Carbon Nanotubes part 2 CNT s s as a toy model for basic science. Niels Bohr Institute School 2005

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1 Carbon Nanotubes part 2 CNT s s as a toy model for basic science Niels Bohr Institute School

2 Carbon Nanotubes as a model system 2 Christian Schönenberger University of Basel B. Babic W. Belzig M. Buitelaar C. Bruder M. Calame M. Choi J. Furer A. Levy-Yeati M. Gräber C. Hoffmann S. Ifadir T. Kontos T. Nussbaumer S. Sahoo Swiss National Science Foundation national center BBW László Forró Ecole Polytechnique Fédérale de Lausanne R. Egger very fruitful collaboration

3 Basic Science low-dimensional conductor pronounced interaction effects, which can be explored both in 1d and 0d many electron physics as for example: Luttinger liquid behaviour in 1d Kondo physics in 0d 3

4 great hybrid system 4

5 Goodies no Peierls instability suppressed backscattering robust (unlike real molecules) high Young s modulus, low specific weight 5 best conductor on the 1 nm-scale great model system to study physics in 1d and 0d

6 controversial results ballistic at room temperature? 6

7 de Heer s experiment T=300 K they conclude that R < 0.2 kω/μm 7 Frank et al. Science Vol 280, 1744 (98)

8 8 mean-free path >> 1 μm Multiwalled carbon nanotubes can have large elastic scattering length exceeding one micrometer even at room temperature

9 Perpendicular Magnetotransport diffusive or at best quasi-ballistic 9

10 UCF in MWNTs long tube (7.4 μm) agrees with UCF self-averaging for a wire δg ( l / L) ϕ 3/ 2 l ϕ T 1/3 10

11 T~1 limit (SWNTs) 5 mv -5 mv gate voltage V 11 W. Liang et al., Nature 411, p 665 (2001)

12 12 MWNTs (high G limit)

13 SWNT s s <-> < > MWNT s s today SWNT MWNT D. Codben et al. M. Buitelaar et al. SWNTs are not perfect even if claimed by experts SWNT are ballistic and MWNTs are diffusive is a wrong statement ballistic is not the same for all 13

14 Summarizing some facts SWNTs 1d TDOS E F -0.3 ev (tunable) n cm -2 (1 dopant per 10 3 C atoms) #modes = 2 (not counting spin) l mfp > 1 μm D > 10 4 cm 2 /s (v F = 10 6 m/s) μ > 10 5 cm 2 /Vs MWNTs quasi 1d E F -0.3 ev (tunable) n cm -2 (1 dopant per 10 3 C atoms) #modes 10 (not counting spin) l mfp = 10 nm... 1 μm typically 100 nm D ~ 10 3 cm 2 /s (v F = 10 6 m/s) μ 10 4 cm 2 /Vs 14

15 controversial results ballistic at room temperature? intrinsic superconductivity? 15

16 Superconductivity TaAu 16

17 Superconductivity TaAu PtAu 17

18 Spectroscopy multi-wall nanotube 18

19 controversial results ballistic at room temperature? intrinsic superconductivity? is it a Luttinger liquid? 19

20 20 Quantum Dot Physics

21 21 Quantum Dots

22 22 Quantum Dots

23 ( conventional )) Quantum dots planar dot planar double-dot vertical dot 23

24 Quantum Dot? 2 μm 24

25 CNT as a 1d quantum dot 2 μm Γ < δe for any quantum dot 25 Note: if δe < (Γ,eV,kT) transition to wire regime. Note also that there is more to Γ than just the coupling to the leads!

26 Three Regimes (simplified...) ideal quantum dot has: δe easy, if U << Γ resonant tunneling easy, if U >> Γ single-electron tunneling not easy, if U Γ correlated electron transport 26

27 Open Nanotube Dot Γ>> U (Normal Leads) 27

28 T~1 Limit (interaction-free wire) bias voltage 5 mv -5 mv gate voltage V 28 W. Liang et al., Nature 411, p 665 (2001)

29 T~1 Limit (wire) T ( E) = 2 1+ R C + TC 2R cos( Θ + C 2δ R ) Θ = L( k + k ) round-trip phase depends on E I V = e 2 { T ( ev / 2) + T ( ev / 2) } 29 W. Liang et al., Nature 411, p 665 (2001)

30 Fabry-Perot (interference) 5 mv -5 mv their data gate voltage V simple model 30

31 quantum interference L = 800nm d = 1.5 nm J. King et al. PRL 87, (2001) origin of beating? disorder, other tube, second shell, new intrinsic mechanism? 31

32 quantum interference works even for MWNTs 32

33 quantum interference 33 B. Babic et al. (unpublished)

34 Closed Nanotube Dot Γ<< U (Normal Leads) 34

35 Single-Electron Electron-Tunneling if tunneling probability p of each junction is small : N+1 N current determined by accessible levels in dot i.e. by level spacing and Coulomb charging energy uncorrelated sequential tunneling dominates. Current I p 35

36 Tutorial on di/dv Plots V sd (mv) ΔE add add addition energy, i.e. sum of: single-electron charging energy U C V g level-spacing δe 36 Change V sd Change V g

37 even even even odd odd odd filling of states according to S = 1/2 0 1/2... odd number of electrons: ΔΕ add = UC even number of electrons: ΔΕ add = UC + δe 37

38 shell structure δe level spacing N=3 E F shell (shell pattern) K K MWNT s SWNT s Mark Buitelaar et al., PRL 88, (2002) Wenjie Liang et al. PRL 88, (2002) 38

39 state filling as spin pairs B δe δε S = 1/2 0 1/2... g=2 MWNTs Mark Buitelaar et al., PRL 88, (2002) 39

40 Open Nanotube Dot Γ U Correlated Transport (Normal Leads) 40

41 Co-Tunneling if tunneling probability p of each junction is large : if tunneling probability p of each junction is small : uncorrelated sequential tunneling dominates. Current I p coherent 2nd (and higher) order processes add substantially p 2 we call this co-tunneling 41

42 V sd (mv) δe ~ 0.55 mev U C ~ 0.45 mev V g Elastic co-tunneling Inelastic co-tunneling 42

43 V sd (mv) Correlated Transport When the number of electrons on the quantum dot is odd, spin-flip processes (which screen the spin on the dot) lead to the formation of a narrow resonance in the density-of-states at the Fermi energy of the leads. This is called the Kondo effect V g 43 or the Abrikosov Suhl resonance

44 50 mk δe ~ 0.6 mev U C ~ 0.4 mev Γ 0.3 mev 44 Mark Buitelaar et al. PRL 89, (2002)

45 Kondo Effect resistance R(T) of a piece of metal log( T / T K ) conductance G(T) through a single magnetic impurity (e.g. a spin ½ quantum dot) unitary limit 2e 2 /h T K ideal metal, e.g. Au wire superconductor, e.g. Pb magnetic impurities in ideal metal e.g. (ppm)fe:au Kondo system T K 45 from L. Kouwenhoven & L. Glazman, Physics World, June 2001

46 Kondo first Kondo in Nanotubes, what about shell structure? 46 first in CNTs: J.Nygard et al, Nature 408, 342 (2000)

47 Kondo and shell pattern in CVD-grown SWNTs 47

48 Kondo and shell structure in SWNTs B=0 T B=5 T Kondo peak T K = K G(e 2 /h) G(T) ~ -ln(t/t K ) 0.6 T(K) T(K) groundstate? in particular at N=2 48 δe 5 mev U C 5 mev and Γ 0.5 mev B. Babic et al. Phys. Rev. B 70, (2004)

49 N=1,3 ground-state B=0T δe N=1 δe δε δε X 1 =δε V g 2X 1 δε = 0.9 mev δe N=3 δe δε δε 2X 3 δε = 0.8 mev X 3 =δε 49 B. Babic et al. Phys. Rev. B 70, (2004)

50 N=2 ground-state B=0T δe N=2 δe T δε δε X 2 (S)=δε-J 2X 2 V g X 2 = 0.9 mev δe N=2 δe δε δε 2X 3 δε = 0.8 mev X 2 (T)=J-δε J is small, i.e. J < 0.1 mev 50 B. Babic et al. Phys. Rev. B 70, (2004)

51 Kondo effect at N=2? B=0T δe N=2 δe T δε δε X 2 (S)=δε-J V g Why Kondo for N=2? (it has been argued that this is a spin triplet Kondo) key point: the two states S and T can only be distinguished provided that: δε-j > Γ δe δε N=2 δe δε Γ increases (and δe decreases) with larger gate voltage X 2 (T)=J-δε = 51 B. Babic et al. Phys. Rev. B 70, (2004)

52 more in recent literature Phys. Rev. B 71, (2005) 52 Nature 434, 484 (2005)

53 from a quantum dot to an open weak -link 53

54 from charge dot to q-wire T << 1 T <~ 1 T ~ 1 different devices Jespers Nygard, David Cobden and Poul-Eric Lindelof

55 from CB to Fano 55 B. Babic and C. Schönenberger, cond-mat/

56 what is it? e.g. ionization 56 U. Fano, Phys. Rev. 124, 1866 (1961)

57 interference effect G = G nonres + G res ( ε + q) 2 ε transmisson o o o 57 hybridization via the leads bonding- and antibonding state orbital phase M. L. Ladron de Guevara, et al. Phys. Rev. B, 67, , (2003).

58 resonance 2 58 B. Babic and C. Schönenberger, cond-mat/

59 59 is it intrinsic or extrinsic Fano?

60 60 more?

61 number of possible exp. X NT Y where X,Y = {N,S,F} NT = {m-swnt,s-swnt,mwnt...} = {bad,intermediate,good} amounts to 200 about 14 have been done beyond that, there are additional opportunities, for example: suspended tubes, multiterminal 61 N-NT-N, S-NT-S, F-NT-F

62 Kondo physics and supreconductivity 62

63 Kondo Physics + Superconductivity Kondo effect Superconductivity Al Δ Δ Kondo effect and superconductivity are many-electron effects can Kondo and superconductivity coexist or do they exclude each other? 63

64 Spin 1/2 Kondo + S-LeadsS normal case superconducting case U E F 1. a gap opens in the leads 2. Cooper pairs have S=0 Kondo effect is the screening of the spin-degree of the dot spin by exchange with electrons from Fermi-reservoirs (the leads) Hence: Kondo effect suppressed, but Mark Buitelaar et al. PRL 89, (2002)

65 Kondo effect Superconductivity Δ Δ Cooper pair S = 0 Cooper pair S = 0 Energy scale : ~ k b T K Energy scale : ~ Δ A cross-over expected at k b T K ~ Δ 65 Mark Buitelaar et al. PRL 91, (2003)

66 Kondo <-> > Proximity Kondo effect & Superconductivity 2Δ Energy scale : ~ k b T K Energy scale : ~ Δ A cross-over over at k B T K ~Δ exp.: Phys. Rev. Lett. 89, (2002) 66 plus further theory work: cond-mat/

67 Limiting cases: (Glazman & Matveev JETP Lett. `89): Weak coupling (T K << Δ): strongly suppressed supercurrent [due to Coulomb interaction] Open contact (T K >> Δ): full ballistic supercurrent [transport through Kondo-(Abrikosov-Suhl) resonance] Weak coupling: (Spivak & Kivelson PRB `91) π shift due to interchange of electrons 67

68 NRG results Pair amplitude on dot: π-junction Sign change of pair-amplitude amplitude 0-π transition Open channel limit: (T K >>Δ) Weakly coupled dot: (T K < Δ) Choi et al., cond-mat/

69 phase diffusion model resistively & capacitively shunted Josephson junction: overdamped limit: β c < 1 C E J R Relation between resistance and supercurrent: theory experiment Δ = 0.1 mev or 1.16 K Buitelaar et al. PRL 88, (2002) 69 Choi et al., cond-mat/

70 MAR in a q-dotq another recent result 70

71 MAR in a Quantum Dot experiment theory di/dv sd 2Δ di/dv sd dot level (center) 2Δ/ 2 2Δ/ 3 2Δ/ 4 V gate V sd Multiple Andreev Reflection (MAR) trough a single level 71 M. Buitelaar et al. Phys. Rev. Lett. 91, (2003)

72 72 more?

73 number of possible exp. X NT Y where X,Y = {N,S,F} NT = {m-swnt,s-swnt,mwnt...} = {bad,intermediate,good} amounts to 200 about 14 have been done beyond that, there are additional opportunities, for example: suspended tubes, multiterminal 73 N-NT-N, S-NT-S, F-NT-F

74 NT-Spintronics NT H 63 Pd PdNi PdNi Pd TMR V g R (kω) S = -2%/T -0.2%/T 200 nm TMR + 3.8% + 2.9% 0.0 V -3.1 V %/T -0.2%/T - 3.1% - 3.5% -5.0 V -3.3 V H (T) 74

75 NT-Spintronics G (e 2 /h) TMR (%) experiment 1 theory V g (V) T = 1.85K 75

76 NT-Spintronics 76

77 77 many thanks to

78 this should be clear now, right! our nanotubes have such a diameter! voilà, here is the prove! 78

79 Kondo physics and supreconductivity 79

80 Q-dot between S-contacts Device parameters: Leads Al/Au bilayers: 135/45nm T c ~1.2K Distance between electrodes: 250nm Length of CNT: 1.5 mm 250

81 V sd (mv) ΔE ~ 0.55 mev U C ~ 0.45 mev V g Elastic co-tunneling Inelastic co-tunneling

82 results: normal contacts 50 mk ΔE ~ 0.6 mev UC ~ 0.4 mev Γ 0.3 mev Transport through single levels Coulomb blockade diamonds conductance ~e 2 /h! good contacts Zero bias anomaly for odd number of electrons Kondo effect Buitelaar, Nussbaumer, Schönenberger, PRL 89, (2002)

83 Experimental results: superconducting contacts N Kondo ridge A : 0.75 K Kondo ridge B : 1.11 K V g Kondo ridge C : 0.96 K S A: decreasing conductance B: increasing conductance C: decreasing conductance V g Questions: Origin of fine structure? Zero-bias conductance? Buitelaar, Belzig, Nussbaumer, Babic, Bruder, Schönenberger PRL 91, (2003)

84 cross-over over & supercurrent Kondo effect Superconductivity 2Δ Energy scale : ~ k b T K S = 0 Cooper pair Energy scale : ~ Δ S = 0 A cross-over over expected at k B T K ~Δ

85 Phys. Rev. Lett. 89, (2002) cross-over over

86 different Kondo temperatures T K =0.71 K T K =0.96 K T K =1.11 K T K =1.86 K

87 Low TK High TK normal state superconducting state Δ = 0.1 mev or 1.16 K Buitelaar et al. PRL 88, (2002)

88 Limiting cases: (Glazman & Matveev JETP Lett. `89): Weak coupling (T K < Δ): strongly suppressed supercurrent [due to Coulomb interaction] Open contact (T K > Δ): full ballistic supercurrent [transport through Kondo-(Abrikosov-Suhl) resonance] Weak coupling: (Spivak & Kivelson PRB `91) π shift due to interchange of electrons

89 NRG results Pair amplitude on dot: π-junction Sign change of pair-amplitude amplitude 0-π transition Open channel limit: (T K >>Δ) Weakly coupled dot: (T K < Δ) Choi et al., cond-mat/

90 phase diffusion model resistively & capacitively shunted Josephson junction: overdamped limit: β c < 1 C E J R Relation between resistance and supercurrent: theory experiment Δ = 0.1 mev or 1.16 K Buitelaar et al. PRL 88, (2002) Choi et al., cond-mat/

91 I V B S-CNT-N Aluminum Gold V G M.R. Buitelaar et al., PRL 88, (2002) & PRL 89, (2002)

92 Electrical Transport Bias Voltage (mv) V sd [mv] T = 90 mk Gate Voltage (V) N-1 O N N+1 E M. Gräber et al. (unpublished) V gate [V] Gate Voltage (V) Conductance (e 2 / h) Conductance (e 2 / h) B = 25 mt, T = 90 mk B = 0 mt, T = 90 mk B = 25 mt, T = 90 mk Bias Voltage (mv) Ridge A T K ~ 0.8 K Bias Voltage (mv) Ridge A T K ~ 0.8 K

93 Kondo (N-S-case) B = 0 mt, T = 90 mk B = 25 mt, T = 90 mk Ridge A T K ~ 0.8 K N LDOS S Conductance (e 2 / h) μ Ν k B T K 2Δ 1.00 Bias Voltage (mv) 0.75 Bias Voltage (mv) Gate Voltage (V) M. Gräber et al. (unpublished)

94 MAR in a q-dotq another recent result 94

95 95 finite bias structure

96 finite bias structure Andreev reflection 96

97 finite bias structure multiple Andreev reflection (MAR) 97

98 MAR has been explored in weak links and in single atom contacts (break junctions) theory: Cuevas et al. PRB 54, 7366 (99) 98

99 MAR has been explored in weak links and in single atom contacts (break junctions) but not in quantum dots dot level dot level Γ Δ (Γ 3Δ) 99

100 100 theory (non-interacting)

101 101 2Δ/2 2Δ/3

102 2Δ 2Δ/ 2 2Δ/ 3 2Δ/ 4 102

103 experiment theory 103

104 experiment 104 theory