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 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. ( 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 P+P+ e-e- Conventional Picture CB Donor EDED E D (P) = meV E D (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 EDED 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 ESES ETET e n A(ε) |  (ε, r 0 )| 2 Contact HF: => Nuclear spin site => Impurity site ∆A(ε)/A(0) =  2 ε 2 (bulk) Theory: Rahman et al. PRL. 99, (2007) Exp: Bradbury et al., PRL 97, (2006) BMB TB ∆A(ε)/A(0) = (  2 ε 2 +  1 ε) (interface) D oxideDonor

Rajib Rahman Why linear Stark Effect near interfaces? Asymmetry in wf 1 st order PT: Oxide-Si-impurity Small Depth: Large Depth: Even symmetry broken Rahman et al. PRL. 99, (2007) Stark Shift of Hyperfine Interaction Quadratic Stark Coefficients MethodDepth(nm)  2 (µm 2 /V 2 ) EXP (Sb) x EMT (P) ∞ -2x BMB (P) x TB (P) x x EMT: Friesen, PRL 94, (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 29 Si (S=1/2) 28 Si (S=0)Si isotopes: Observables in QM: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 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 DeviceE1 (meV)E2 (meV)E3 (meV) 10G G G HSJ GLG GLJ 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

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+ 15 nm Vs1Vb1Vb2Vs2 V=0V>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: 1 st 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- D 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 Mayur, et al., PRB 48, No. 15 (1993) EsEs EpEp EdEd E s* 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 2 nd 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