stark tuning of electronic properties of impurities for ......rajib rahman central issues 1. single...
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Rajib Rahman
Stark Tuning of Electronic Properties of Impurities for Quantum Computing Applications
Rajib Rahman
Advisors:
Gerhard Klimeck
Lloyd Hollenberg
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Rajib Rahman
Single Donors in Semiconductors
Motivation
• Shrinking device size • Quantum mechanics of donors • Donors provide 3D confinement
to electrons • Analogous to Quantum Dots • Can we control quantum
properties of single donors ?
Devices with few impurities
Lansbergen, Delft Andresen, UNSW
Kane Qubit
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Rajib Rahman
Quantum Computing
Idea: • Encode information in quantum states. • Manipulate information by controlled
perturbation of states. • Classical Computing: |0> or |1> • Quantum Computing: a|0> + b|1>
Bloch Sphere
Advantages: • Quantum parallelism (speed) • Algorithms: Quantum search, Fourier
Transform • Applications: cryptography, simulations,
factoring, database search, etc.
Design criteria (DiVincenzo): • Isolation of the qubit Hilbert Space • Decoherence times • Ease of measurement
• Scalability (Hollenberg, PRB 74) • Fault-tolerant designs
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Rajib Rahman
Quantum Computing Implementations
Vandersypen et al., July 2000 PRL
NMR 5 qubit (IBM) Ion Traps
http://www.uni-ulm.de/qiv/ forschung/ControlAndMeasurementE.html
Quantum Optics
Gasparani et al., PRL 93, No. 2 (2004)
Cavity QED Mckeever, Science Express Reports (Feb 26, 2004)
SQUID Oliver etal., Sceince 310, 1653 (2005)
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Rajib Rahman
Solid State Qubits
Ion Trap, eg. (http://www.uni-ulm.de/qiv/)
Scalability ?
Solid State (QDs, Donors, Si QW)
Donor Qubits Benefits: • Industry experience in Si:P • Long coherence • Scalability
Problems: • Precise donor placement (1 nm) • Control is sensitive
Donor Charge Qubit (Hollenberg)
Electron Spin (Vrijen)
Si – SiGe Quantum Wells (Friesen)
Nuclear spin qubit (Kane)
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Rajib Rahman
P Donor Qubits in Si
Charge Qubit (Hollenberg)
Charge Qubit • Molecular states of P2+ • Control electron localization by S & B gates • Information transport - CTAP
Spin Qubits (Kane, Vrijen, Hill)
Spin Qubit • Single Qubit: Hyperfine (A ) + Zeeman (g) • Two-qubit: Exchange J(V) • Tunable by gates
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Rajib Rahman
Si
Si
Si
P+
Si
Si
Si
Si
Si
e- Conventional Picture
CB
Donor ED
ED(P) = -45.6 meV
ED(As) = -54 meV
Simple Model
• Coulomb potential screened by Si
• Hydrogen analogy: 1s, 2s, 2p …
• Si Band Structure: Bloch Functions, valley degeneracy
• Valley-orbit interaction – binding energy varies from donor to donor
Quantum Picture
CB
ED Donor QD
Donor Physics 101
EMT: Kohn-Luttinger, Das Sarma, Koiller, Hollenberg, Friesen, …
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Rajib Rahman
Central Issues
1. Single Donor Spin Control A. Hyperfine Interaction B. g-factor control
2. Control of Charge States A. Orbital Stark Effect B. CTAP
3. Two Electron Interactions A. D- Modeling B. Exchange Interaction
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Rajib Rahman
Central Issues
1. Single Donor Spin Control A. Hyperfine Interaction
• Can we engineer the donor hyperfine interaction? • Can we resolve discrepancies between theory and exp.? • Is it possible to generate an experimentally detectable spatial map of a wf?
B. g-factor control • How does an E-field modify the Zeeman interaction in donors? • How does multi-valley structure affect g-factor? • Can we verify ESR measurements?
2. Control of Charge States A. Orbital Stark Effect B. CTAP
3. Two Electron Interactions A. D- Modeling B. Exchange Interaction
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Rajib Rahman
Stark Shift of Hyperfine Interaction
ES
ET e
nA(ε) |Ψ(ε, r0)|2
Contact HF:
€
HA = I • ˆ A (ε,r0) • S
€
r0 => Nuclear spin site => Impurity site
∆A(ε)/A(0) = η2ε2 (bulk) Theory: Rahman et al. PRL. 99, 036403 (2007) Exp: Bradbury et al., PRL 97, 176404 (2006)
BMB
TB
∆A(ε)/A(0) = (η2ε2 + η1ε) (interface)
D
oxide Donor
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Rajib Rahman
Why linear Stark Effect near interfaces?
Asymmetry in wf
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) η2(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:
€
E = ψ Hψ Hyperfine:
Application: Experimentally mapping WF deformations (idea: L. Hollenberg)
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Rajib Rahman
Central Issues
1. Single Donor Spin Control A. Hyperfine Interaction
• Can we engineer the donor hyperfine interaction? • Can we resolve discrepancies between theory and exp.? • Is it possible to generate an experimentally detectable spatial map of a wf?
B. g-factor control • How does an E-field modify the Zeeman interaction in donors? • How does multi-valley structure affect g-factor? • Can we verify ESR measurements?
2. Control of Charge States A. Orbital Stark Effect B. CTAP
3. Two Electron Interactions A. D- Modeling B. Exchange Interaction
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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)
Interface: g||-g|_=8e-3
1e-5
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Rajib Rahman
Central Issues 1. Single Donor Spin Control
A. Hyperfine Interaction B. g-factor control
2. Control of Charge States A. Orbital Stark Effect
• Can we explain single donor tunneling expt? • Can we infer info about donor species and location in devices through atomistic
modeling? • Can we indirectly observe symmetry transition of a 3D electron to 2D?
B. CTAP • Can we control tunnel barriers between donors by realistic gates? • Does there exist adiabatic pathways connecting end states for transport? • Can we develop a framework to guide expts?
3. Two Electron Interactions A. D- Modeling
B. Exchange Interaction
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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-impurity Oxide-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 states Device 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
Central Issues 1. Single Donor Spin Control
A. Hyperfine Interaction B. g-factor control
2. Control of Charge States A. Orbital Stark Effect
• Can we explain single donor tunneling expt? • Can we infer info about donor species and location in devices through atomistic
modeling? • Can we indirectly observe symmetry transition of a 3D electron to 2D?
B. CTAP • Can we control tunnel barriers between donors by realistic gates? • Does there exist adiabatic pathways connecting end states for transport? • Can we develop a framework to guide expts?
3. Two Electron Interactions A. D- Modeling
B. Exchange Interaction
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Rajib Rahman
Vs1=0.05V Vs1=0.1V
E1
E2
E1
E2
E1
E2
Vs1=0.3V Vs1=0.0V
E1
E2
Vs1=0.4V
E1
E2
P P+ P+ 15 nm
15 nm
Vs1 Vb1 Vb2 Vs2 V=0 V>0
Electrostatic gating of single donors
Nano-TCAD+TB
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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 control Objective: • 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
Central Issues
1. Single Donor Spin Control A. Hyperfine Interaction B. g-factor control
2. Control of Charge States A. Orbital Stark Effect B. CTAP
3. Two Electron Interactions A. D- Modeling
• Can we interpret the D- state probed by expts? • How does the charging energy vary with donor depth and field?
B. Exchange Interaction • Does the exchange coupling for two qubit operations suffer from
controllability issues, as shown by EMT?
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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
Central Issues
1. Single Donor Spin Control A. Hyperfine Interaction B. g-factor control
2. Control of Charge States A. Orbital Stark Effect B. CTAP
3. Two Electron Interactions A. D- Modeling
• Can we interpret the D- state probed by expts? • How does the charging energy vary with donor depth and field?
B. Exchange Interaction • Does the exchange coupling for two qubit operations suffer from
controllability issues, as shown by EMT?
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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 CI Approach: • 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
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Rajib Rahman
Methods and Details
Tight-binding and NEMO3D
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Rajib Rahman
Methods & Some Details
• Tight Binding: sp3d5s* NN model (NEMO3D)
• Typical Domain: 3-4 M atoms
• Typical Resources: 40 processors
• Compute Times: Single electron 6-8 hours
• Solver – parallel Lanczos / Block Lanczos (degenerate or closely spaced states)
• Electrostatic modeling – TCAD + NEMO
• Two electron integrals: STOs, Monte Carlo, off-site coulomb from Ohno formula.
NEMO Scaling (G. Klimeck)
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Rajib Rahman
TB parameterization of Donor
6
1
2 3
Mayur, et al., PRB 48, No. 15 (1993)
Es Ep
Ed Es*
Orbital based shift:
On-site energy corrections
Shift all orbitals by U0
TB
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Rajib Rahman
Conclusions
Hyperfine Interaction: • Verified ESR measurements • Characterized E-field control and interface effects • Proposed expt. to measure wf at different lattice sites
G-factor Control: • Verified ESR measurements • Characterized E-field control, interface and band-structure effects • Showed dimensional transition can probe single valley g-factors
Orbital Stark Effect: • Used atomistic modeling to interpret transport data • Performed dopant metrology through modeling • Demonstrated indirect symmetry transition and quantum control
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Rajib Rahman
Conclusions Coherent Tunneling: • Demonstrated Gate control of single donors with TCAD • Found adiabatic path for electron transfer • Developed framework to guide future CTAP expts
Charge Qubit Design: • Established the engineering variables for a donor charge qubit • Established the effect of excited states on performance limits
D- state Modeling: • Established the effect of field and depth on the 2nd bound donor electron • Understanding of the D- states may lead to realization of spin-dependent
tunneling in donor.
Exchange Interaction: • Atomistic exchange calculation also verify the basic EMT exchange results