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
Rajib Rahman
Andresen, UNSW
Kane Qubit
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:
Design criteria (DiVincenzo):
• Quantum parallelism (speed)
• Algorithms: Quantum search, Fourier
Transform
• Applications: cryptography, simulations,
factoring, database search, etc.
• Isolation of the qubit Hilbert Space
• Decoherence times
• Ease of measurement
Rajib Rahman
• Scalability (Hollenberg, PRB 74)
• Fault-tolerant designs
Quantum Computing Implementations
NMR 5 qubit (IBM)
Ion Traps
http://www.uni-ulm.de/qiv/
forschung/ControlAndMeasurementE.html
Vandersypen et al., July 2000 PRL
Quantum Optics
Gasparani et al., PRL 93, No. 2 (2004)
SQUID
Oliver etal., Sceince 310, 1653 (2005)
Cavity QED
Rajib Rahman
Mckeever, Science Express
Reports (Feb 26, 2004)
Solid State Qubits
Electron Spin (Vrijen)
Nuclear spin qubit (Kane)
Scalability ?
Solid State
(QDs, Donors,
Si QW)
Ion Trap, eg. (http://www.uni-ulm.de/qiv/)
Donor Qubits
Benefits:
• Industry experience in Si:P
• Long coherence
• Scalability
Problems:
• Precise donor placement (1 nm)
• Control is sensitive
Rajib Rahman
Si – SiGe Quantum Wells
(Friesen)
Donor Charge Qubit (Hollenberg)
P Donor Qubits in Si
Spin Qubits (Kane, Vrijen, Hill)
Spin Qubit
• Single Qubit: Hyperfine (A ) + Zeeman (g)
• Two-qubit: Exchange J(V)
• Tunable by gates
Charge Qubit (Hollenberg)
Charge Qubit
• Molecular states of P2+
• Control electron localization by S & B gates
• Information transport - CTAP
Rajib Rahman
Donor Physics 101
Si
Si
Si
P+
Si
Si
Si
e-
Si
Si
Conventional Picture
Quantum Picture
CB
CB
ED
Donor
ED(P) = -45.6 meV
ED
Donor QD
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
Rajib Rahman
EMT: Kohn-Luttinger, Das Sarma, Koiller, Hollenberg, Friesen, …
Central Issues
Rajib Rahman
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
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
Contact HF:
e
ET
n
r0
A(ε)
ES
=> Nuclear spin site
ˆ (,r )  S
HA  I  A
0
=> Impurity site
oxide
|(ε, r0)|2
Donor


BMB
D
TB
∆A(ε)/A(0) = 2ε2 (bulk)
Exp: Bradbury et al., PRL 97, 176404 (2006)
Rajib Rahman
∆A(ε)/A(0) = (2ε2 + 1ε) (interface)
Theory: Rahman et al. PRL. 99, 036403 (2007)
Stark Shift of Hyperfine Interaction
Method
How good are the theories?
Quadratic Stark Coefficients
EMT: Friesen, PRL 94, 186403 (2005)
Depth(nm)
EXP (Sb) 150
2(µm2/V2)
-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
Why linear Stark Effect near interfaces?
Asymmetry in wf
Large Depth:
1st
order PT:
Ecorrection    y  y 0 
Even symmetry broken
Rajib Rahman
Small Depth:
Rahman et al. PRL. 99, 036403 (2007)
Oxide-Si-impurity
Hyperfine Map of Donor Wave-functions
Usefulness of HF – an example
Observables in QM:
E   H
Hyperfine:
A(,r0 )  C | (,r0 ) |2
ESR Experiments can measure A => Direct measure of WF

Si isotopes:
28Si
(S=0)
29Si
(S=1/2) 
Application: Experimentally mapping WF deformations (idea: L. Hollenberg)
Rajib Rahman
Park, Rahman, Klimeck, Hollenberg (submitted)
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
Si:P
1e-5
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.
Rajib Rahman
Interface:
g||-g|_=8e-3
Rahman, Park, GK, LH (to be submitted)
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
•
•
•
3.
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?
Two Electron Interactions
A. D- Modeling
B. Exchange Interaction
Rajib Rahman
Orbital Stark Shift of donor-interface states
ε=0
Oxide-Si-impurity
ε
Oxide-Si-impurity
Donor-interface system
Smit et al. PRB 68 (2003)
Martins et al. PRB 69 (2004)
Calderon et al. PRL 96 (2006)
Rajib Rahman
Lansbergen, Rahman, GK, LH, SR, Nature Physics, 4, 656 (2008)
Orbital Stark Shift of donor-interface states
Exp. Measurements
Transport through donor states
Energies w.r.t. ground state (below CB)
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 different from a bulk donor (21, 23, 44)
• Donor states – depth & field dependent
Rajib Rahman
Rajib Rahman
Orbital Stark Shift of donor-interface states
Si:As (Depth 7a0)
A
Si:P (Bulk)
Features found
• 3 regimes
• Interface effects
• anti-crossing
• p-manifold
• valley-orbit
B
C
Friesen, PRL 94 (2005)
A (Coulomb bound)
Rajib Rahman Rahman,
B (Hybridized)
C (Surface bound)
Lansbergen, GK, LH, SR (Orbital Stark effect theory paper, to be submitted)
Stark Effect in donor-interface well
Exp data with TB simulations
Where are the exp. points?
• Interpretation of Exp.
• Indirect observation of symmetry transition
• P vs As Donor distinction
Rajib Rahman
Lansbergen, Rahman, GK, LH, SR, Nature Physics (2008), IEDM (2008)
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
•
•
•
3.
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?
Two Electron Interactions
A. D- Modeling
B. Exchange Interaction
Rajib Rahman
Electrostatic gating of single donors
Vs1
Vb1
Vb2
Vs2
V=0
V>0
15 nm
P
P+
P+
15 nm
Nano-TCAD+TB
E2
E2
E2
E2
E2
E1
E1
E1
E1
E1
Vs1=0.0V
Rajib Rahman
Vs1=0.05V
Vs1=0.1V
Vs1=0.3V
Vs1=0.4V
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.
Rajib Rahman
Rahman, Park, GK, LH ( to be submitted)
Objective:
• Control & design issues: donor
depths, separation, gate placement.
• Feasible S and B gate regimes.
• Effect of excited states: charge state
superposition.
Charge qubit control
Molecular Spectrum
+ Tunnel barriers
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
Rajib
Rahman
RR, SHP,
GK,
LH (to be submitted)
Surface gate response of tunnel barriers
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 Efields and donor depths: important in spindependent 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.
Rajib Rahman
D-, D0 vs E
D7a0
D0
-45.6
D- vs charging energy
D-
-4
Ec comparison
Rahman, Arjan, Park, GK, LH, Rogge (in prep)
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) J(V) for various impurity separations along [100]
• 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.
Sensitivity of J(V) to donor placement
• 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?
Rajib Rahman
Methods and Details
Tight-binding and NEMO3D
Rajib Rahman
Methods & Some Details
NEMO Scaling (G. Klimeck)
• 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.
Rajib Rahman
TB parameterization of Donor
e2
V (r ) 
40 k Si | r  r0 |
r  r0
V (r )  U 0
r  r0
6
2
3
1
Mayur, et al., PRB 48, No. 15 (1993)
Rajib Rahman
On-site energy
corrections
TB
Shift all orbitals
by U0
Orbital based
shift:
Ep
Es
Ed
Es*
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
Rajib Rahman