Electrical Control of Single Spins in Semiconductor Quantum Dots Jason Petta Physics Department, Princeton University g Q 2 m T + S Mirror U 3 U 1 U 2 U 3 Mirror Detector See Hanson et al., Rev. Mod. Phys. 79, 1217 (2007) for an exhaustive reference list.
Quantum computing hardware Trapped ion/atom QC NMR QC Optical QC Superconducting QC Spin based QC Topological QC Bertet Petta Raussendorf
DiVincenzo Criteria 1 Loss & DiVincenzo Proposal Phys. Rev. A 57, 120 (1998) 0 Efficient initialization Readout Universal set of gates Long decoherence times (T 1,T 2 ) Scalable Prepare spin in ground state Spin to charge conversion ESR for single spin rotations Exchange interaction T 1 ~ 1 second, T 2 ~ 200 s Standard semiconductor fabrication
Quantum condensed matter physics- The goals 1. Build quantum systems with experimentally tunable Hamiltonians H= Z J(V g ) B nuc Z B 0 nuc J ( (ev)2 1 0-2 -1 0 (mv) 2. Determine the limits of quantum coherence in the solid state 3. Extend coherence by tailoring the solid state environment (isotopic purification.) 4. Build artificial quantum materials Spin chains & lattice Atom Tri-atom Molecule Ciorga et al. (2000) Petta et al. (2005) Gaudreau, Schroer (2007)
Quantum condensed matter physics- The challenges 1. Quantum system must be isolated from the environment to preserve coherence 2. Hamiltonian must be experimentally tunable 3. Desire strong and controllable qubit coupling (not always compatible with 1.) Electron-phonon coupling quantum bit Charge dephasing Hyperfine interactions Spin-orbit interactions
Lecture 1: Introduction to quantum dots Loss and DiVincenzo PRA 57, 120 (1998) Motivation for using spins in semiconductors Introduction to quantum dot physics Isolating and detecting single electrons Spin in quantum dots Spin-orbit and hyperfine interactions ti Measurements of the spin lifetime (T 1 )
Electron spin as a quantum bit Single spin qubit Motivation for using electron spin: Relatively easy to trap single charges Weakly interacting Spin-orbit interactions (~1 sec. T 1 ) Qubit states: Hyperfine interactions (~ 200 s T 2 ) ħ 1 E Zeeman Manipulate single spins using ESR Couple spins using exchange interactions 0 Use electron spin resonance to drive coherent evolution Loss and DiVincenzo PRA 57, 120 (1998)
Semiconductor technology: scaling and Moore s law First transistor 1947 410 Million Transistors (2009) ~10 9 Feature e size (Ǻ) 10 10 Quad-Core Itanium 10 5 10 9 Itanium 2 10 8 Pentium 4 10 4 10 7 486 286 10 3 8080 quantum quantum 10 2 effects? 1970 1975 1980 1985 1990 1995 2000 2005 Year 2010 10 6 10 5 10 4 10 3 # of tran nsistors Source: Intel
Types of quantum dots Single molecules Vertical semiconducting dots Groups: McEuen, Ralph Groups: Kouwenhoven, Tarucha Metal nanoparticles Lateral semiconducting dots Groups: Ralph, Petta Groups: Marcus, Kouwenhoven, many others
Quantum dots- Artificial atoms Kouwenhoven and Marcus, Phys. World (1998)
Quantum dots- Artificial atoms Atoms Quantum dots 3D confinement V(x) x Discrete spectrum He Cu nanoparticle Petta&Ralph (2002) GaAs double dot Molecular bonding Livermore (1996)
Some differences between quantum dots and atoms Atoms Quantum dots V(x) x Size d=1 Ǻ d=1000 Ǻ Level spacing ~ ev ~ 100 s of ev-mev # of nuclei N nuc =1 N nuc =10 5-10 6 Spin-orbit L S Material dependent Structure dependent Anisotropic
Lateral quantum dot fabrication GaAs/AlGaAs heterostructure Quantum point contact ohmic contact depletion gate
Evolution of quantum dot devices Many-electron quantum dots Few-electron quantum dots N e ~100 400 nm N e ~1 450 nm Marcus group (1996) Ciorga et al., PRB 61, 16315 (2000) Double dots Double-double dots Triple dots Elzerman (2003) Petta (2004) Petta (2006) Gaudreau, Schroer (2007)
Coulomb blockade and charging R 1,C 1 R 2,C 2 N-1 N N+1 N+2 I V G Energy V SD Charging energy E 2 c =e /2C V G E F e 2 /C Kastner, RMP 64, 849 (1992) V G V G
Non-interacting electrons-in-a-box model V E ev Can resolve discrete states if E>>kT di/dv QD level spacing E 0.8 R 1,C 1 R 2,C 2 04 0.4 I (n na) B Z = 100 mt T e ~ 80 mk I Quantum dot 0 di I/dV (ns) 300 200 V 100 Petta and Ralph, PRL 87, 266801 (2001) Cu #1 0 5 10 15 20 25 V (mv)
Finite bias spectroscopy di/dv color-scale plot Coulomb diamonds L. P. Kouwenhoven et al., Science 278 1788 (1997).
Trapping and detecting single electrons Transport measurements Charge detection empty dot I QPC charged dot I QPC 450 nm Ciorga et al., PRB 61, 16315 (2000) -2.88 V g -2.82 Field et al., PRL 70, 1311 (1993) Hanson et al., PRL 91, 196802 (2003)
Spin: Zeeman splitting E I (n na) 0.8 04 0.4 B Z = 100 mt T e ~ 80 mk o B z ev Specific case: B Bzˆ di I/dV (ns) 0 300 200 100 Cu #1 0 5 10 15 20 25 V (mv) 9 Cu#1 B z E g o 1 B B 2 z Petta and Ralph, PRL 87, 266801 (2001) 6 3 0 2 5 8 E (mev)
DiVincenzo Criteria 1 Loss & DiVincenzo Proposal Phys. Rev. A 57, 120 (1998) 0 Efficient initialization Readout Prepare spin in ground state Spin to charge conversion Long decoherence times (T 1,T 2 ) T 1 ~ 1 second, T 2 ~ 200 s Universal set of gates Scalable ESR for single spin rotations Exchange interaction Standard semiconductor fabrication
Spin qubit initialization Load Empty spin-up py dot electron ect L E R E E Z E F E E Z 5 kt P P e E Z 10 L >> R
DiVincenzo Criteria 1 Loss & DiVincenzo Proposal Phys. Rev. A 57, 120 (1998) 0 Efficient initialization Readout Prepare spin in ground state Spin to charge conversion Long decoherence times (T 1,T 2 ) T 1 ~ 1 second, T 2 ~ 200 s Universal set of gates Scalable ESR for single spin rotations Exchange interaction Standard semiconductor fabrication
Spin-to-charge conversion (Readout) measure spin-up L R manipulate qubit L R E F E F? ħ measure spin-down L R E F Fujisawa, Nature (2002) Elzerman, Nature (2004)
DiVincenzo Criteria 1 Loss & DiVincenzo Proposal Phys. Rev. A 57, 120 (1998) 0 Efficient initialization Readout Prepare spin in ground state Spin to charge conversion? Long decoherence times (T 1,T 2 ) T 1 ~ 1 second, T 2 ~ 200 s Universal set of gates ESR for single spin rotations Exchange interaction Scalable Standard semiconductor fabrication
Charge relaxation -vs- spin relaxation time (T 1 ) charge relaxation Phonon emission E F E F 1 spin relaxation Phonon emission +spin flip E 0 E Requires: energy loss phonon emission inel ( 1-0 )~100 MHz Requires: energy loss + spin flip phonon emission + spin-orbit interactions SF ~khz (field dependent) phonon emission + hyperfine interactions SF ~khz-mhz (field dependent)
Spin-orbit interactions in quantum dots: electric field S L H p V H SO B = effective B field Effective S O field E B = effective B field Structural inversion asymmetry (Rashba spin-orbit term) Effective S-O field x p y y p x R H Structural inversion asymmetry (Rashba spin orbit term) Bulk inversion asymmetry (Dresselhaus spin-orbit term) y p y x p x D H
Hyperfine interactions in quantum dots: H I S H j e B ExtS e b, j js I,, j Zeeman Hyperfine H e( BExt BNuc) S GaAs nuclear spins 75 As, I=3/2, 100% 69 Ga, I=3/2, 60% 71 Ga, I=3/2, 40% Fully polarized nuclei, B Nuc =5.3 T Unpolarized nuclei, B Nuc ~2 mt S B Ext B Tot B Nuc
Single spin relaxation time measurements (T 1 ) Elzerman et al., Nature 430, 431 (2004). See also Fujisawa et al., Nature 419, 278 (2002).
Spin relaxation time measurements (T 1 ) T 1 ~0.5 ms at 10 T Hanson et al., Nature 430, 431 (2004). Optical measurements Kroutvar et al., Nature 432, 81 (2004). T 1 B 5 spin-orbit limited at high field
High field spin relaxation mechanism E F spin relaxation E Phonon emission +spin flip 2 2 nl" " H nl" ", e, ph n l DE Z E phonon density of states H E SO Z, e ( ),, iq H q j M r b b e ph q j q, j q, j 2 Z E piezeoelectric field scales as 1/ q 1/ E Z electron-phonon coupling E Z E 3/2 z 2 E2 E5 z z Erlingsson, Nazarov, Falko, Phys. Rev. B (2001)
Energy dependence of the relaxation rate hc E ph ph 16 nm c 4000 m/ s ph E mev ph 1 >d ~d <d T. Meunier et al., Phys. Rev. Lett. 98, 126601 (2007)
Beyond single quantum dots Loss and DiVincenzo PRA 57, 120 (1998) Single spin rotations and spin-spin interactions, e.g. are sufficient for spin based quantum computation Requires coupled quantum dots SWAP single dot double dot triple dot
Single dots -vs- double dots kt E Z kt Require E Z >>kt Freq. = 4 GHz/Tesla Interdot tunneling linewidth is independent of kt