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