collaboration D. G. Austing (NTT BRL, moved to NRC) Y. Tokura (NTT BRL) Y. Hirayama (NTT BRL, CREST-JST) S. Tarucha (Univ. of Tokyo, NTT BRL,

Transcription

1 This is the viewgraph with the recorded talk at the 26th International Conference on Physics of Semiconductor (ICPS26, Edinburgh, 22). By clicking the upper-left button in each page, you can hear the talk for the page. Inelastic spin relaxation in quantum dots Toshimasa Fujisawa e- NTT Basic Research Laboratories collaboration D. G. Austing (NTT BRL, moved to NRC) Y. Tokura (NTT BRL) Y. Hirayama (NTT BRL, CREST-JST) S. Tarucha (Univ. of Tokyo, NTT BRL, ERATO-JST)

2 Outline. Introduction (energy relaxation inside a QD) 2. Momentum relaxation phonon emission 3. Spin & momentum relaxation cotunneling effect spin-orbit interaction ~ ns (experiment) ~ 2 µs (experiment) 4. Spin relaxation between Zeeman sub-levels (spin-orbit interaction) 5. Summary ~ ms (estimation)

3 Environment surrounding a quantum dot electromagnetic filed (photon) fluctuations of the confinement potential or magnetic field Artificial atom orbital degree of freedom spin-orbit coupling (relativistic correction) semiconductor lattice (phonon) spin degree of freedom coupling with electrodes, impurities (higher order tunneling, Coulomb interactions) nuclear spin (dynamic nuclear polarization) other excitations (plasmon, polaron,...) dissipation (T) & dephasing (T2) important characteristics for dynamics, coherence, quantum computing,...

4 Energy relaxation in a quantum dot momentum relaxation, which changes the orbital but does not change the spin theory: electron-phonon interaction τe-ph ~ ns (for GaAs QD, ~ mev) U. Bockelmann, Phys. Rev. B5, 727 (994). optical studies: discussions about phonon bottleneck effect J. Urayama et al., Phys. Rev. Lett. 86, 493 (2). R. Heiz et al., Phys. Rev. B 64, 2435 (2). spin relaxation, which changes the spin (and can changes the orbital) theory: spin-orbit interaction (+ electron-phonon interaction) τso > µs (for GaAs QD, ~ mev) A. V. Khaetskii and Yu. V. Nazarov, Phys. Rev. B6, 2639 (2) optical studies: longer than measurable range at low temperature (>2 M. Paillard et al., Phys. Rev. Lett. 86, 634 (2).

5 A vertical quantum dot (artificial atom) B drain quantum dot Energy GaAs 9 nm AlGaAs InGaAs (5%) 2 nm AlGaAs 7.5 nm gate SEM picture of mesa after gate metallization non-circular dot (nominally circular, d =.5 µmφ) no orbital degeneracy (y) (x) z GaAs source S. Tarucha et al., Phys. Rev. Lett. 77, 363 (996). L. P. Kouwenhoven et al., Science 278, 788 (997). ΓS ~ (3 ns)- Source electrode ΓD ~ ( ns)- Drain electrode measured at B = - 8 T and T =.5 - K

6 Energy spectrum of one- and two-electron QD two-electron QD (artificial helium atom) t excited state T (triplet) VG (V) dc excitation spectrum N=3 -.5 N=2 -.2 one-electron QD (artificial hydrogen atom) t excited state ground state S (singlet) -.25 T -.35 state -.4 state N= disd/dvg evsd = 2.8 mev -.3 S Energy ground state ES GS 4 2 B (T) evsd 6 I VG

7 Electrical pump-and-probe experiment I (one-electron QD) VG Vh tl th Vl z VG(t) x G pulse generator D QD ISD VSD S t µs ε- ΓS µd ΓD τ- rise time of the pulse: ~.7 ns empty QD (N = ) pump into state (~ 2 ns), and probe by drain current (max. ~ ns) average number of tunneling electrons during one pulse <nt> ~ ΓDτ-[ - exp(-th/τ-)] ΓS- < τ- < ΓD- ~ 2 ns ~ ns T. Fujisawa et al., Phys. Rev. B 63, 834(R) (2).

8 Pump-and-probe current I B=2T Vh Vl tl 2 th t VG = Vg,static + Vp(t) Current (fa) VG Vp ε- pump&probe current Ip VSD =.2 mv Vp = Vh - Vl ~ 6 mv th = 3 ns th + tl = 2 ns Vg,static (V) ΓS ΓD τ- <nt> = Ip/efrep = Ip(tl + th)/e -.46

9 .4.2 τ = ns B=T.4 <nt> ~ ΓDτ-[ - exp(-th/τ-)] τ- τ = 7 ns.2 B=2T.4.2 τ = 4 ns B=3T.4 τ = 3 ns.2 τ- (ns) average number of tunneling electrons during one pulse, <nt> B-dependence of the relaxation time B = 4.5 T Pulse width (ns) B (T) 5 6

10 Spontaneous emission of a phonon At effectively zero-temperature (kt << ε-), spontaneous emission of a phonon dominates the relaxation process. T. Fujisawa et al., Science 282, 932 (998). Wphonon = 2π <ψ h H' ψ> 2 ρ δ(e - E + εphonon) H' ε/2edef eikr (deformation) H' ε/2c /k eikr (piezoelectric) GaAs : polar semiconductor piezo εphonon = ε- : acoustic phonon energy λphonon = hv/ε- : phonon wavelength wavelength of the phonon λ ~ nm < ~ εphonon ~ 2 mev sound velocity: v = 5 m/s (GaAs LA phonon) size of the quantum dot a = 2 nm, lx/y ~ 2 nm The phonon bottleneck effect is expected.

11 τ- (ns) wavelength, ε- (mev) size (nm) B- or λ-dependence of phonon emission rate 3 2 hω = 2.5 mev x hωy = 5.5 mev 2 a lx wavelength of the phonon λ- = hv/ε- (v = 5 m/s) size of the dot z: a = 2 nm F x,y: lx=y = ~=m (!x=y +!c =4) =" lx a ε- = εphonon approx. elliptic dot ly ly λ- experiment calculation (Fermi s golden rule) approximated to a circular dot F ~!ef f = ~!x!y ( +!c =(!x +!y ) ) standard GaAs material parameters Deformation potential: 6.8 ev Piezoelectric constant:.6 C/m2 U. Bockelmann, Phys. Rev. B 5, 727 (994). calc B (T) 5 6 a shorter wavelength gives longer lifetime (bottleneck effect)

12 Two-electron QD (artificial helium atom) t excited state ground state T (triplet) S= The relaxation should change both orbital and spin parts of the wavefunction. S (singlet) Forbidden in the Russel-Saunders approx. (if the total spin is a good quantum number) S= The previous pulse technique used for one-electron QD was unsuccessful. ΓD- (~ ns) < τs-t

13 Electrical pump-and-probe experiment VG Vh Vm Vl tl µs tm VG = Vh VG = Vm εs-t S Vg g t t VSD ~.3 mv VG = Vl T th tl = ns th = ns ~ µs tm = 3 ns µd T Γd T Γs N= VSD double-step pulse gen. b rise time at sample: ~.7 ns (< Γd -, Γs -, τs - T ) evsd τs-t S pump into T state, wait for relaxation empty states (become N = ) I S probe unrelaxed electron average number of tunneling electrons during one pulse <nt> = Aexp(-th/τS-T) T S = /2 S S= A~: related to the injection efficiency S= τs-t: spin-flip energy relaxation time

14 Relaxation time from the triplet to the singlet QD-2 B=T T = 7 mk <nt> <nt> = Aexp(-th/τ) τ: energy relaxation time.5 τ = 2 µs th (µs) 8

15 εs-t (mev) B- and T- dependence of τs-t thermal excitation from the QD to the electrode (T > K) τs-t ~ 2 µs τs-t (µs) τs-t (µs) T ~. K. B (T) T S. B = T 2 3 no measurable B-dependence.5 T (K) no T-dependence for T <.5 K

16 Vh- dependence of τs-t εs-t =.6 mev B=T Total energy: (en - CgVg + q)2 U= + Echem(N, S) 2CΣ (electrostatic energy + chemical energy). 3 (mev) 2 Vm (mev) Vh (V) Vh-dependence indicates the interaction with the electrodes. thermal excitation - Γtot = 3.5 ns T = 7 mk Total energy τs-t (µs) T 3 ES S ε N= GS GS N=3 GS Vg

17 Inelastic cotunneling process virtual states assuming zero-bias voltage (VSD = ) and zero-temperature N=3 GS τ-cot = h T S ( ) (hγtot)2 εs-t εs-t =.6 mev Γtot = Γs + Γd = (7 ns)- N= GS (N=2 - N=3) (obtained from an independent meas.) M. Eto, Jpn. J. Appl. Phys. 4, 929(2). E.V. Sukhorukov, G. Burkard, D. Loss, Phys. Rev. B 63, 2535 (2). εs-t =.6 mev B=T τs-t (µs) Total energy 3 ES ε GS S S3 Vg cotunneling, τco. 3 (mev) 2 Vm Vh (V) (mev) -.6

18 Spin-orbit effect on the spin relaxation Spin-orbit interaction: (dominant spin relaxation mechanism in 2D system) lack of crystal inversion symmetry, local electric field, impurities, interfaces, etc. Spin relaxation in quantum dots reduction of spin relaxation time (large energy spacing) E2 τso,theory = τe-ph z ~ 6 µs δεs-t (lack of crystal inversion symmetry) Γe-ph: phonon emission rate, ~ (3 ns)- δ: spin splitting energy, ~ 4 µev ε: energy spacing, ~ mev Ez: vertical confinement energy, ~ 3 mev A. V. Khaetskii and Yu. V. Nazarov, Phys. Rev. B 6, 2639 (2). Our electrical measurements: τs-t ~ 2 µs (cotunneling process) The spin-orbit interaction should have weaker effect. τso > τco ~ 2 µs

19 Comparison between artificial and real atoms - transition from the t excited state to the ground state - artificial atoms (quantum dots) one-electron system - allowed for phonon emission τ = 3 - ns (hν ~ 2 mev) ()() 3P - ()2 S two-electron forbidden by spin conservation system τ ~ 2 µs (hν ~.5 mev) τforbidden τallowed > 3x4 real atoms H K He Ca 4p - 4s - (Lyman α) allowed for photon emission allowed for photon emission τ =.6 ns (hν =.2 ev) τ = 2.6 ns (hν =.6 ev) ()(2s) 3S - ()2 S forbidden by spin & parity τ = 786 s (hν = 9.8 ev) 4.5x2 (4s)(4p) 3P - (4s)2 S forbidden by spin τ = 385 µs (hν =.89 ev).5x5 Spin states in artificial atom are almost ideal, comparable to real atoms.

20 Spin relaxation time between Zeeman sub-levels Estimate the spin relaxation time by only considering spin-orbit interactions and phonon emission. ( gµbb ) 2 ε τphonon(gµbb) τspin,so = - so τs-t,so = at B = 5 T (ε- ~.2 mev, gµbb ~. mev) ) τs-t,so > 2 µs τphonon ~ 3 ns using so < 4 µev, τspin,so > ms ( εs-t 2 τphonon(εs-t) so so < 4 µev : spin-orbit coupling energy between and orbitals τspin,so > µs at B = 9 T (ε- ~.8 mev, gµbb ~.8 mev) cf. T ~ µs at B = 9 T (ESR study of donor state in GaAs) M. Seck, M. Potemski, P. Wyder, Phys. Rev. B 56, 7422 (997). W.P. Halperin, Rev. Mod. Phys. 58, 533 (986).

21 Summary I. Momentum relaxation phonon emission ~ ns (experiment) II. Spin & momentum relaxation cotunneling effect spin-orbit interaction ~ 2 µs (experiment) III. Spin relaxation between Zeeman sub-levels (spin-orbit interaction) > B = 5 T(estimation)