Spin Qubits in Semiconductor and Graphene Quantum Dots

Transcription

1 ICQFT, Shanghai, 18 July 2009 Spin Qubits in Semiconductor and Graphene Quantum Dots Guido Burkard University of Konstanz, Germany!!

2 Spin Qubits qubit natural realization: electron spin 1/2 = = spins localized in QDs and coupled via exchange quantum dot 1 2 gate electrodes 2D electron gas V g1 V g2 exchange coupling from virtual electron hopping Loss & DiVincenzo, Phys. Rev. A (1998).

3 Spin Hamiltonian 1 2 Loss & DiVincenzo, Phys. Rev. A (1998). CNOT SU(2) exchange coupling Burkard et al. PRB QDs: Petta et al Zeeman term, g-factor ESR: Koppens et al optical Stark effect: Berezovsky et al. 2008

4 Experimental breakthroughs: mostly GaAs T 1 (s -1 ) single spin read-out, T 1 Elzerman et al., PRB (2003); --- Nature (2004) sqrt-swap, spin echo, T 2 * Petta et al., Science (2005). spin state preparation with high fidelity Atatüre et al., Science (2006). long T 1 at low fields Amasha et al. (2007). single-spin ESR Koppens et al., Nature (2006). News&Views: GB, Nature ( 06).

5 Challenges for Spin Qubits in GaAs = = nuclear-spin induced spin decoherence, T 2 2. spin-orbit coupling to phonons: spin relaxation, T 1 3. long-distance coupling

6 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B x B z

7 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B x B z

8 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B x B z

9 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B x B z

10 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t

11 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ (gµ BB + ˆB n )

12 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + +

13 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + +

14 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + +

15 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + + T 2 *

16 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + + T 2 *

17 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + + Ŝx = cos(t) exp [ (t/t2 ) 2] T 2 *~10 ns: Petta et al., Science 2005 T 2 *

18 Decoherence due to nuclear spins External (classical) field H = gµ B Ŝ B B z x <S X > t Nuclear spins: Overhauser field (has quantum fluctuations) H = Ŝ ˆB n = (gµ BB + ˆB n ) A i Î i i + + Ŝx = cos(t) exp [ (t/t2 ) 2] T 2 *~10 ns: Petta et al., Science 2005 T 2 * not good as qubit anymore

19 How to deal with nuclear spin decoherence? Two ideas 1. projection into Overhauser eigenstate (suppressing quantum fluctutations) Klauser, Coish, Loss, PRB (2006). Stepanenko, GB, Giedke, Imamoglu, PRL (2006). + + Reilly et al., Science 321, 817 (2008) Ribeiro & Burkard, PRL (2009)

20 How to deal with nuclear spin decoherence? Two ideas 1. projection into Overhauser eigenstate (suppressing quantum fluctutations) Klauser, Coish, Loss, PRB (2006). Stepanenko, GB, Giedke, Imamoglu, PRL (2006). + + Reilly et al., Science 321, 817 (2008) Ribeiro & Burkard, PRL (2009) 2. materials without nuclear spin (Si, Ge, C) CNTs, graphene Trauzettel, Bulaev, Loss & Burkard, Nature Physics (2007) Recher, Nilsson, Burkard & Trauzettel, PRB 79, (2009)

21 Outline Introduction Nuclear spin preparation in semiconductor QDs Spin qubits in graphene quantum dots Summary

22 Singlet-triplet qubits Qubit basis : Hyperfine interaction in double quantum dots commutes with spin Hamiltonian Dynamics affected by fluctuations of Overhauser difference operator

23 Singlet-triplet qubits Dynamics in external magnetic field Spin decoherence time σ (z) state narrowing natural nuclear state σ (z) N Coish & Loss, PRB (2005). Khaetskii, Loss & Glazman, PRL (2002). Petta et al., Science (2005).

24 Polarizing nuclear spins Nuclear spin polarization enhances electron spin coherence G. Burkard, D. Loss & D. P. DiVincenzo Phys. Rev. B 49, 2070 (1999) Sizable enhancement of electron spin coherence for P > 99% W. A. Coish, D. Loss, Phys. Rev. B 72, (2005) Recent breakthrough experiment D. J. Reilly et al., Science 321, 817 (2008) Measurement after cycles : 1% polarization but T 2,f * = 70 T 2,i *

25 Landau-Zener-Stückelberg Effective Hamiltonian in S / T + subspace Ribeiro & GB, Phys. Rev. Lett. 102, (2009). Fix χ : two-level system with Hamiltonian probability for adiabatic transition L. D. Landau, Phys. Z. 2, 46 (1932) C. Zener, Proc. R. Soc. London A (1932) E. C. G. Stückelberg, Helv. Phys. Acta 5, 370 (1932)

26 Generalized LZS many nuclear spins nuclear states highly degenerate apply Morris-Shore transformation G. S. Vasilev, S. S. Ivanov, N. V. Vitanov, Phys. Rev. A 75, (2007) unphysical: infinite energy, phase finite time propagation N. V. Vitanov, B. M. Garraway, Phys. Rev. A 53, 4288 (1996)

27 Results for n=200, I=1/2, A=const. Ribeiro & GB, Phys. Rev. Lett. 102, (2009). (a) σ (z) [nev] 65 T * 2 [ns] T 2,f /T 2,i =1.34 final polarization ~ 8% cycle (b) 1 (c) 0.05 P S p(δh z ) cycle δh z / A cycling induces narrowing of nuclear distribution mechanism: weak measurement

28 Monte Carlo algorithm What happens to and to? Initial state: nuclear spins: State after k + 1 cycles : Idea : sample with a Monte Carlo algorithm Random number Averaging over many runs leads to the same result as the first method

29 Weak measurement toy model: a unique nuclear spin in one of the dots 1 2 incoherent mixture of up- down-nuclear spin Case of singlet outcome: > σ (z) 1 = Ā 2 cosh(η/2)

30 Scaling to bigger systems Simulation with n=200 spins/qd T 2,f /T 2,i =1.34 Initial fluctuations Final fluctuations σ (z) f const. n 10 6 T 2,f /T 2,i 90 experiment (Reilly et al.): T 2,f T 2,i 70

31 Outline Introduction Nuclear spin preparation in semiconductor QDs Spin qubits in graphene quantum dots Summary

32 Why Graphene? B 2D monolayer of carbon in hexagonal lattice A realized experimentally B A unusual electronic properties Geim & Novoselov, The rise of graphene, Nature Materials (2007). Castro Neto et al., Rev. Mod. Phys. (to appear), arxiv: few nuclear spins ( 13 C): expect long electron spin coherence time (T2) low atomic mass (Z=6): expect long spin lifetime (T1)

33 tight-binding model Bloch states: pseudo-spin Electrons in graphene B A B δ 1 A δ 3 δ 2 a 2 a 1 Linearize band structure at K, K points: E/t Dirac-Weyl equation for the envelope function: K a k x a k y K slowly varying potiential

34 Klein paradox Oskar Klein, Z. Phys. (1929) Dombay, Calogeracos, Phys. Rep. (1999) Katsnelson, Novoselov, Geim, Nature Phys. (2006) regular semiconductor (parabolic conduction band) graphene: gapless semiconductor (with linear spectrum) E(k) k V(x) dot barrier bound state particle can tunnel out through valence band of barrier region!

35 arxiv: v1 [cond-mat.mes-hall] electronic properties, such as massless carriers, electronhole symmetry near the charge neutrality point, and weak spin-orbit coupling [3] makes graphene interesting for high mobility electronics [4, 5], for tracing quantum electrodynamics in 2d solids, and for the realization of spin-qubits [6]. Whereas diffusive transport in graphene and the anomalous quantum Hall effect have been investigated intensively [7, 8], graphene quantum dots are still in their infancy from an experimental point of view [9]. This is mainly due to difficulties in creating tunable quantum dots in graphene because of the absence of an energy gap. Also phenomena related to Klein tunneling make it hard to confine carriers laterally using electrostatic potentials [10, 11]. Here we report on Coulomb blockade and Coulomb diamond measurements on an etched graphene quantum dot tunable by graphene side gates [12]. spectroscopy, the SFM step height in Fig. 1(d), proves also the singlegraphene flake and shows that the attack the SiO2. The fabricated device consists o row graphene constrictions conne drain (D) electrodes to a graphen A 0.06 µm2. The twoelectrostat and the island, respectively. For a electrodes see Fig. 1(a). All three have been patterned closer than 1 graphene regions, as shown in Fig Dots exist despite the Klein paradox mesoscopic devices and are due to quantum interference (1 4, 18 20). Smooth variations in the CB peak height (Fig. 1A) are attributed to interference-induced changes in the barriers transparency, as shown by studying individual QPCs (13). Furthermore, we have measured the dependence of CB on applied bias Vb and, from the standard stability diagrams (Coulomb diamonds), found the charging energy Ec. The lower inset in Fig. 1B shows such diamonds for D 250 nm, which yields Ec 3 mev and the total capacitance C = e2/ec 50 af. The rather large Ec implies that the CB oscillations in Fig. 1B graphene nanoribbon boundary conditions: gap, electrodes: QD Trauzettel, Bulaev, Loss, GB, Nature Phys. (2007) A ownloaded from on April 27, 2008 Theas nanodevice, schematically in Fig. 1(a), has FIG. 2: Source-drain current a function of the shown two barbeen fabricated from graphene, which has rier gate voltages VSG1 and VSG2 for constant bias, Vbiasbeen = extracted from bulk graphite by mechanical exfoliation onto 300 nm 200µV. The dashed lines indicate transmission modulations SiO/2 nanostructured on n-si substrate as described in Ref. [13]. Raman carved graphene imaging [14] is appliedconstrictions to verify the single-layer and oscillations attributed to the graphene (hor- character of the investigated devices [15, 16, 17] nm PMMA Ponomarenko et al., Science izontal and vertical lines) and to the island (diagonal line). (positive e-beam resist) is then spun onto the samples Stampfer et(e-beam) al., APL 2008 Measurements are preformed atelectron-beam VBG = 6 V and VP G = 0isV. and lithography used to pat- tern the etch mask for the graphene devices. Reactive ion etching (RIE) based on an Ar/O2 (9:1) plasma is introduced to etch away unprotected graphene. A scanning FIG. 1: (color online) Nanostructured force microscope (SFM) image of the etched graphene tional back gate (BG) is used to adjust the overall Fermi device. (a) Schematic illustration of structure after removing the residual PMMA is shown in quantum dot. (b) Scanning force micr Fig. 1(b). Finally, the graphene device is contacted by energy. the investigated graphenecurrent device aftert δ FIG. Source-drain e-beam patterned 2 nm Ti and 50 nm Au electrodes as 3: ter contacting the graphene structure Transport measurements a ontructure shownhave in Fig. been 1(c). A performed Raman spectrumin recorded the function of The thedashed plung sizeas is approx. 50 nm. lin final device taken the location of the graphene island of the graphene areas. (d) shows a SF variable temperature He cryostat at aatbase temperature Coulomb resonances are observ is plotted in Fig. 1(e). It is an unambiguous fingerprint path x [marked in (b)] averaged over of 1.7 K. Before the cool-down the sample haswidth been modulations. to the path proving the selective etch of single-layer graphene with a line of the 2Dscale line conductance of approx. 33 cm 1 [15, 16, 17]. The elevated backraman spectra recorded on spacing the final of (a) and in (c) the peak baked in vacuum at 135 C for 12 h. We have measured island with a spot size of approx. 400 ground originates from the nearby metal electrodes and δ B magnetic confinement De Martino, Dell Anna, Egger, PRL (2007) biased bilayer graphene δ Nilsson et al., PRB 2007 Milton Pereira et al., Nano through Letters the 2007 the two-terminal conductance dot by applying a small (symmetric) DC or AC bias voltage Recher, Nilsson, GB, Trauzettel, PRB 2009Vbias, and Corresponding measuring the current through the author, dot withstampfer@phys.ethz.ch a resolution peaks.single-layer Measurements character ofare the prefor investiga information on the D, G and 2D (als VBG = 6 V, VSG1 = 25 mv, an refer to Ref. [17].

36 Valley degeneracy problem exchange coupling with 1) singlet hybridization with virtual state dot 1 dot 2 2) triplet single orbital (e.g., one K point): Pauli principle degenerate orbital (e.g., both K points) K K K K dot 1 dot 2

37 Graphene ribbons solve both problems! i. Klein paradox Prohibits confinement of particles, e.g., in a quantum dot. ii. Valley (K-K ) degeneracy Prohibits Heisenberg exchange coupling, required for two-qubit operations. Chen, Lin, Rooks & Avouris, Physica E (2007). Important: Most other proposed solutions do not appear to break valley (K-K ) degeneracy!

38 K Graphene ribbons armchair boundaries M=4 3 M = 3N: M = 3N±1: B A metallic semiconducting x K x=w K =(M+1/2)a K zigzag boundaries A K K 2 1 mixes K and K x=0 a y B K finite gap K-K valley degeneracy lifted for all modes k y K Brey & Fertig, PRB (2006) no gap K-K degeneracy not broken

39 K Graphene ribbons armchair boundaries M=4 3 M = 3N: M = 3N±1: B A metallic semiconducting x K x=w K =(M+1/2)a K zigzag boundaries A K K 2 1 mixes K and K x=0 a y B K finite gap K-K valley degeneracy lifted for all modes k y K Brey & Fertig, PRB (2006) no gap K-K degeneracy not broken

40 Quantum dots: Bound states solve transcendental equation for ε L=5/q 0 L=2/q 0 barrier1 E(k) dot bound states exists only in the energy window barrier2 V barrier k V gate

41 Quantum dots: Bound states solve transcendental equation for ε L=5/q 0 L=2/q 0 barrier1 E(k) dot bound states exists only in the energy window barrier2 V barrier k V barrier V gate V gate

42 Double dot Two electrons in two neighboring quantum dots Exchange coupling based on Pauli principle with singlet-triplet splitting J (for t U) In the regime of weak tunneling: t can be increased by lowering the barrier (both ways!)

43 Alternative: gapped graphene Recher, Nilsson, GB & Trauzettel, PRB 79, (2009) single layer: use substrate to open gap G. Giovannetti et al., Phys. Rev. B 76, (2007). S. Y. Zhou et al., Nature Mat. 6, 770 (2007). bilayer: use electrical gating to open gap E. McCann, PRB (2006). T. Ohta et al., Science (2006). magnetic field + mass term -> gap in both cases (figures: single layer) 0 T 3.4 T (R=25nm) parameters for plots: 165 mev/r[nm]

44 Summary spin qubits in semiconductor QDs: nuclear spin preparation enhances spin coherence Ribeiro & GB, PRL (2009) graphene as an interesting material for spin qubits Trauzettel, Bulaev, Loss & GB, Nature Physics 3, 192 (2007) Recher, Nilsson, GB & Trauzettel, PRB (2009) + ongoing work (graphene spin T1 and T2) open PhD / postdoc positions

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