Tutorial on Electronic Transport. Jesper Nygård Niels Bohr Institute University of Copenhagen

Transcription

1 Tutorial on Electronic Transport Jesper Nygård Niels Bohr Institute University of Copenhagen

2 ...for newcomers in the field... Prior knowledge: What is a carbon nanotube (Some) nanotube band structure Ohm s first law

3 Our aim: Understand the two-terminal electrical transport Metal lead V I Rich on exciting physics and surprising phenomena Key for interpreting wide range of experiments Many possible variations over this theme: - electrostatic gates - contact materials (magnetic, superconducting) - sensitive to environment (chemical, temperature...) - interplay with nano-mechanics, optics,...

4 Outline Electronic structure (1D, 0D) density of states Electron transport in 1D systems (general) quantization, barriers, temperature Transport in nanotubes (1D) contacts, field-effect,... Nanotube quantum dots (0D) Coulomb blockade, shells, Kondo,... Nanotube electronics, various examples Problem session Wellcome party

5 Nanoscale electrical transport Electrons Charge -e Single-electron effects Wave function Size quantization Mesoscopic transport; quantum transport...? nm µm

6 Electronic structure Density of states (DOS) 1D: semiconductor quantum wires conducting polymers nanotubes 0D: atoms molecules nanocrystals metal nanoparticles quantum dots short nanotubes

7 Tunneling spectroscopy (STM) van Hove singularities: di/dv ~ DOS ~ (E) -1/2 Cees Dekker, Physics Today, May 1999

8 Electronic transport

9 Resistance and conductance Ohm s first law: V = R. I Ohm s second law: R = V / I [Ω] Bulk materials; resistivity ρ: R = ρ L / A L R is additive Nanoscale systems (coherent transport): R is a global quantity, cannot be decomposed into local resistivities (see why later) ρ A Conductance G: G = 1 / R = I / V [unit e 2 /h] (Not conductivity σ)

10 Conductance of 1D quantum wire E 1D ballistic channel (k > 0) L Contacts: Ideal reservoirs ev µ 1 µ 2 Chemical potential µ ~ E F (Fermi level) Channel: 1D, ballistic (transport without scattering) velocity I V 1D density spin k > 0 Conductance is fixed, regardless of length L, no well defined conductivity σ

11 Conductance from transmission electron wavefunction barrier T Landauer formula: Rolf Landauer ( ); - G controversial issue in 80 ies With N parallel 1D channels (subbands):

12 Quasi-1D channel in 2D electron system High mobility 2D electron gas ( ballistic ) Depletion by electrostatic gates 2D Quantum Point Contact d L 2D Narrow constriction; quasi-1d (width d ~ Fermi wavelength λ F ) 2e 2 /h Limited conductance 2e 2 /h even without scattering, regardless of length L: contact resistance

13 Finite temperature Electrons populate leads according to Fermi-Dirac distribution: E E F Conductance at finite temp. T: eg. thermal smearing of conductance staircase (previous slide) Higher T: incoherent transport (dephasing due to inelastic scattering, phonons etc)

14 Where s the resistance? One-channel case again: T 2D d L 2D R=1-T scattering from barriers (zero for perfect conductor) quantized contact resistance

15 Resonant transport Two identical barriers in series: t r r L Total transmission: t Coherent transport; complex transmission and reflection amplitudes: t = t e iφ t r = r e iφ r Resonant transmission when round trip phase shift φ = 2πn: Perfect transmission even though transmission of individual barriers T = t 2 < 1! Resonance condition (if φ r =0): kl = πn ( particle-in-a-box states, n(λ/2)=l)

16 Resonant transport, II Two identical barriers in series: t t r r L Quasi-bound states: ( electron-in-a-box ) kl = πn Resonant transmission: E T Spectrum (0D system) Transport measurement...spectroscopy by transport... (quantum dots, see later)

17 Incoherent transport Separation L > phase coherence length L φ : r1 t1? r2 t2 Electron phase randomized between barriers Total transmission (no interference term): L Resistors in series: Ohmic addition of resistances from independent barriers

18 Transport regimes (simplified) Length scales: λ F Fermi wavelength (only electrons close to Fermi level contribute to G) L m momentum relaxation length (static scatterers) L φ phase relaxation length (fluctuating scatterers) L sample length Ballistic transport, L << L m, L φ no scattering, only geometry (eg. QPC) whenλ F ~ L: quantized conductance G~e 2 /h Diffusive, L > L m scattering, reduced transmission Localization, L m << L φ << L R ~ exp(l) due to quantum interference at low T Classical (incoherent), L φ, L m << L ohmic resistors

19 Brief conclusion - prediction: Conductance of one-dimensional ballistic wire is quantized: With perfect contacts: G = current / voltage I V = 2*2e 2 /h (two subbands in NT) Quantum of resistance: h/e 2 = 25 kohm

20 Outline Electronic structure (1D, 0D) density of states Electron transport in 1D systems (general) quantization, barriers, temperature Transport in nanotubes (1D) contacts, field-effect,... Nanotube quantum dots (0D) Coulomb blockade, shells, Kondo,... Nanotube electronics, various examples Problem session Wellcome party

21 Tutorial on Electronic Transport Metal lead II. Electronic transport in nanotubes

22 Electrical measurements on individual tubes source Au drain Nanotubes deposited or grown (CVD) Localize nanotubes (AFM) Electron-beam lithography to define electrodes Evaporate metal contacts on top (Au, Pd, Al, Ti, Co, ) SiO 2 p++ Si gate V g I V 100 µm

23 Typical device fabrication PMMA Catalyst islands Lift-off Electron beam lithopgraphy SiO 2 p+ + Si e-beam Development Lift off SWNT CVD PMMA Catalyst material sourc e drain Cr/Au c ontac ts gate Another (UV) lithography step for bond pads before mounting

24 AFM and alignment marks; EFM Atomic Force Microscopy (AFM) Electrostatic Force Microscopy (EFM) SWNTs EFM AFM 0.1 mm 12 µm Bockrath et al, NanoLetters (2002) Jespersen et al, NanoLetters (2005) Poster T.S. Jespersen

25 Henrik I. Jørgensen, NBI Poster

26 Room temperature transport source gate drain Type I. Metallic tube Type II. Semiconducting tube E F ON E F OFF (Naive sketch) Wire Field-effect transistor Seen first by Tans et al, Nature (1998) Data from Appl. Phys. A 69, 297 (1999).

27 Chirality determines bandstructure E g ~ ev (N, M) = (5, 5) (N, M) = (10, 5) In reality, also Type III: Small-gap semiconducting tubes (zigzag metals)

28 First interpretation of Semiconducting nanotube FET [Field Effect Transistor] E F G (e 2 /h) On Saturation due to contact resistance, ~ e 2 /h V g (V) Off Off at positive V g p type FET (intrinsic p doping?) Linear increase in G-V g ie limited by mobility (diffusive transport) Slope: dg/dv g = µc g /L 2 (from Drude σ = neµ) µ 10,000 cm 2 /Vs (For silicon µ 450) Mobility: µ = v d / E electric field E Later work showed importance of (Schottky) contacts! drift velocity v d

29 Schottky barriers in nanotube FETs Oxygen exposure Band structure Schottky barrier V g =-0.5 V V g =0 Different workfunctions (eg due to O 2 exposure) No shift of mid gap; contacts (shape) modulated Heinze et al (IBM), PRL 89, (2002) Barriers thinnest for Vg < 0, ie largest current when p type No change of gap Asymmetry due to modulation of contact Schottky barriers (Gas sensors)

30 Ballistic transport in metal tubes 4e 2 /h 1D conductor Near the theoretical limit 4e 2 /h (with two subbands) Close to Fermi level backscattering is suppressed in armchair (metal) tubes by symmetry Kong et al, PRL 87, (2001) McEuen et al, PRL 83, 5098 (1999) (Ballistic transport also possible in very short semiconducting tubes, otherwise mostly diffusive)

31 Quantized current limiting Metallic tube with good contacts I 0 ~ 25 µa High electric field transport - electrons are accelerated - emit (optical) phonons when E~E phonon - electron-phonon scattering for high bias Steady state current: I 0 = 4e/h E phonon ~ 4e/h * 160 mev ~ µa Z. Yao, C. Kane and C. Dekker PRL 84, 2941 (2000)

32 Metal contacts; rarely ideal Room temperature resistance of CVD grown SWNT devices Babic et al, AIP Conf. Proc. Vol. 723, (2004); cond-mat

33 Room temperature transport Ballistic transport possible (near 4e 2 /h) - ideal wires - current limiting ~ 25 µa Field effect transistors - high performance - optimised geometries (not shown) NB: Nanotube quality and contact transparency are important

34 Outline Electronic structure (1D, 0D) density of states Electron transport in 1D systems (general) quantization, barriers, temperature Transport in nanotubes (1D) contacts, field-effect,... Low temperature transport, quantum dots (0D) Coulomb blockade, shells, Kondo,... Nanotube electronics, circuits, examples Problem session Wellcome party

35 Transport in metallic tube oscillations at low T Conductance (µs) K 129 K 40 K 16 K 8 K 1.6 K G (µs) T (K) Gate voltage (V) Seen first in data by Bockrath et al and Tans et al (1997)

36 Power laws in tunneling Typical T dependences Conductance (µs) 10 G ~ T 0.6 tunneling into end of Luttinger liquid α = (g -1-1)/4 g = Luttinger parameter 1 b) Bockrath et al, Nature (1999) T (K) α = 0.6 g ~ 0.29 Evidence for Luttinger liquids Predicted: Egger & Gogolin (1997), Kane, Balents & Fischer (1997)

37 Initiator for breakthroughs Availability of high quality single walled nanotube material (Smalley group, Rice University, )

38 It is cryostat, it exists, OK, let s get on Low temperature transport Dilution refrigerator 20 mk sample

39 Low temperature data 0.4 G (e 2 /h) 300 K K 0.34 K V g (V) G (e 2 /h) G peak ~ 1/T KK K 4.2 K V g (V) Coulomb blockade in the quantum regime below ~4 K - Nanotube quantum dot First by Tans et al. (1997), Bockrath et al. (1997)

40 Coulomb blockade Electrons can tunnel only at V g for which E(N+1,V g ) = E(N,V g ) ± kt Cost for adding one electron: Bias V Total energy E(N,V g ) N electrons charging energy: E C ~ Q 2 /C ~ e 2 /C gate Quantum dots (artificial atoms) Two energy parameters: V g U charging energy e 2 /C (e-e interaction strength) E single-particle level spacing U E

41 Single-electron charging at low T 0.3 T ~ 300 mk G (4e 2 /h) Add 1 e Gate voltage V g (V) 1) large electrostatic charging energy U = e 2 /C total ~ 10 mev > k B T 2) tunnel contacts well-defined number N of electrons N electrons fixed number N of electrons i.e. zero current - Coulomb blockade except when E(N)=E(N+1)...

42 Published data not always typical... T = 4K

43 Non-linear characteristics T = 4.2 K Coulomb blockade peaks (zero bias) Nonlinear I-V curve (fixed gate volt.) Differential conductance high di / dv Bias spectroscopy, Coulomb diamonds, 0 Appl. Phys. A 69, 297 (1999).

44 Ohmic resistor Color map of di/dv I V Bias voltage V Gate voltage V g

45 Color map of di/dv (white: high R, blue: low R) Bias voltage V Measurement at T = 100 mk Gate voltage V g inelastic level spacing process Coulomb blockade (spectroscopy) single many-body state electron tunneling (Kondo effect) Electron transport governed by: - tunneling processes - discrete electron charge - orbitals of the molecule - electron-electron interactions and many-body effects

46 Transport spectroscopy of a tube quantum dot V (mv) B = 0 T T = 100 mk A B C V g (V) ev E U+ E A B C

47 Shell filling in closed dot Confirmed by Zeeman splitting: E C + E E C E C + E V g (V) T ~ 300 mk Count V g (V) G (e 2 /h) V g V g (V) Molecular spectroscopy by electrical measurements PRL 89, (2002)

48 4-electron shells and excited states Experiment Level structure Model 5 parameters: - charging energy U - level spacing - subband mismatch δ < (small) - exchange energy J (small) - residual Coulomb energy du (Liang et al, PRL 88, (2002)) Sapmaz et al, PRB 71, (2005)

49 Semiconducting quantum dot with electron-hole symmetry Small-gap semiconducting tube (Type III., zigzag metal) First hole enters Empty dot First electron enters Few-electrons dots can be made Jarillo-Herrero et al, Nature 429, 389 (2005).

50 From closed to open quantum dots G (e 2 /h) G (T = 300 K) 0.3 e 2 /h Coulomb blockade peaks Tunnel contacts weak coupling limit G (e 2 /h) 1.7 e 2 /h G (e 2 /h) 3.1 e 2 /h Metallic contacts - strong coupling limit Dips rather than peaks

51 Gray scale plots of the diffential conductance di/dv vs. V g and V 1D quantum dot Tunnel contacts weak coupling limit V V V Metallic contacts - strong coupling limit 1D molecular Fabry-Perot etalon Liang et al, Nature (2001).

52 Fabry-Perot resonances in nanotube waveguide Generally high conductance a coherent electron waveguide Dips in conductance due to interference in the resonant cavity Liang et al, Nature 411, 665 (2001)

53 With medium-transparency contacts V (mv) G (e 2 /h) mk X mk V g (V) Y Standard Coulomb Blockade di/dv G (e 2 /h) X Y 0.1 T (K) 1 1) Alternations 2) Peaks in di/dv 3). G ~ -logt - Key signatures of the Kondo effect Nature 408, 342 (2000)

54 M Cotunneling and Kondo imagine E C E 0 Ground state Virtual Okay Net result: - transport - spin flip Heisenberg: t ~ h/e 0 Co-tunneling due to more open contacts (higher order process) New ground state - Kondo state : Ψ = + + Coherent superposition + T < T K resonance T > T K blockade

55 The Kondo effect, correlations Anderson model: Normal ground state: e 2 /C contact localised state on nanotube coupling For strong coupling (large V), new ground state: Ψ = Coulomb blockade EXPERIMENT Even N, S=0: no correlated state, suppression of conductance 2.0 I (a.u.) 75 mk 740 mk coherent superposition (when T low enough) predicts resonance for transport (for S=1/2) Even Odd Even Odd N, S=1/2: correlated state at really low T, conductance restored! 0.0 Highly tuneable system V g (V)

56 ...recent data...

57 Superconductor-SWCNT-Superconductor junction Normal (B=180mT), 30mK Superconducting leads Four-period shell filling. Ec~ E ~ 3-4meV Normal (B=180mT) Bias (mv) SWCNT contacted to Ti/Al/Ti 5/40/5 nm leads (T c = 760mK) Bias (mv) Superconducting Vgate (V) Clear sign of Multiple Andreev Reflections, i.e., structure in di/dv vs bias at V n =2 /en, n=1,2,3,... Vgate (V) di/dv (e^2/h) Kasper Grove-Rasmussen et al Poster

58 Ferromagnetic contacts FM-N-FM system FM N FM Tunnel barriers Difference between tunnel resistance for parallel (R p ) and antiparallel (R ap ) magnetised contacts (Julliere, 1975): 2 R Rap Rp 2P = = 2 R R 1 P ap ap + P: fraction of polarised conduction electrons in the FM.

59 Multiwall tubes with magnetic leads K. Tsukagoshi, B.W. Alphenaar, H. Ago, Nature 401, 572 (1999) Two-terminal resistance vs. B-field for three different devices at 4.2 K Diameter 30 nm SiO 2 Si In the best case: R/R a ~ 9 % Multiwall tubes: diffusive conductors intrinsic magnetoresistance

60 Spin-polarized transport? Au FM Tube Au FM Micromagnet electrodes, single-walled Current G (e 2 /h) (na) Parallel (P) P AP B (T) Anti Parallel (AP) T = 4.2 K T=4.2K P External magnetic field (T) Parallel (P) Sweep directions: The simplified picture Fe Fe Fe Fe Fe Fe source drain source drain source drain B B B Jensen et al, PRB (June 2005) Spin-tronics Use the electronic spin rather than charge as carrier of information

61 SiO 2 Au SiO 2 Gold nanoparticle single-electron transistor with carbon nanotube leads AFM manipulation 7 nm gold particle (e 2 /C~60 mev) C. Thelander et al, APL 79, 2106 (2001).

62 Electrons in nanotubes Attach leads to 1D electron system Low T measurements Normal metal Ferromagnet Superconductor Nanoparticle Single-electron effects - spectroscopy, shells (2, 4) - Fabry-Perot resonances Correlated states -Kondo effect, long-range interactions, Luttinger liquid Magnetic contacts - Spin transport, spin transistors? Superconducting contacts - supercurrents? AFM manipulation of nanoscale objects - Gold particle transistor, 1D-0D-1D Au FM Tube Au FM Spin

63 Outline Electronic structure (1D, 0D) density of states Electron transport in 1D systems (general) quantization, barriers, temperature Transport in nanotubes (1D) contacts, field-effect,... Low temperature transport, quantum dots (0D) Coulomb blockade, shells, Kondo,... Nanotube electronics, circuits, examples Problem session Wellcome party

64 Crossed Nanotube Devices Optical micrograph showing five sets of leads to crossed nanotube devices AFM image of one pair of crossed nanotubes (green) with leads (yellow)

65 Crossed Nanotube Junctions 100 I (na) MM (4 probe) SS (2 probe) MS (2 probe) MS (2 probe) V (mv) MM junctions: R = kΩ Τ = SS junctions: R = kΩ T = MS junctions: R = 30-50MΩ T = 2 x 10-4 Fuhrer et al., Science (2000)

66 Metal-Semiconductor Nanotube Junction T=50K V g =-25V Junction A Junction B I (na) V (mv) (applied to SC) A (leaky) Schottky diode Forward Reverse ZeroBias Bias E g /2 = meV (expect: 250meV) Fuhrer et al., Science (2000)

67 Nanotube logic Bachtold et al, Science 2001

68 Integration of CNT with Si MOS UC Berkeley and Stanford, Tseng et al., Nano Lett. 4, 123 (2004). Random access nanotube test chip (switching network): - N-channel MOS-FET circuit (standard Si IC processing) - Nanotubes grown by Chemical Vapor Deposition -Growth from CH 4 +H 2 at 900 C (compatible with MOS) - Contacted by Mo electrodes 22 binary inputs to probe 2 11 =2048 nanotube devices on single chip Proof of concept (only 1% showed significant gate dependence)

69 Nanotube grown expitaxially into a semiconductor crystal - Epitaxial overgrowth by MBE (single crystal) - Nanotubes survive being buried - Hybrid electronics from molecular and solid state elements NanoLetters 4, 349 (2004) J.R. Hauptmann et al Poster

70 Single-electron transistors at low T Room T I-V: R = 125 kohm

71 V. New aspects of tube electronics Optoelectronics Nanoelectromechanical systems (NEMS)

72 Optical emission from NT-FET Effective p-n junction in semiconducting CNT (Schottky barriers + appropriate bias) SiO2 IR image emission peak due to recombination ambipolar nanotube FET moving LET Misewich et al, Science 300, 783 (2003). GATE Avouris group (IBM), PRL 2004

73 M Transport in suspended tubes nanotube SiO 2 Si (gate) Au Cr Before V g ~70mV and after V g ~160mV V g ~ e/c g, gate capacitance decreases Appl. Phys. Lett. 79, 4216 (2001)

74 Nanotube Electromechanical Oscillator Resonance Electrostatic interaction with underlying gate electrode pulls tube towards gate Actuation and detection of vibrational modes Employing sensitive semiconducting tube Resonance (tension) tuned by DC gate voltage: Put AC signal on source and gate Sazonova et al, Nature 431, 284 (2004)

75 S.W. Lee et al, Nano Letters 4, 2027 (2004) Nanorelay Switch based on nanotube beam suspended above gate and source electrodes: Electrostatic attraction to gate Reversible operation of switch

76 Outline Electronic structure (1D, 0D) density of states Electron transport in 1D systems (general) quantization, barriers, temperature Transport in nanotubes (1D) contacts, field-effect,... Low temperature transport, quantum dots (0D) Coulomb blockade, shells, Kondo,... Nanotube electronics, circuits, examples Problem session Wellcome party

77 Anti-conclusion - what we have not covered High-performance FET transistors Behavior in magnetic field Luttinger liquid behavior, correlated electrons in 1D Small-gap tubes Sensors (chemical, bio, mechanical) Problems in separation and positioning Bottom-up fabrication of devices, self-assembly Many other recent developments (see NT05)... Focused on the basic understanding of transport and electrons in NT

78 Recommended reading Electronic transport (general) S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge Uni. Press, 1995) C. Kittel, Introduction to Solid State Physics (Wiley, 2005) Chapter 18 by P.L. McEuen in 8th edition only! Nanotubes and transport R. Saito et al, Physical Properties of Carbon Nanotubes (Imperial College, 1998) M.S. Dresselhaus et al, Carbon Nanotubes (Springer, 2001) S. Reich et al, Carbon Nanotubes (Wiley-VCH, 2004) P.L. McEuen et al, "Single-Walled Carbon Nanotube Electronics," IEEE Transactions on Nanotechnology, 1, 78 (2002) Ph. Avouris et al, Carbon Nanotube Electronics, Proceedings of the IEEE, 91, 1772 (2003)

79 Carbon gives biology, but silicon gives geology and semiconductor technology. In: C. Kittel, Introduction to Solid State Physics

80 Acknowledgements CNT (Copenhagen Nanotube Team, Niels Bohr Institute 1998-) David Cobden, Uni. Washington, Seattle

81 Enjoy the conference!