

Transport through donor statesDevice E1 (meV) E2 (meV) E3 (meV)

10G16 2 15 23

11G14 4.5 13.5 25

13G14 3.5 15.5 26.4

HSJ18 5 10 21.5

GLG14 1.3 10 13.2

GLJ17 2 7.7 15.5

Energies w.r.t. ground state (below CB)

Exp. Measurements

• Energies different from a bulk donor (21, 23, 44)

• Donor states – depth & field dependent

Orbital Stark Shift of donor-interface states

Rajib Rahman

Rajib Rahman

Friesen, PRL 94 (2005)

Si:P (Bulk)

A B

C

Si:As (Depth 7a0)

Features found• 3 regimes • Interface effects• anti-crossing• p-manifold• valley-orbit

Orbital Stark Shift of donor-interface states

A (Coulomb bound)

Rahman, Lansbergen, GK, LH, SR (Orbital Stark effect theory paper, to be submitted)

B (Hybridized) C (Surface bound)

Rajib Rahman

Stark Effect in donor-interface well

Lansbergen, Rahman, GK, LH, SR, Nature Physics (2008), IEDM (2008)

• Interpretation of Exp.• Indirect observation of symmetry transition• P vs As Donor distinction

Exp data with TB simulations Where are the exp. points?

Rajib Rahman

Stark Shift of Hyperfine Interaction

ES

ETe

nA(ε) |(ε, r0)|2

Contact HF:

€

HA = I • ˆ A (ε,r0) • S

€

r0 => Nuclear spin site => Impurity site

∆A(ε)/A(0) = 2ε2 (bulk)

Theory: Rahman et al. PRL. 99, 036403 (2007) Exp: Bradbury et al., PRL 97, 176404 (2006)

BMB

TB

∆A(ε)/A(0) = (2ε2 + 1ε) (interface)

D

oxide Donor

Rajib Rahman

Why linear Stark Effect near interfaces?

Asymmetry in wf

0yyEcorrection 1st order PT:

Oxide-Si-impurity

Small Depth:

Large Depth:

Even symmetry broken

Rahman et al. PRL. 99, 036403 (2007)

Stark Shift of Hyperfine Interaction

Quadratic Stark Coefficients

Method Depth(nm) 2(µm2/V2)

EXP (Sb) 150 -3.7x10-3 -3

EMT (P) ∞ -2x10-2 -2

BMB (P) 10.86 -2.74x10-3 -3

TB (P) 10.86 -2.57x10-3 -3

21.72 -2.76x10-3 -3

EMT: Friesen, PRL 94, 186403 (2005)

How good are the theories?

Rajib Rahman

Hyperfine Map of Donor Wave-functions

Park, Rahman, Klimeck, Hollenberg (submitted)

ESR Experiments can measure A => Direct measure of WF

Usefulness of HF – an example

€

A(ε,r0) = C | Ψ(ε,r0) |2

29Si (S=1/2)28Si (S=0)Si isotopes:

Observables in QM:

€

E = ψ Hψ Hyperfine:

Application: Experimentally mapping WF deformations (idea: L. Hollenberg)

Rajib Rahman

Stark Shift of the donor g-factor

Zeeman effect:

€

HZ = g(ε)μB B B-field response => g-factor

€

HSO = Cσ • ∇V × pSpin-orbit (LS) interaction: very important in QC

€

→ ψ → L → Hso → S ⇒ Δgg-factor Stark shift: Indirect measure of SO

ε [010] Si:P

Anisotropic Zeeman Effect

Rajib Rahman

Stark Shift of the donor g-factor

Multi-valley g to single-valley g transition in Si (g||-g|_=8e-3)

Rahman, Park, GK, LH (Gate induced g-factor control, to be submitted)

Impurity E || valley-axis 2

Si:P B|| -1.0x10-5

Ge:P B|| 1.43x10-1

GaAs:Si B|| -9.4x10-3

Quadratic Stark Shift (bulk):

∆g(ε)/g(0) = 2ε2

Conclusions

• SO strength• valley-structure• anisotropic Zeeman• single-valley anisotropy• Exp. Magnitude verified

Rajib Rahman

Vs1=0.05V Vs1=0.1V

E1

E2

E1

E2

E1

E2

Vs1=0.3VVs1=0.0V

E1

E2

Vs1=0.4V

E1

E2

P P+ P+15 nm

15 nm

Vs1 Vb1 Vb2 Vs2V=0 V>0

Electrostatic gating of single donors

Nano-TCAD+TB

Rajib Rahman

Coherent Tunneling Adiabatic Passage (CTAP)

• Solid-state analogue of STIRAP (Quantum Optics), Greentree et al., PRB 70 (2004)• Molecular states: no middle donor occupation• Pathways in Eigen-space connecting end states• Spin state transport• Many-donor chain: Less gating, more robust

Purpose (NEMO-CTAP):• Relax assumptions• Real solid-state system: bands, interfaces, excited states, gate-cross talk, realistic donor models• Does the adiabatic path exist ?

Quantum Info Transport

Hollenberg et al., PRB 74 (2006)

Rajib Rahman

Anti-crossing gap => tunneling times

Barrier gate modulation

Rahman, Park, GK, LH (Atomistic simulations of CTAP, in prep.)

|Ψ2|2 at various voltages

Left localized

Middle stage

Right localized

No population at center donor any time

Atomistic simulations of CTAP3

Rajib Rahman

Donor Based Charge Qubits

S B

P+

P0

TCAD Gate Molecular States

Sensitivity to impurity positioning• Molecular states of P2+ encode info

• Proposal: Hollenberg • EMT work: X. Hu, B. Koiller, Das Sarma

• Tunnel Coupling: 12 = E2 - E1

• Excited states ignored so far: 23 = E3 – E2

TB result similar to EMT

Rajib Rahman

Control of Charge Qubits

Goal: • Establish limiting conditions

for operation• Characterize gate control• Explore design parameter space

Molecular Spectrum

Surface Gate Control

Some Findings • R > 8 nm• Smooth Control• Surface Ionization• Saturated regime• 12 = 23

Rajib Rahman

Surface Gate Control of Charge Qubits

V=0

Wf 1 Wf 2

V=0.2

V=-0.2

V=0.5

Wf 1 Wf 2

V=-0.5

Saturation

Linear

Ionization

Bonding

Rajib Rahman

Many-body Interactions in Donor Qubits

e e

P+P+R=|R2-R1|

2e Hamiltonian

Koiller, Hu, Das Sarma, PRL 88, No 2 (2002)

Kane Qubit: Two qubit interaction

• Exchange coupling J between donors

• Modify WF overlap by gate voltage

Known facts:• J oscillates with R (Koiller)• SiGe strain can reduce oscillations (conditionally)

(Koiller)• Gate control smooth (mostly) – Wellard, Hollenberg • A BMB (Wellard) work showed reduced oscillatons.

Goal: • TB wfs, extended band structure, VO interaction• Beyond Heitler-London (CI)• Effect of strain, interfaces, gates• Other systems: spin-measurement

Rajib Rahman

Exchange Interaction in Heitler-London Formalism

)]()()()([2

1)]()()()([),( 21212121212 ssssrrrrNrr LRRLsS ψ Singlet:

Triplet:

Many-body wfs must be anti-symmetric w.r.t. interchange of r and s

)]()()()([2

1)]()()()([),( 212121212212 ssssrrrrNrr LRRLTT ψ

P

Vb

L Dot

P

Vb

R Dot

Basis: 2P

Vb

2 electron system

HL valid for “large” donor separations

TB

Voltage Controllability Problem.

Similar result in EMT: Wellard et al., J. Phys.: Condens. Matter 16 5697–5704 (2004).

Rajib Rahman

Conf. 1 Conf. 2 Conf. 3

Conf. 4 Conf. 5 Conf. 6

• Example: 4 states

• 4 choose 2 Many Body configurations

• 6 x 6 CI Hamiltonian

Possible Future Work with CI• P-P Molecular Spectrum

• D- State of P: Charging Energy

• Donor-Interface 2e problem (spin read-out prop. by Kane)

Exchange Interaction in FCI Formalism (on-going)

New goal: Refine HL by including spin, HM states (TCI)

Rajib Rahman

Hyperfine Stark Effect of P-Impurities Objective:• Study Stark Shift of hyperfine coupling• Compare with experiment, BMB & EMT• Investigate interface effects • Establish the physics of quadratic and

linear Stark coefficientsApproach:• Use 3.5 M atomistic domain P impurity

under E-fields• TB approach optimized for P donors• Vary impurity depth from interface• Solve the 20 band spin Hamiltonian by

parallel Lanczos algorithm Results / Impact:• Quadratic Stark coefficient from TB, BMB &

experiment agree well• EMT estimate differs by an order of

magnitude• Proximity of impurity to interface produces

significant linear Stark effect

Method Depth (nm)

2 (µm2/V2)

EXP (Sb) 150 -3.7x10-3

EMT (P) ∞ -2x10-2

BMB (P) 10.86 -2.74x10-3

TB (P) 10.86 -2.57x10-3

21.72 -2.76x10-3

Quadratic Stark Coefficients

Rahman et al. PRL. 99, 036403 (2007)

wavefunction change with E field

Hyperfine coupling in E field / depths

Rajib Rahman

Hyperfine maps of donor wave functions

Challenge / Objective:• Can a single impurity donor wavefunction(wf) be

experimentally mapped?

Approach:• Indirectly probe wfs by measuring Hyperfine

tensors (idea: L. Hollenberg).• Use Si-29 as a single probe atom or a sample of

probe atoms • Calculate donor wfs in realistic geometries and

electric fields• Propose experiment:

Distort wf by electric fields and interfaces => distort HF => measure HF based on lattice symmetries=> map the wavefunction

Results / Impact:• Probe local values of WF instead of global

expectation values• Demonstrated distortion of the WF through its

hyperfine map• Verified feasibility of detecting such distortions.

Fermi term Dipolar term

Park, Rahman, GK, LH, Rogge (paper submitted)

28Si host, 29Si probe

Rajib Rahman

Gate control of donor g-factors and dimensional isotropy transition

Objective:• Investigate Stark Shift of the donor g-factor. • g-factor shift for interface-donor system.• Probes spin-orbit effects with E-fields and symmetry transition.

• Relative orientations of B and E field.Approach:• The 20 band nearest neighbor sp3d5s* spin model captures SO interaction of the host.

• Same atom p-orbital SO correction• g-factor obtained from L and S operators. • Donor wfs with E-field are obtained from NEMO

Results / Impact:

• Quadratic trend with E-field for bulk donors.• Stark parameter larger in Ge and GaAs• Anisotropic Zeeman effect – E and B field• Dimensional transition- multi-valley to single valley g-factors.

• Exp. Quadratic coef. matches in magnitude.

Si:P

Rahman, Park, GK, LH (to be submitted)

Rajib Rahman

Coherent Tunneling Adiabatic Passage (CTAP)

Objective:• Investigate CTAP in realistic setting.• Include Si full band-structure, TCAD gates, interfaces, excited states, cross-talk.

• Verify that adiabatic path exists: 3 donor device.

Approach:• TCAD gates coupled with a 3 donor TB. Hamiltonian: obtain molecular states in the solid state.

• Simulate 3-4 M atoms for a realistic device.• Compute time of 4-5 hours on 40 procs.• Fine tune gate voltages to explore the CTAP. regime.

Results / Impact:• Demonstrated that the CTAP regime exists for a 3 donor test device.

• Verification of results (under relaxed assumptions)

• CTAP despite noisy solid-state environment.• Developed the framework to guide future CTAP expt.

Rahman, Park, GK, LH ( to be submitted)

Rajib Rahman

Charge qubit controlObjective:• Control & design issues: donor

depths, separation, gate placement. • Feasible S and B gate regimes.• Effect of excited states: charge state

superposition.

Approach:• S and B gates - TCAD potentials• Empirical Donor model + TB+ TCAD:

bound molecular states. • Lanczos + Block Lanczos solver

Results:• Smooth voltage control• excited states at higher bias mingle

with operation.• Placement of S and B gates important

relative to donors.• Comparison with EMT

RR, SHP, GK, LH (to be submitted)

Surface gate response of tunnel barriers

Molecular Spectrum + Tunnel barriers

Rajib Rahman

D- Modeling for As/P Donor

Objective:• Obtain 2e binding energy of donors with E-fields and donor depths: important in spin-dependent tunneling and measurement.

• D- ground and excited states : Analyze measured Coulomb diamonds from Transport Spectroscopy measurements.

Approach:• 1st approximation: SCF Hartree method.• Use a domain of 1.4 M atoms with 1 donor. • SCF: 1. Obtain wf from NEMO 2. Calculate electron density and Coulomb repulsion potential 3. Repeat NEMO with the new potential. 4. Stop when D- energy has converged.

• On-going: D- from configuration interaction Results:• D- energy for a bulk donor within 2 meV from measured value.

• D- vs. Depth & field calculations. • Explains charging energy of some samples• Screening likely to play a role.

D-, D0 vs E

D7a0

D- vs charging energy

D-

D0

-45.6

-4

Ec comparison

Rahman, Arjan, Park, GK, LH, Rogge (in prep)

Rajib Rahman

Control of exchange for adjacent qubits Objective:• Investigate gate control of exchange(vs EMT)• Reconfirm controllability issues (from BMB)• Treatment of interfaces & strain• From Heitler London to Full CIApproach:• atomistic basis for exchange calculations• orbital interactions for short distances• Interpolate TCAD potential on atomistic

lattice • Heitler-London scaled and tested for 4 M

atoms removing previous computational bottlenecks.

• FCI is still a computational challenge

Results / Impact:• Similar exchange trends obtained as BMB• Controllability issues at some specific

angular separations verified• Magnitude an order less from EMT• Basis functions for short range interactions?

J(V) for various impurity separations along [100]

Sensitivity of J(V) to donor placement