1. Nanoelectronics Part II Many Electron Phenomena Chapter 10 Nanowires, Ballistic Transport, and Spin Transport 1Q.Li@Physics.WHU@2015.3

2. 2 Nickel nanowires Si nanowires

3. A review of nanowire technology Q.Li@Physics.WHU@2015.33 Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

4. Q.Li@Physics.WHU@2015.34 Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

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10. Q.Li@Physics.WHU@2015.310 Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

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14. Q.Li@Physics.WHU@2015.314 Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

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16. Q.Li@Physics.WHU@2015.316 Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

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24. 10.1 Classical and Semi-classical Transport 10.1.1 classical theory of conduction – free electron gas model A particle has 3 degrees of freedom Q.Li@Physics.WHU@2015.324 Mean thermal velocity

25. 10.1.1 classical theory of conduction – free electron gas model At T = 0 K, v T =0 At room tem.: However, this motion is random and does not result in net current. Now consider applying a voltage Q.Li@Physics.WHU@2015.325

26. 10.1.1 classical theory of conduction – free electron gas model (Drude model) will velocity increase infinitely as time increases? Collision between electrons and material lattice Mean time between collision is τ It is called relaxation time Q.Li@Physics.WHU@2015.326

27. 10.1.1 classical theory of conduction – free electron gas model Therefore, velocity is accelerated from 0 to τ: More precisely, mean time between collision and momentum relaxation time are different. µ is electrical mobility Q.Li@Physics.WHU@2015.327

28. 10.1.1 classical theory of conduction – free electron gas model For example, for copper at room temperature, τ=2.47×10 -14 s, E = 1V/m: v d = 4.35×10 -3 m/s This is much smaller than thermal velocity: ~ 10 5 m/s. Despite the small value of drift velocity, electrical signals propagate as electromagnetic waves at the speed of light. (EM signal propagates in the medium exterior to the wire, such air or other dielectric insulation.) Q.Li@Physics.WHU@2015.328

29. 10.1.1 Classical theory of conduction – free electron gas model Classical current density So Conductivity: For electron and hole: Q.Li@Physics.WHU@2015.329

30. 10.1.1 classical theory of conduction – free electron gas model For copper: Velocity: Q.Li@Physics.WHU@2015.330

31. 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model Classical model is simply a classical free electron gas model with the addition of collisions. It is better to use quantum physics principles. We should consider Fermi velocity. All momentum states within Fermi sphere are occupied, and outside the Fermi sphere are empty Q.Li@Physics.WHU@2015.331

32. 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model Q.Li@Physics.WHU@2015.332

33. For copper: Fermi velocity is many orders of magnitude larger than drift velocity, in general. It is also randomly directed in the absence of an applied field. From the viewpoint of Fermi surface: Q.Li@Physics.WHU@2015.333 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model

34. The increase in momentum in a time increment dt is Fermi gas model: use Fermi velocity rather than thermal velocity to describe conduction Q.Li@Physics.WHU@2015.334 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model

35. Q.Li@Physics.WHU@2015.335 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model

36. Q.Li@Physics.WHU@2015.336 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model Mean-free path: the mean distance an electron travels before a collision with the lattice. Copper: Using Fermi velocity is better than thermal velocity, in fact, the electron can pass through a perfect periodic lattice without scattering, where the effect of lattice merely leads to the use of effective mass.

37. Mobility is a strong function of temperature. As T decreases, mobility increases due to diminished phonon scattering. At sufficiently low temperature, scattering is mostly due to impurities. For relatively pure copper at 4K, it is possible to obtain long τ = 1 ns. Q.Li@Physics.WHU@2015.337 10.1.2 Semiclassical theory of electrical conduction – Fermi gas model

38. 10.1.3 Classical Resistance and Conductance Q.Li@Physics.WHU@2015.338 ab

39. 10.1.3 Classical Resistance and Conductance Ohm’s law If S = w 1 X w 2 and length L, field E is uniform throughout the material with magnitude ϵ Q.Li@Physics.WHU@2015.339

40. 10.1.3 Classical Resistance and Conductance For a two-dimensional flat wire having length L and width w: where σ s is the sheet conductivity in S (not S/m). Q.Li@Physics.WHU@2015.340

41. 10.1.4 Conductivity of Metallic Nanowires- the influence of Wire Radius Wire resistance increases with decreasing radius For radius between 1-20nm, the wire resistance increases significantly, due to scattering from wire surface, from grain boundaries, defects/impurities. For 5-10 nm, a quantum wire model that accounts for transverse quantization is needed. Q.Li@Physics.WHU@2015.341

42. 10.1.4 Conductivity of Metallic Nanowires- the influence of Wire Radius Q.Li@Physics.WHU@2015.342 As the Cu nanowire was being oxidized, the Cu became more and more like Cu 2 O and the wire acted like a p-type semiconductor.

43. 10.2 Ballistic Transport When L is large, conductivity is derived assuming a large number of electrons and a large number of collision between electrons and phonons, impurities, imperfections. As L becomes very small L << L m, mean free path, will the classical model of resistance works? When L << L m, one would expect that no collisions would take place – rendering the classical collision- based model useless Q.Li@Physics.WHU@2015.343

44. 10.2 Ballistic Transport At very small length scales, electron transport occurs ballistically. It is very important in nanoscopic devices. Q.Li@Physics.WHU@2015.344

45. 10.2.1 Electron Collisions and Length Scales An electron can collide with an object such that there is no change in energy – elastic collision. another type, the energy of electron changes – inelastic collision. L is system length L m is mean free path L φ is the length over which an electron can travel before having an inelastic collision. It is also called phase-coherence length, since it is the length over which an electron wavefunction retains its coherence (i.e., its phase memory) Q.Li@Physics.WHU@2015.345

46. So, inelastic collisions are called dephasing events. L φ is about tens – hundreds nm During ballistic transport, no momentum or phase relaxation. Thus, in a ballistic material, the electron wavefunction can be obtain from Schrodinger’s equation. Q.Li@Physics.WHU@2015.346 10.2.1 Electron Collisions and Length Scales

47. 10.2.2 Ballistic Transport Model The reservoir is large and it energy states form essentially a continuum: infinite source and sink for electrons. Q.Li@Physics.WHU@2015.347 reservoir

48. 10.2.2 Ballistic Transport Model The ballistic channel and subbands Q.Li@Physics.WHU@2015.348 y and z dimensions are small

49. 10.2.2 Ballistic Transport Model Let w 1 =w 2 =w The total number of subbands at or below the Fermi energy: Q.Li@Physics.WHU@2015.349 We assume:

50. 10.2.3 Quantum Resistance and Conductance Fermi energy Left reservoir: E F – eV Right reservoir: E F the electrons in the wire have wavefunction: With an associated probabilistic current density Q.Li@Physics.WHU@2015.350

51. Q.Li@Physics.WHU@2015.351 10.2.3 Quantum Resistance and Conductance Using:

52. Assume the wavefunction can be represented by a traveling state, indicating left-to-right (positive k) movement of electron Such that: Because spin up and down: Q.Li@Physics.WHU@2015.352 10.2.3 Quantum Resistance and Conductance

53. We don’t know if a certain state will be filled or not. The probability that the electron makes it into the channel from the left reservoir, out of the channel into the right reservoir must be considered. – Fermi-Dirac probability: f(E, E f - eV, T) and f(E, F F, T) – Transmission probability: T n (E) Q.Li@Physics.WHU@2015.353 10.2.3 Quantum Resistance and Conductance

54. Current flowing from left to right Right to left Q.Li@Physics.WHU@2015.354 10.2.3 Quantum Resistance and Conductance

55. Q.Li@Physics.WHU@2015.355 10.2.3 Quantum Resistance and Conductance Total current flowing Because

56. So, the temperature-dependent conductance: At very low temperatures Q.Li@Physics.WHU@2015.356 10.2.3 Quantum Resistance and Conductance

57. If there are N electronic channels, and the transmission probability is one for each channel. This is Landauer formula Since N is number of conduction channels Resistance of each channel is Q.Li@Physics.WHU@2015.357 10.2.3 Quantum Resistance and Conductance

58. Q.Li@Physics.WHU@2015.358 10.2.3 Quantum Resistance and Conductance reservoir

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60. Note: Landauer formula can also be applied to tunnel junctions T(E F ) is the transmission coefficient obtained from solving Schrodinger’s equation. As T increases, the observed quantization tends to vanish. (k B T becomes large) Q.Li@Physics.WHU@2015.360 10.2.3 Quantum Resistance and Conductance

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62. 10.2.4 Origin of the Quantum Resistance The resistance quantum, R 0, arises from perfect (infinitely wide) reservoirs in contact with a single electronic channel (i.e., a very narrow physical channel). Indeed, the resistance of a ballistic channel is length independent, as long as L << L m, L φ Ballistic metal nanowires have been shown to be capable of carrying current densities much higher than bulk metals, due to absence of heating in the ballistic channel itself. Q.Li@Physics.WHU@2015.362