Rajib Rahman
Stark Tuning of Electronic Properties of Impurities for Quantum Computing Applications
Rajib Rahman
Advisors:
Gerhard Klimeck
Lloyd Hollenberg
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
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
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)
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)
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
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, …
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
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
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
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?
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
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
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
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
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)
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
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
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
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
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 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
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?
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
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?
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
Rajib Rahman
Methods and Details
Tight-binding and NEMO3D
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)
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
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
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
