1. 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

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

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

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

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

6. Experiment: magnetic NCs [H. Zeng et. al., PRB 73, 020402 (2006)] [H. Xing et. al., J. Appl. Phys. 105, 063920 (2009)] Fe 3 O 4 NCs Co / CoO NCs

7. Experiment: superconductor NCs [Zolotavin and Guyot-Sionnest, ACS Nano 6, 8094 (2012)] superconductor/insulator transition tuned by B-field or insulating barrier [talk by Philippe Guyot-Sionnest]

8. Experiment: granular films Indium evaporated onto SiO 2 [Belobodorov et. al., Rev. Mod. Phys. 79, 469 (2007)] [Y. Lee et. al., PRB 88, 024509 (2013)] [talk by Allen Goldman] Disordered Indium Oxide

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

10. electron wavefunction d a High tunneling barriers a << d G/(e 2 /h) << 1 Tunneling between NCs is weak: Model of an array of metal NCs

11. Single-NC energy spectrum E g1(E)g1(E) - EcEc ground state energy levels: e-e- A single, isolated NC: EfEf Coulomb self-energy: E c = e 2 /2C 0

12. Single-NC energy spectrum E g1(E)g1(E) + 2Ec2Ec ground state energy levels: e-e- A single, isolated NC: EfEf Coulomb self-energy: E c = e 2 /2C 0

13. Single-NC energy spectrum - Multiple-charging: Coulomb self-energy: E c = e 2 /2C 0

14. Single-NC energy spectrum -2 ground state energy levels: e-e- Multiple-charging: E g1(E)g1(E) 2Ec2Ec EfEf each NC has a periodic spectrum of energy levels Coulomb self-energy: E c = e 2 /2C 0 → (2e) 2 /2C 0 Same spectrum that gives rise to the Coulomb blockade

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

16. Variable-range hopping rate of phonon- assisted tunneling: ξ = localization length r D’

17. Variable-range hopping ξ = localization length rate of phonon- assisted tunneling: D’

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

19. Variable-range hopping ξ = localization length R ij i j rate of phonon- assisted tunneling: D’ Miller-Abrahams resistor network

20. Simulate a (2D) lattice of NCs with random interstitial charges δq (-Q max, Q max ) Search for the electron occupation numbers {n i } 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 Hamiltonian and computer model q i = (δq) i - en i

21. DOGS - results Main features: 1. g(E) vanishes near E = 0 2. g(|E| > 2) = 0 3. Perfect symmetry 1. Distribution is universal at sufficiently large disorder

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

23. Absence of deep energy states Usual situation: (lightly-doped semiconductors) Here: E g1(E)g1(E) w1w1 small disorder, w 1 : No deep energy states for any value of disorder.

24. Absence of deep energy states Usual situation: (lightly-doped semiconductors) Here: E g1(E)g1(E) small disorder, w 1 : E large disorder, w 2 : Coulomb gap is less prominent Here, deep states are not possible: E-E- E+E+ E i + = E i - + 2E c w1w1 w2w2 No deep energy states for any value of disorder.

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

26. Miller-Abrahams resistor network R ij i j k l R kl R jl R ik R jk R il... ρ is equated with the minimum percolating resistance.

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

28. Variable-range hopping ξ = localization length R ij i j rate of phonon- assisted tunneling: D’

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

30. Model of an insulating array of superconductor NCs

31. Uniform superconducting pairing energy, 2 Δ Weak Josephson coupling J ~ Δ ∙ G/(e 2 /h) << E c  heavily insulating, with decoherent tunneling Focus on the case where Δ and E c are similar in magnitude pairing energy # pairs in NC i [Mitchell et. al., PRB 85, 195141 (2012))]

32. Single-electron energy spectrum E g1(E)g1(E) - EcEc single electron density of ground states: An isolated NC with Cooper pairing (and an even number of electrons): EfEf e-e- Coulomb self-energy: E c = e 2 /2C 0

33. Single-electron energy spectrum single electron density of ground states: e-e- + E g1(E)g1(E) EcEc EfEf Ec+2ΔEc+2Δ Coulomb self-energy: E c = e 2 /2C 0 Binding energy of pair: 2∆ An isolated NC with Cooper pairing (and an even number of electrons):

34. Pair energy spectrum pair density of ground states: E g2(E)g2(E) 4Ec4Ec EfEf Coulomb self-energy: (2e) 2 /2C 0 = 4E c +/- 2 Can also have hopping of pairs: 2e -

35. DOGS - results Δ = 0: singles pairs Δ = 2E c : Δ = E c : e  2ee  2e e  √2e 4(1 – Δ) 2(Δ – 1)

36. Miller-Abrahams network for singles and pairs i j ρ 1 is the percolating resistance of the singles network. ρ 2 is the percolating resistance of the pair network.

37. Effective charges in hopping transport ES hopping:

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

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

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

41. MR peak: single versus pair conduction? [Steiner and Kapitulnik, Physica C 422, 16 (2005)] granular InOx: pairs? singles? In principle, our model can produce a MR peak : ξ 1 = D’ ξ 2 = 10D’ k B T = 0.1 E c Δ/EcΔ/Ec [Thin Solid Films 520, 1242 (2010)]...but is not a satisfactory explanation of experiment

42. Conclusions 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 e* = e e* = √2e e* = 2e coordinate energy E = Thank you. [PRL 109, 126805 (2012)] [PRB 109, 045135 (2012)]

43. Reserve Slides

44. 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).

45. 2D and 3D DOGS - metal 2D: 3D:

46. 2D and 3D DOGS - SC 2D: 3D:

47. Disorder +e+e +e+e +e+e

48. Some impurities are effectively screened out by a single NC +e+e +e+e -e-e +e+e -e-e

49. Disorder Some impurities are effectively screened out by a single NC +q A +q B +e+e -qB-qB -qA-qA +e+e -e-e +e+e -e-e Others get “fractionalized”

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

51. Tunneling conductance intra-NC density of states: tunneling conductance:

52. Electron energy spectrum of a semiconductor nanocrystal Electron energy spectrum has two components: 1) quantum confinement energy: ΔEQΔEQ

53. Electron energy spectrum has two components: 2) electrostatic charging energy: EcEc U(Q) = Q 2 /κD total Coulomb self-energy: energy to add one electron: E c = U(Q - e) - U(Q) E c = (e 2 - 2Qe)/κD e2/κDe2/κD 3e2/κD3e2/κD 5e2/κD5e2/κD -5e 2 /κD -3e 2 /κD -e2/κD-e2/κD 0 Electron energy spectrum of a semiconductor nanocrystal

54. Random doping of NCs Regular lattice of equal-sized NCs Donor number N i is random:

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

56. Typical case: ν = 5, ΔE Q = 5 E c E N = 1 E N = 5 E N = 9 E 1S 1P N = 0 N = 6 E N = 4 E N = 10 E... 1D

57. Typical case: ν = 5, ΔE Q = 5 E c E N = 1 E N = 5 E N = 9 E 1S 1P N = 0 N = 6 E N = 4 E N = 10 E... 1D -2 +1 +2

58. Density of states: ν = 5, Δ = 5 e 2 /κD 1S 1P 1D