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Quantum Confinement BW, Chs. 15-18, YC, Ch. 9; S, Ch. 14; outside sources

Quantum Confinement BW, Chs. 15-18, YC, Ch. 9; S, Ch. 14; outside sources. Overview of Quantum Confinement. Brief History In 1970 Esaki & Tsu proposed fabrication of an artificial structure, which would consist of alternating layers of 2 different semiconductors with Layer Thicknesses

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Quantum Confinement BW, Chs. 15-18, YC, Ch. 9; S, Ch. 14; outside sources

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  1. Quantum ConfinementBW, Chs. 15-18, YC, Ch. 9; S, Ch. 14; outside sources

  2. Overview of Quantum Confinement Brief History • In 1970 Esaki & Tsu proposed fabrication of an artificial structure, which would consist of alternating layers of 2 different semiconductors with Layer Thicknesses  1 nm = 10 Å = 10-9 m “Superlattice” Physics • The main idea was that introduction of an artificial periodicity will “fold” the BZ’s into smaller BZ’s  “mini-BZ’s” or “mini-zones”. The idea was that this would raise the conduction band minima, which was needed for some device applications.

  3. Modern Growth Techniques • Starting in the 1980’s, MBE & MOCVD have made fabrication of such structures possible! • For the same reason, it is also possible to fabricate many other kinds of artificial structures on the scale of nm Nanometer Dimensions “Nanostructures”

  4. Modern Growth Techniques • Starting in the 1980’s, MBE & MOCVD have made fabrication of such structures possible! • For the same reason, it is also possible to fabricate many other kinds of artificial structures on the scale of nm Nanometer Dimensions “Nanostructures” Superlattices = “2 dimensional” structures Quantum Wells = “2 dimensional”structures

  5. Modern Growth Techniques • Starting in the 1980’s, MBE & MOCVD have made fabrication of such structures possible! • For the same reason, it is also possible to fabricate many other kinds of artificial structures on the scale of nm Nanometer Dimensions “Nanostructures” Superlattices = “2 dimensional” structures Quantum Wells = “2 dimensional”structures Quantum Wires = “1 dimensional”structures

  6. Modern Growth Techniques • Starting in the 1980’s, MBE & MOCVD have made fabrication of such structures possible! • For the same reason, it is also possible to fabricate many other kinds of artificial structures on the scale of nm Nanometer Dimensions “Nanostructures” Superlattices = “2 dimensional” structures Quantum Wells = “2 dimensional”structures Quantum Wires = “1 dimensional”structures Quantum Dots = “0 dimensional” structures!!

  7. Modern Growth Techniques • Starting in the 1980’s, MBE & MOCVD have made fabrication of such structures possible! • For the same reason, it is also possible to fabricate many other kinds of artificial structures on the scale of nm Nanometer Dimensions “Nanostructures” Superlattices = “2 dimensional” structures Quantum Wells = “2 dimensional”structures Quantum Wires = “1 dimensional”structures Quantum Dots = “0 dimensional” structures!! • Clearly, it’s not only the electronic properties that can be drastically altered in this way. Also, vibrational properties (phonons). Here, only electronic properties & only an overview! • For many years, quantum confinement has been a fast growing field in both theory & experiment! It is at the forefront of current research! • Note that I am not an expert on it!

  8. ky kx nz Quantum Confinement in Nanostructures: Overview 1. For Electrons Confined in 1 Direction Quantum Wells(thin films)  Electrons can easily move in 2 Dimensions!  1 DimensionalQuantization! “2 Dimensional Structure”

  9. ky kx nz ny kx nz Quantum Confinement in Nanostructures: Overview 1. For Electrons Confined in 1 Direction Quantum Wells(thin films)  Electrons can easily move in 2 Dimensions!  1 DimensionalQuantization! “2 Dimensional Structure” 2. For Electrons Confined in 2 DirectionsQuantum Wires Electrons can easily move in 1 Dimension!  2 DimensionalQuantization! “1 Dimensional Structure”

  10. nz nx ny For Electrons Confined in 3 Directions Quantum Dots  Electrons can easily move in0 Dimensions! 3 Dimensional Quantization! “0 Dimensional Structure”

  11. nz nx ny For Electrons Confined in 3 Directions Quantum Dots  Electrons can easily move in0 Dimensions! 3 Dimensional Quantization! “0 Dimensional Structure” Each further confinement direction changes a continuous k component to a discrete component characterized by a quantum number n.

  12. Physics Recall the Bandstructure Chapter • Consider the 1st BZfor the infinite crystal. • The maximum wavevectors at the BZ edge are of the order km  (/a) a = lattice constant. The potentialV is periodic with period a. In the almost free e- approximation, the bands are free e- like except near the BZ edge. That is, they are of the form: E  (k)2/(2mo) So, the energy at the BZ edge has the form: Em  (km)2/(2mo) or Em  ()2/(2moa2)

  13. Physics Superlattice Alternating layers of 2 or more materials. • Layers are arranged periodically, with periodicityL(= layer thickness). Let kz = wavevector perpendicular to the layers. • In a superlattice, the potential Vhas a new periodicity in the z direction with periodicity L >> a.  In the z direction, the BZ is much smaller than that for an infinite crystal. Themaximumn wavevectors are of the order: ks  (/L)  At the BZ edge in the z direction, the energy is Es  ()2/(2moL2) + E2(k) E2(k) =the 2 dimensional energy fork in the x,y plane. Note that:()2/(2moL2) << ()2/(2moa2)

  14. Primary Qualitative Effects of Quantum Confinement • Consider electrons confined along one direction (say, z) to a layer of width L: Energies • The Energy Bands are Quantized(instead of continuous) in kz& shifted upward. So kzis quantized: kz = kn = [(n)/L], n = 1, 2, 3 • So, in the effective mass approximation (m*), The Bottom of the Conduction Band is Quantized (like a particle in a 1 d box) & also shifted: En = (n)2/(2m*L2), Energies are quantized! • Also, the wavefunctions are 2 dimensional Bloch functions(traveling waves)for kin the x,yplane &standing waves in the z direction.

  15. Quantum Confinement Terminology Quantum Well  QW  A single thin layer of material A(of layer thickness L), sandwiched between 2 macroscopically large layers of material B. Usually, the bandgaps satisfy: EgA < EgB Multiple Quantum Well  MQW  Alternating layers of materials A(thickness L) & B(thickness L). In this case: L >> L So, the e- & e+ in one Alayer are independent of those in other Alayers. Superlattice  SL  Alternating layers of materials A& B with similar layer thicknesses.

  16. Brief Elementary Quantum Mechanics & Solid State Physics Review • Quantum Mechanicsof a Free Electron: • The energies are continuous: E = (k)2/(2mo) (1d, 2d, or 3d) • Wavefunctions are Traveling Waves: ψk(x) = A eikx(1d), ψk(r) = A eikr(2d or 3d) • Solid State Physics:Quantum Mechanics of an Electron in a Periodic Potential in an infinite crystal : • Energy bands are(approximately)continuous: E = Enk • At the bottom of the conduction band or the top of the valence band, in the effective mass approximation, the bands can be written: Enk  (k)2/(2m*) • Wavefunctions = Bloch Functions = Traveling Waves: Ψnk(r) = eikrunk(r); unk(r) = unk(r+R)

  17. More Basic Solid State Physics • Density of States(DoS)in 3D:

  18. QM Review: The 1d (Infinite) Potential Well(“particle in a box”)In all QM texts!! • We want to solve the Schrödinger Equation for: x < 0, V  ; 0 < x < L, V = 0; x > L, V   -[2/(2mo)](d2 ψ/dx2) = Eψ • Boundary Conditions: ψ = 0at x = 0 & x = L (V  there) • Solutions • Energies:En = (n)2/(2moL2), n = 1,2,3 • Wavefunctions:ψn(x) = (2/L)½sin(nx/L) (standing waves!) • Qualitative Effects of Quantum Confinement: • Energies are quantized • ψchanges from a traveling wave to a standing wave.

  19. For the 3D Infinite Potential Well R • But, Real Quantum Structures Aren’t So Simple!! • In Superlattices & Quantum Wells, the potential barrier is obviously not infinite! • In Quantum Dots, there is usually ~ spherical confinement, not rectangular! • The simple problem only considers a single electron. But, inreal structures, there are many electrons& alsoholes! • Also, there is oftenan effective mass mismatch at the boundaries. That is the boundary conditions we’ve used may be too simple!

  20. QM Review: The 1d (Finite) Potential Well In most QM texts!!  Analogous to a Quantum Well • We want to solve the Schrödinger Equationfor: We want the bound states: ε < Vo [-{ħ2/(2mo)}(d2/dx2) + V]ψ = εψ(εE) V = 0, -(b/2) < x < (b/2); V = Vo otherwise

  21. (½)b -(½)b Vo • Solve the Schrödinger Equation: [-{ħ2/(2mo)}(d2/dx2) + V]ψ = εψ (εE) V = 0, -(b/2) < x < (b/2) V = Vo otherwise The Bound States are in Region II Region II:ψ(x) is oscillatory Regions I & III:ψ(x) is decaying V= 0

  22. The 1d (Finite) Rectangular Potential WellA brief math summary! Define:α2  (2moε)/(ħ2); β2  [2mo(ε - Vo)]/(ħ2) The Schrödinger Equationbecomes: (d2/dx2) ψ + α2ψ = 0, -(½)b < x < (½)b (d2/dx2) ψ - β2ψ = 0, otherwise. Solutions: ψ = C exp(iαx) + D exp(-iαx), -(½)b < x < (½)b ψ = A exp(βx), x < -(½)b ψ = A exp(-βx), x > (½)b Boundary Conditions: ψ & dψ/dxare continuous SO:

  23. Algebra (2 pages!) leads to: (ε/Vo) = (ħ2α2)/(2moVo) ε, α, βare related to each other by transcendental equations. For example: tan(αb) = (2αβ)/(α2- β2) • Solve graphically or numerically. Get:Discrete Energy Levels in the well (a finite number of finite well levels!)

  24. Vo • Even Eigenfunction solutions (a finite number): Circle,ξ2 + η2 = ρ2, crosses η = ξ tan(ξ) b

  25. Odd Eigenfunction solutions (a finite number): • Circle,ξ2 + η2 = ρ2, crosses η = - ξ = cot(ξ) Vo b

  26. Quantum Confinement Discussion Review ky kx nz ny kx nz nz nx ny 1.For Confinement in 1 Direction  Electrons can easily move in 2 Dimensions!  1 DimensionalQuantization! “2 Dimensional Structure” Quantum Wells(thin films) 2. For Confinement in 2 Directions Electrons can easily move in1 Dimension!  2 DimensionalQuantization! “1 Dimensional Structure” Quantum Wires 3. ForConfinement in 3 Directions  Electrons can easily move in0 Dimensions! 3 Dimensional Quantization! “0 Dimensional Structure” Quantum Dots

  27. That is: Each further confinement direction changes a continuous k component to adiscrete component characterized by a quantum number n.

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