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Hopping transport and the “Coulomb gap triptych” in nanocrystal arrays

Hopping transport and the “Coulomb gap triptych” in nanocrystal arrays. Brian Skinner 1,2 , Tianran Chen 1 , and B. I. Shklovskii 1 1 Fine Theoretical Physics Institute University of Minnesota 2 Argonne National Laboratory 2 September 2013. UMN MRSEC. Electron conduction in NC arrays.

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Hopping transport and the “Coulomb gap triptych” in nanocrystal arrays

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  1. Hopping transport and the “Coulomb gap triptych” in nanocrystal arrays Brian Skinner1,2, TianranChen1, and B. I. Shklovskii1 1Fine Theoretical Physics Institute University of Minnesota 2Argonne National Laboratory 2 September 2013 UMN MRSEC

  2. Electron conduction in NC arrays Each site has one energy level: . filled or empty. energy Conventional hopping models: Conductivity is tuned by: • spacing between sites • insulating material • disorder in energy/coordinate • Fermi level μ μ coordinate In nanocrystal arrays: Energy level spectrum is tailored by: • size • composition • shape • surface chemistry • magnetism • superconductivity • etc. Each “site” is a NC, with a spectrum of levels:

  3. Electron conduction in NC arrays energy μ coordinate Conductivity reflects the interplay between individual energy level spectrum and global, correlated properties.

  4. Experiment: semiconductor NCs CdSe NCs, diameter 2 nm – 8 nm [Bawendi group, MIT] [talk by Philippe Guyot-Sionnest] [talk by Alexei Efros] 15 nm [JJ Shianget al, J. Phys. Chem. 99:17417–22 (1995)]

  5. Experiment: metallic NCs range of shapes: precise control over size/spacing: tuneable metal/insulator transition: cubes stars rods wires hollow spheres core/ shell [Kagan and Murray groups, UPenn] [Aubin group, ESPCI ParisTech] [Talapin group, Chicago]

  6. Experiment: magnetic NCs Co / CoO NCs Fe3O4 NCs [H. Zeng et. al., PRB 73, 020402 (2006)] [H. Xing et. al., J. Appl. Phys. 105, 063920 (2009)]

  7. Experiment: superconductor NCs superconductor/insulator transition tuned by B-field or insulating barrier [Zolotavin and Guyot-Sionnest, ACS Nano6, 8094 (2012)] [talk by Philippe Guyot-Sionnest]

  8. Experiment: granular films Disordered Indium Oxide Indium evaporated onto SiO2 [Belobodorov et. al., Rev. Mod. Phys.79, 469 (2007)] [Y. Lee et. al., PRB 88, 024509 (2013)] [talk by Allen Goldman]

  9. Model of an array of metal NCs Uniform, spherical, regularly-spaced metallic NCs with insulating gaps insulating gaps Large internal density of states: spacing between quantum levels δ 0 metallic NCs

  10. Model of an array of metal NCs High tunneling barriers a << d Tunneling between NCs is weak: G/(e2/h)<<1 d electron wavefunction a

  11. Single-NC energy spectrum A single, isolated NC: e- Coulomb self-energy: Ec= e2/2C0 - E Ec ground state energy levels: Ef g1(E)

  12. Single-NC energy spectrum A single, isolated NC: Coulomb self-energy: e- Ec= e2/2C0 + E 2Ec ground state energy levels: Ef g1(E)

  13. Single-NC energy spectrum Multiple-charging: Coulomb self-energy: Ec= e2/2C0 -

  14. Single-NC energy spectrum Multiple-charging: e- Coulomb self-energy: Ec= e2/2C0 -2 → (2e)2/2C0 E each NC has a periodic spectrum of energy levels 2Ec ground state energy levels: Same spectrum that gives rise to the Coulomb blockade Ef g1(E)

  15. Density of Ground States • Disorder randomly shifts NC energies: +e E +e -e Ec Ef g1(E) -e -e +e E “Density of ground states” (DOGS): distribution of lowest empty and highest filled energies across all NCs Ef g1(E)

  16. Hamiltonian and computer model • Simulate a (2D) lattice of NCs with random interstitial charges δq (-Qmax, Qmax) • Search for the electron occupation numbers {ni} that minimize the total energy • Calculate DOGS by making a histogram of the single-electron ground state energies at each NC: • Calculate resistivity ρ as a function of temperature T by mapping the ground state arrangement to a resistor network qi = (δq)i - eni

  17. DOGS - results • Main features: • g(E) vanishes near E = 0 • g(|E| > 2) = 0 • Perfect symmetry 1. Distribution is universal at sufficiently large disorder

  18. The Coulomb gap E+ E- in 2D Efros-Shklovskii conductivity: Typical hop length:

  19. Absence of deep energy states Here: • Usual situation: • (lightly-doped semiconductors) g1(E) small disorder, w1: E w1 No deep energy states for any value of disorder.

  20. Absence of deep energy states Here: • Usual situation: • (lightly-doped semiconductors) g1(E) small disorder, w1: E w1 large disorder, w2 : Coulomb gap is less prominent No deep energy states for any value of disorder. E w2 Ei+ = Ei-+ 2Ec Here, deep states are not possible: E+ E-

  21. “Triptych” symmetry [orthodoxy-icons.com] Ei+ = Ei-+ 2Ec DOGS is completely constrained by symmetry and Coulomb gap. g(E) is invariant in the limit of large disorder.

  22. Miller-Abrahams resistor network ... • ρ is equated with the minimum percolating resistance. Rij j i Rjk ... ... Rik Rjl Ril k l Rkl ...

  23. Variable-range hopping rate of phonon-assisted tunneling: D’ ξ = localization length ξ ~ a D’/d >> a

  24. Variable-range hopping rate of phonon-assisted tunneling: i Rij D’ ξ = localization length j

  25. Efros-Shklovskii conductivity low T ρ(T) is largely universal at sufficiently large disorder higher T (T*)-1/2

  26. Model of an insulating array of superconductor NCs

  27. Model of an insulating array of superconductor NCs Uniform superconducting pairing energy, 2Δ Weak Josephson coupling J ~ Δ ∙ G/(e2/h) << Ec  heavily insulating, with decoherent tunneling Focus on the case where Δ and Ecare similar in magnitude # pairs in NC i [Mitchell et. al., PRB 85, 195141 (2012))] pairing energy

  28. Single-electron energy spectrum An isolated NC with Cooper pairing (and an even number of electrons): Coulomb self-energy: e- Ec= e2/2C0 - E Ec Ef g1(E) single electron density of ground states:

  29. Single-electron energy spectrum An isolated NC with Cooper pairing (and an even number of electrons): Coulomb self-energy: Ec= e2/2C0 e- Binding energy of pair: 2∆ + E Ec Ef g1(E) single electron density of ground states: Ec+2Δ

  30. Pair energy spectrum Can also have hopping of pairs: Coulomb self-energy: 2e- (2e)2/2C0 = 4Ec +/- 2 E 4Ec Ef g2(E) pair density of ground states:

  31. DOGS - results e √2e singles pairs 4(1 – Δ) Δ = 0: e 2e 2(Δ – 1) Δ = 2Ec: Δ = Ec:

  32. Miller-Abrahams network for singles and pairs • ρ1 is the percolating resistance of the singles network. • ρ2 is the percolating resistance of the pair network. j i

  33. Effective charges in hopping transport ES hopping:

  34. Effective charges in hopping transport Slope gives ES hopping:

  35. Effective charges in hopping transport Slope gives ES hopping: e*= 2e e*= √2e e*= e

  36. Magnetoresistance 1.5 1 0.5 0 single e- hopping is gapped pair hopping is gapped [Lopatin and Vinokur, PRB 75, 092201 (2007)] • Superconducting gap is reduced by a transverse field: ( For example, Zeeman effect: ) Δ/Ec increasing magnetic field:

  37. Conclusions E energy = • In NC arrays, single-particle spectrum and global correlations combine to determine transport • For metal NCs, the “Coulomb gap triptych” is a marriage between the Coulomb blockade and the Coulomb gap •  Disorder-independent transport • For superconducting NCs, the gap changes the “effective charge” for hopping coordinate [PRL 109, 126805 (2012)] e*= 2e e*= √2e e*= e [PRB 109, 045135 (2012)] Thank you.

  38. Reserve Slides

  39. Publications • metal NCs:Tianran Chen, Brian Skinner, and B. I. Shklovskii, Coulomb gap triptych in a periodic array of metal nanocrystals, Phys. Rev. Lett. 109, 126805 (2012). • superconducting NCs:Tianran Chen, Brian Skinner, and B. I. Shklovskii, Coulomb gap triptychs, √2 effective charge, and hopping transport in periodic arrays of superconductor grains, Phys. Rev. B 86, 045135 (2012). • semiconductor nanocrystal arrays: Brian Skinner, Tianran Chen, and B. I. Shklovskii, Theory of hopping conduction in arrays of doped semiconductor nanocrystals, Phys. Rev. B 85, 205316 (2012).

  40. 2D and 3D DOGS - metal • 2D: 3D:

  41. 2D and 3D DOGS - SC 3D: 2D:

  42. Disorder +e +e +e

  43. Disorder • Some impurities are effectively screened out by a single NC +e -e +e +e -e

  44. Disorder +qA +qB • Some impurities are effectively screened out by a single NC -qB +e -qA -e +e Others get “fractionalized” +e -e

  45. Disorder +qA -e + qB • Some impurities are effectively screened out by a single NC -qB +e -qA -e +e Others get “fractionalized” +e -e Result is a net fractional charge on each NC

  46. Tunneling conductance • intra-NC density of states: tunneling conductance:

  47. Electron energy spectrum of a semiconductor nanocrystal • Electron energy spectrum has two components: 1) quantum confinement energy: ΔEQ

  48. Electron energy spectrum of a semiconductor nanocrystal • Electron energy spectrum has two components: Ec total Coulomb self-energy: 2) electrostatic charging energy: 5e2/κD U(Q) = Q2/κD 3e2/κD energy to add one electron: e2/κD Ec = U(Q - e) - U(Q) 0 -e2/κD Ec = (e2 - 2Qe)/κD -3e2/κD -5e2/κD

  49. Random doping of NCs Regular lattice of equal-sized NCs Donor number Ni is random:

  50. Electron energy spectrum of a single nanocrystal N = 9 N = 5 no donors N = 1 N = 2 N = 3 E E E E E E 1D 1P ... ... 1S EQ1S

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