Current-Voltage Relation in 1-D Wires/Nanotubes: Ballistic, Quasi-Ballistic Transport and Optical Transitions

1. L3 ECE-ENGR 4243/6243 09132016 F. Jain NOTES L4 (p. 89) and L5 (102B) Overview Derivation of current-voltage relation in 1-D wires/nanotubes (pp. 90-102A) Ballistic, quasi-ballistic transport—elastic and inelastic length and phase coherence (pp. 89-90) Universal Conductivity Fluctuation (UCF) in in 1-D wires/nanotubes in the presence of quasi-ballistic transport (pp. 102B-104). Optical transitions- Quantum Confined Stark effect (QCSE) and optical modulators Optical transitions- Photon absorption and emission Optical transitions- LEDs and Lasers fundamentals: spectral width, gain etc.

2. Energy levels in nanowire (discrete due to nx and nz)Freedom of movement along Y-axis gives energy width 1. Discrete value of nx=1 and then add nz= 1, 2, 3 2. discrete value of nx=2 and then add nz= 1, 2, 3 3. Add to each discrete value of nx, nz an energy width as shown in density of state plot (for one level).

3. Derivation of current due to 1D subband or channels (I >1) [page 91-92] Show conductance quantized as e2/h, g is 2 due to spin

4. Density of state (quantum wires) review L2 p85 Density of state (quantum wells) review L3 p85 Carrier density n or p Density of states (12) Without f(E) we get density of states Quantization due to carrier confinement along the x and z-axes. Looking at the integration 13 This simplifies 19 Go back to Eq. (12), the density of states in a nano wire is where E=E-Enx-Enz

5. Density of states in 0-D (Quantum dots)The k values are discrete in all three directions

6. (1) n = half of the number of carriers per unit length (carrier density) e = electron charge = increase in velocity due to constriction. and f(Ef)=1 at 0°K. Where, N(E) = r1D(E), Here, (1b) Fermi energy is expressed in terms of carrier velocity (Fermi velocity) Vf (1c) Substituting Eq. 1c in Eq. 1b, 2(a) Now we need an expression for the increase in velocity due to constriction or applied voltage V. (2b) e*external voltage V = eV. This gives rise to change in velocity

7. Derivation of current-voltage relation in 1-D wires/nanotubes continued e * external voltage V = eV = (3) If << Vf , eV = as we neglect the first term in Eq. 3. = Or (4) Substituting Eq. 2(a) and Eq. 4 into Eq. 1: (5a) (non-magnetic field case where gs = 2) (5b) Multiple sub-bands or multiple channels The total resistance R is expressed as: G = (6)

8. Resistance in 1-D wires/nanotubes as a function of gate voltage Vg/

9. Ballistic, quasi-ballistic, and diffusive Transport L is the length and W the width of the quantum wire or constriction; le is the elastic mean free path between impurity scattering processes, and l is the inelastic or phase-breaking mean free path between phonon scattering events.

10. Ballistic: l >> le >> L, W. Electrons can only experience the boundaries of the wire, and quantum states extend from one end to the other. Any occupied states carry current from one end to the other. When there is no applied voltage, the left and the right currents cancel each other. If a small voltage V is applied the only states present on the left and the right ends are those who have chemical potentials L and R on the left and on the right respectively. This imbalance gives a net current that is proportional to the chemical potential difference, LR = eV. For the electrons in the quantum wire, their paths are determined by scattering on the potential walls of the wire in a perfectly ballistic way. If W is small compared with the Fermi wavelength, then only one of a few channels can be occupied. Quasi-ballistic and Universal Conductance Fluctuation (UCF) regimes: Refer to figure 1(b). In this regime, there are a few impurities in the wire, and transport is via channel but scattering introduced by the impurities mixes the modes, and increase the reflection probability of electrons entering the wire. It is also possible for electrons via multiple scattering on the walls and on a few impurities to be trapped in states that are localized on the scale of le. These states have no contact to her reservoirs, and they do not contribute to the transport. The conductance in this regime depends on the precise positions of the impurities and the potentials that define the wire, and conductance changes in the order of e2/h when the potentials or the sample is changed. Quasi-ballistic regime: l >> L >> le; UCF: l >> L >> le >> W.

11. Weakly Localized Regime: l >> L >> W >> le In this situation, multiple scattering on the impurities dominates, so wire modes no longer have meaning. Electrons are localized both longitudinally and transversely on the length scale le. The electrons no longer see the one-dimensionality of the wire. NO states exist that extend from one end of the wire to the other end. This 2D weakly localized regime has no conductivity at low temperatures. Diffusive regime: L, W >> l Electrons diffuse through the wire, and the transport of electrons through the wire is by scattering between the localized states, and this requires inelastic scattering. Mobility is determined by the average density of impurities, but at high temperatures when inelastic length is smaller than elastic length, phonon scattering determine mobility. Refer to Figure 1(a).

12. L5 Universal conductance fluctuation (UCF) pp. 120B Fig. 1. Biased quantum wire between two reservoirs (top). The location of Fermi levels (bottom). Fig. 2. Density of states in wire and point contact (represented by quantum well like reservoirs). When energy states extend from left hand reservoir to the right hand reservoir all across the wire, conductance is determined by quantum mechanical transmission probability of state between µR +µL wire represent a barrier between two reservoir

13. Scattering of carriers: Impurity/defect/grain boundary scattering, Ionized impurity –scattering is elastic (no energy exchange and maintain phase relationship) Phonon scattering (acoustical phonons and optical phonons—quantum of sound waves)-electrons scattering with phonon is inelastic. Energy can increase or decrease. Electron wave no longer the same. It is known as phase breaking scattering. No longer ballistic.

16. Ballistic or quasi-ballistic transport is wave like. Like microwaves in a waveguide. Non-locality or global nature of transport. (page 103)

17. Carrier Mobility [Chapter 2 ECE 4211] , it depends on the scattering processes . The carrier mobility μn =

18. Notes Chapter L6 (page 105-128). Optical transitions in quantum confined systems Photon absorption or emission involves electronic transitions which depend on the density of states. In turn density of states depend on the physical structure of the absorbing layer if it is thick (>10-15nm), or thin. In terms of thin, we need to know if it comprise of quantum wells, multiple quantum wells, quantum wires, and quantum dots. Absorbing or emitting layer is made of direct gap or indirect gap semiconductors. Upward transitions involve photon absorption. These transitions are with and without phonons. In indirect semiconductors, phonon participation is essential to conserve momentum. Downward transitions result in photon emission. Transitions involve valence band-to-conduction band, band to impurity band or levels, donor-to-acceptor, and intra-band. Band-to-band transitions could involve free carriers or excitonic transitions.

19. Eg Ehh1 Fig. 1(a) E = 0 Exciton Transitions, Optical Modulators and Nanowire/Nanotube We cover basics of exciton formation in quantum wells and related changes in the absorption coefficient and index of refraction. This phenomena is called confined Stark effect. Photon energy needed to create an exciton is given by   h = Eg + (Ee1 + Ehh1) - Eex (Bulk Band gap) (Zero field energy of (Binding energy Electron and holes) of exciton) If the photon energy is greater than above equation, free electron and hole pair is created. Ee1

20. Electron wave function Hole wave function Fig. 1(b) Quantum well in the presence of E Application of perpendicular Electric Field When a perpendicular electric field is applied, the potential well tilts. Its slope is related to the electric field. • As E increases Ee & Eh decreases. As a result photon energy at which absorption peak occurs shifts to lower values (Red shift) . • The interaction between electron and hole wavefunctions (and thus, the value of the optical matrix elements and absorption coefficeint) also is reduced as magnitude E is increased. This is due to the fact that electron, hole wave functions are displaced with respect to each other. Therefore, the magnitude of absorption coefficient decreases with increasing electronic filed E.

21. MQW Modulator based on change in absorption due to quantum confined Stark effect Fig.3. Responsivity of a MQW diode (acting as a photodetector or optical modulator).

22. Change in index of refraction

23. Phase Modulator: A light beam signal (pulse) undergoes a phase change as it transfers an electro-optic medium of length ‘L’ --------------- (1) In linear electro-optic medium …………………(2) E= Electric field of the RF driver; E=V/d, V=voltage and d thickness of the layer. r= linear electro-optic coefficient n= index

24. Figure compares linear and quadratic variation Dn as a funciton of field. MQWs have quadratic electro-refractive effect. Figure shows a Fabry-Perot Cavity which comprises of MQWs whose index can be tuned (Dn) as a funciton of field.

25. Fabry-Perot Modulator (pp. 178-179)

26. Fabry-Perot Modulator (pp. 180-181, ECE 5212

27. Fabry-Perot Modulator (pp. 180-181)

28. Asymmetric InGaAs-GaAs Modulator

29. Mach-Zehnder Modulator Here an optical beam is split into two using a Y-junction or a 3dB coupler. The two equal beams having ½ Iin optical power. If one of the beam undergoes a phase change , and subsequently recombine. The output Io is related as Mach-Zehnder Modulator comprises of two waveguides which are fed by one common source at the input (left side). When the phase shift is 180, the out put is zero. Hence, the applied RF voltage across the waveguide modulates the input light.

30. Absorption and Emission of Photons in Nanostructures Absorption coefficient a(hn) depends on type of transitions involving phonons (indirect) or not involving phonons (direct). Generally, absorption starts when photon energy is about the band gap. It increases as hn increases above the band gap Eg. It also depends on effective masses and density of states. (p.140, Eqs 1-2) The absorption coefficient in quantum wires is higher than in wells. It also starts at higher energy than band gap in bulk materials. Absorption coefficient is related to rate of emission. Rate of emission in quantum wells is higher than in bulk layers. Rate of emission in quantum wires is higher than in quantum wells. Rate of emission in quantum dots is higher than in quantum wires. Rate of emission has two components: Spontaneous rate of emission Stimulated rate of emission.

31. Absorption coefficient and rate of emission and complex index of refraction Index of refraction nc is complex when there are losses. nc= nr – i k where extinction coefficient k = α/4pn, here alpha is the absorption coefficient. Also, nc is related to the dielectric constant. nc = =

32. Absorption coefficient is related to rate of emission using Van Roosbroeck- Shockley. This enables obtaining an expression for radiative transition lifetime tr. There are ways to compute non-radiative lifetime tnr. The internal quantum efficiency is expressed as follows: Pr is the probability of a radiative transition. [See ECE 4211 notes on LEDs, Chapter 4. Two types of downward transitions. 1. Radiative transitions. 2. Non-radiative transition:

33. Spectral width of emitted radiation Density of states in bulk is N(E)dE = The electron concentration ‘n’ in the entire conduction band is given by (EC is the band edge) This equation assumes that the bottom of the conduction band is =0. Electron and hole concentration as a function of energy

34. Graphical method to find carrier concentration in bulk or thick film (Chapter 2 ECE 4211)

35. Graphical method to find carrier concentration in quantum well

36. Spectral width in quantum well active layer is smaller than bulk thin film active layer What is the spectral width of emitted radiation in a quantum wire or nanotube? How would you approach it?

37. ·Direct and Indirect Energy Gap Semiconductors Semiconductors are direct energy gap or indirect gap. Metals do have not energy gaps. Insulators have above 4.0eV energy gap. Fig. 10b. Energy-wavevector (E-k) diagrams for indirect and direct semiconductors. Here, wavevector k represents momentum of the particle (electron in the conduction band and holes in the valence band). Actually momentum is = (h/2p)k = k

38. Absorption coefficient relations Transitions (upward) involve: • Valence band-to-conduction band A= • Direct band-to-band • Indirect band-to-band • Band to band involving excitons (hn < Eg) • Band to impurity band or levels, • Donor level-to-acceptor level, and • Intra-band or free carrier absorption. Downward direction result in emission of photons.

39. Effect of strain on band gap (page 106) Ref: W. Huang, 1995 UConn doctoral thesis with F. Jain • Under the tensile strain, the light hole band is lifted above the heavy hole, resulting in a smaller band gap. • Under a compressive strain the light hole is pushed away from heavy. As a result the effective band gap as well as light and heavy hole m asses are a function of lattice strain. Generally, the strain is +/- 0.5-1.5%. "+" for tensile and "-" for compressive. • Strain does not change the nature of the band gap. That is, direct band gap materials remain direct gap and the indirect gap remain indirect.

40. DEVICES based on optical transitionsEmission: LEDs Lasers PhotodetectorsAbsorption: Solar cellsOptical modulators and switches (quantum confined Stark effect)Optical logic

41. Why quantum well, wire and dot lasers, modulators and solar cells? Quantum Dot Lasers: • Low threshold current density and improved modulation rate. • Temperature insensitive threshold current density in quantum dot lasers. Quantum Dot Modulators: • High field dependent Absorption coefficient (α ~160,000 cm-1) : Ultra-compact intensity modulator • Large electric field-dependent index of refraction change (Δn/n~ 0.1-0.2): Phase or Mach-Zhender Modulators Radiative lifetime τr ~ 14.5 fs (a significant reduction from 100-200fs). Quantum Dot Solar Cells: High absorption coefficent enables very thin films as absorbers. Excitonic effects require use of pseudomorphic cladded nanocrystals (quantum dots ZnCdSe-ZnMgSSe, InGaN-AlGaN) Table I Computed threshold current density (Jth) as a function of dot size infor InGaN/AlGaN Quantum Dot Lasers (p.11) (Ref. F. Jan and W. Huang, J. Appl. Phys. 85, pp. 2706-2712, March 1999).

42. Transitions in Quantum Wires: The probability of transition Pmo from an energy state "0" to an energy state in the conduction band "m" is given by an expression (derived in ECE 5212). This is related to the absorption coefficient a. It depends on the nature of the transition. The gain coefficient g depends on the absorption coefficient ain the following way: g = -a(1- fe - fh ). The gain coefficient g can be expressed in terms of absorption coefficient a, and Fermi-Dirac distribution functions fe and fh for electrons and holes, respectively. Here, fe is the probability of finding an electron at the upper level and fh is the probability of finding a hole at the lower level. Free carrier transitions: A typical expression for g in semiconducting quantum wires, involving free electrons and free holes, is given by:

43. Excitonic Transitions in Quantum Wires Excitonic Transitions: This gets modified when the exciton binding energy in a system is rather large as compared to phonon energies (~kT). In the case of excitonic transitions, the gain coefficient is:

44. Laser Modulators The gain coefficient depends on the current density as well as losses in the cavity or distributed feedback structure. Threshold current density is obtained once we substitute g from the above equation. For modulators, we need to know: • Absorption and its dependence on electric field (for electroabsorptive modulators) • Index of refraction nr and its variation Dn as a function of electric field (electrorefractive modulators) • Change of the direction of polarization, in the case of birefringent modulators. In the presence of absorption, the dielectric constant e (e1-je2) and index of refraction n (nr –j k) are complex. However, their real and imaginary parts are related via Kramer Kronig's relation. See more in the write-up for Stark Effect Modulators. Lasers and Modulators: In the case of lasers, we need to find the gain coefficient and its relationship with fe and fh, which in turn are dependent on the current density (injection laser) or excitation level of the optical pump (optically pumped lasers).

45. p114 notes 4. Quasi Fermi Levels Efn, Efp, and Dz Quantum Wells Quantum Wires Quantum Dots

46. Laser operating wavelength (p 114) 5. Operating Wavelength l: The operating wavelength of the laser is determined by resonance condition L=ml/2nr Since many modes generally satisfy this condition, the wavelength for the dominant mode is obtained by determining which gives the maximum value of the value of the gain. In addition, the index of refraction, nr, of the active layer is dependent on the carrier concentration, and knowing its dependence on the current density or gain is important. For this we need to write the continuity equation. 6. Continuity equation and dependence of index of refraction on injected carrier concentration The rate of increase in the carrier concentration in the active layer due to forward current density J can be expressed as: