Electron spin coherence exceeding seconds in high-purity silicon Alexei M. Tyryshkin, Shinichi Tojo 2, John J. L. Morton 3, H. Riemann 4, N.V. Abrosimov 4, P. Becker 5, H.-J. Pohl 6, Thomas Schenkel 7, Michael L. W. Thewalt 8, Kohei M. Itoh 2, and Stephen A. Lyon Dept. of Electrical Engineering, Princeton University 2 School of Fundamental Science and Technology, Keio University 3 Dept. of Materials, and CAESR, Clarendon Laboratory, Dept. of Physics, Oxford University 4 Institute for Crystal Growth, IKZ 5 PTB Braunschweig 6 VITCON Projectconsult GmbH 7 Lawrence Berkeley National Laboratory 8 Dept. Physics, Simon Fraser University
Outline Why long coherence? Spin echo method for T 2 measurements Phosphorus donors in high-purity 28 Si (only 50ppm 29 Si) Three decoherence processes identified (all related to donor-donor interactions) Upon suppressing these mechanisms, we demonstrate T 2 ~ 0 seconds for P donors in silicon
Why Long Coherence Times? Quantum bits (qubits) for quantum computation and quantum communication: Kane quantum computer (Nature, 998) Qubits as sensitive probes of the environment: Maze et al, Nature (2008) Expected field sensitivity ~ nt Basic physics of relaxation and dephasing of quantum states: Still many unsolved questions
Three Characteristic Relaxation Times B 0 ψ = cos θ 0 + sin θ e iφ T relaxation time (involves energy exchange with the environment) T 2 * dephasing time (not real decoherence since includes static field inhomogeneities) T 2 coherence time (irreversible dephasing) Typical arrangement: T > T 2 > T 2 *
Measuring T 2 : 2-Pulse (Hahn) Echo Experiment 90 0 80 0 echo 90 0 q 2 echo T T T T Hahn echo experiment 2-pulse echo experiment
Electron Spins in Solid State Devices Electron spins have a magnetic moment sensitive to magnetic field noise 3 Sources of noise: 3 Nuclear spins Defect spins Other (uncontrolled) donors 2 Ideal host material would have no other (uncontrolled) spins
Hahn Echo Decay Example: Magnetic Nuclei in Host Material.2 P donors in natural silicon Quantum dots in GaAs (Bluhm et al, Nat Physics, 20).0 0.8 T 2 = 0.6 ms 0.6 0.4 T 2 = 35 ms 0.2 0.0 0.0 0. 0.2 0.3 0.4 0.5 tau (ms) 4.7% of 29 Si nuclei (spin /2) Fast (non-exponential) decay 00% magnetic nuclei ( 69,7 Ga, 75 As, both spins 3/2) Ideal host would have no magnetic nuclei: Silicon, germanium, carbon, and helium are promising materials
Phosphorus Donors in Very Pure 28 Si kilogram 28 Si sphere for accurate Avogadro number Our samples (came from Avogadro project): 50 ppm 29 Si nuclei Donor densities *0 4 to 3*0 5 P/cm 3 Experimental details: X-band ESR (microwave 9.6 GHz, magnetic field 0.35 T) Spin polarization ~ 0% at 2 K Two-pulse echo experiment to measure T 2 90 0 q 2 echo T T
Relaxation Times, T and T 2 (s) Donor T 2 Depends on Donor Density 0 3 0 2 0 0 0 0-0 -2 0-3 0-4 0-5 T 0 4 0 5 0 6 (Gordon, 958) (Feher, 959) 0 6 0 2 4 6 8 0 2 4 Temperature (K) 90 0 80 0 echo T T Instantaneous diffusion dominates T 2 at low temperatures
Hahn Echo Signal What Is Instantaneous Diffusion? (.20 4 P/cm 3, at 2.K) 90 0 q 2 echo T T 8 6 4 T 2 = 20 ms 2 0 0.00 0.02 0.04 0.06 0.08 Time (s) q 2 = 80 0 gives the best echo signal But also maximal instantaneous diffusion
Suppressing Instantaneous Diffusion Instantaneous diffusion maximized Instantaneous diffusion suppressed 90 0 80 0 echo 90 0 T T T 4 0 T echo Salikhov et al. (JMR, 98) Kurshev, Ichikawa (JMR, 992)
Hahn Echo Signal Hahn Echo Signal Instantaneous Diffusion Is Dominant T 2 Mechanism 8 (.20 4 P/cm 3, at 2.K).2.0 6 4 T 2 = 20 ms 0.8 0.6 0.4 T 2 = 0.45 s 2 0.2 0 0.00 0.02 0.04 0.06 0.08 Time (s) 0.0 0.0 0.2 0.4 0.6 0.8 Time (s) 90 0 80 0 echo 90 0 T T T 4 0 T echo Instantaneous diffusion maximized Instantaneous diffusion suppressed
/T 2 (s - ) Extrapolating to Small Rotating Angles Removes Instantaneous Diffusion Effect 50 40 30 6K 4.8K.9K T 2 = 20 ms 90 0 80 0 echo T T T 2 = 0.7 s 20 0 90 0 T 22 0 T echo Slope = 49 s - corresponds to.2*0 4 P/cm 3 0 0.0 0.2 0.4 0.6 0.8.0 sin 2 (q 2 /2) Salikhov et al. (JMR, 98) Kurshev, Ichikawa (JMR, 992)
Relaxation Times, T and T 2 (s) Extrapolated Donor T 2 (Instantaneous Diffusion Suppressed) 0 3 0 2 0 0 0 T Hahn echo T 2 Extrapolated T 2 0-0 -2 0-3 0-4 0 4 0 6 0 5 0 6 0-5 0 2 4 6 8 0 2 4 Temperature (K)
T 2 (s) Extrapolated T 2 at Different Donor Densities Flip-Flops Donor Flips (T ).2*0 4 T limit Above 8 K: Phonon-relaxation (T ) processes 0..2*0 5 Between 4 K and 8 K: Spectral diffusion by donor T -flips 0.0 3.3*0 5 Below 4 K: Spectral diffusion by donor flip-flops 2 4 6 8 Temperature (K) Solid fit curves: T 2 T T SD T ( T flips) SD( flip flops)
T -Induced Donor Flips Donor spins flip randomly on T timescale B 0 90 0 80 0 echo T T Random flips modulate the dipole fields and thus decohere the central spin Theory: Decay ~ exp time T SD 2 Donors: Characteristic time: T P T 2 SD Dipole interactions: Mims (Phys Rev 68, 370, 968) Hu and Hartmann (PRB 9,, 972)
2-Pulse Echo Signal Simulated Decays for Spectral Diffusion by Donor T -Induced Flips.0 Simulation parameters: 0.8 0.6 0.4 2.K [P] =.2*0 4 P/cm 3 T =.6 s (5.8 K) T = 8 s (4.7 K) 0.2 5.8K 4.7K 0.0 0.0 0.2 0.4 0.6 0.8.0 Mims (PR 68, 370, 968): Time (s) Decay ~ exp time T SD 2 T = 300 s (2. K), where T P T 2 SD
Donor Flip-Flops B 0 90 0 80 0 echo T T Random flip-flops modulate the dipole fields causing decoherence of the central spin Theory: Salikhov et al. (JMR, 98) Kurshev, Ichikawa (JMR, 992) Witzel et al. (PRL, 20) Donors: Dipole interactions:
Donor Flip-Flops Can Be Suppressed Spin flip-flops are induced by dipole interactions: dipole interactions ~ 0 Hz at.20 4 P/cm 3 Flip-flop can only occur if Zeeman energies are close: (n n 2 ) < dipole interaction Flip-flops can be turned off: - By low temperature (our 2K is not low enough) - By very high magnetic field (our 0.35 T is not high enough) - By inhomogeneous broadening (e.g. external magnetic field gradient, 29 Si hyperfine broadening, etc.)
Echo Signal Intensity /T 2 (s - ) ESR Signal Intensity Suppressing Donor Flip-Flops by Magnetic Field Gradient 50.2 0 4 P/cm 3, at 2 K 4 ESR linewidth 40 2 30 90 deg 0 20 3482. 3482.4 3482.7 3483.0 3483.3 Magnetic Field (Gauss) 0.0 Echo decays q 2 = 90 deg 0 0.0 0.2 0.4 0.6 0.8.0 sin 2 (q 2 /2) No gradient T 2 =.3 0. sec Gradient (0.3Hz/nm) T 2 = 2 7 sec 0.5 0.0 0.00 0.05 0.0 Time (s)
Summary Donor T 2 ~ 0 seconds in high-purity 28 Si Three order of magnitude longer than in other materials Comparable to 3 P nuclear T 2 = 50 s at.4 K (Mike Thewalt) Decoherence mechanisms: Above 8 K Phonon-relaxation (T ) processes Below 8 K Instantaneous diffusion Between 4-8 K Spectral diffusion by donor T -induced flips Below 4 K Spectral diffusion by donor flip-flops Still much T 2 is shorter than T ~ 2000 s at 2 K: The new 28 Si sample with lower donor densities (4 0 2 P/cm 3 ) hopefully provides more answers