Orbital Stark Shift of donor-interface states
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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)
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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
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Rajib Rahman
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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)
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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?
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Rajib Rahman
Stark Shift of Hyperfine Interaction
ES
ETe
nA(ε) |(ε, r0)|2
Contact HF:
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HA = I • ˆ A (ε,r0) • S
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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
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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?
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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:
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E = ψ Hψ Hyperfine:
Application: Experimentally mapping WF deformations (idea: L. Hollenberg)
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Rajib Rahman
Stark Shift of the donor g-factor
Zeeman effect:
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HZ = g(ε)μB B B-field response => g-factor
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HSO = Cσ • ∇V × pSpin-orbit (LS) interaction: very important in QC
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→ ψ → L → Hso → S ⇒ Δgg-factor Stark shift: Indirect measure of SO
ε [010] Si:P
Anisotropic Zeeman Effect
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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
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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
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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)
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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
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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
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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
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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
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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
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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).
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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)
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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
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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
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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)
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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)
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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
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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)
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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