1. Quantum Confinement BW, 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. Quantum Confinement in Nanostructures: Overview 1. For Electrons Confined in 1 Direction Quantum Wells (thin films)  Electrons can easily move in 2 Dimensions!  1 Dimensional Quantization! “2 Dimensional Structure” kxkx kyky nznz

9. Quantum Confinement in Nanostructures: Overview 2. For Electrons Confined in 2 Directions Quantum Wires  Electrons can easily move in 1 Dimension!  2 Dimensional Quantization! “1 Dimensional Structure” 1. For Electrons Confined in 1 Direction Quantum Wells (thin films)  Electrons can easily move in 2 Dimensions!  1 Dimensional Quantization! “2 Dimensional Structure” kxkx kyky nznz kxkx nznz nyny

10. For Electrons Confined in 3 Directions Quantum Dots  Electrons can easily move in 0 Dimensions! 3 Dimensional Quantization! “0 Dimensional Structure” nyny nznz nxnx

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

12. Consider the 1 st BZ for the infinite crystal. The maximum wavevectors at the BZ edge are of the order k m  (  /a) a = lattice constant. The potential V 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 /(2m o ) So, the energy at the BZ edge has the form: E m  (  k m ) 2 /(2m o ) or E m  (   ) 2 /(2m o a 2 ) Physics Recall the Bandstructure Chapter

13. Physics Superlattice  Alternating layers of 2 or more materials. Layers are arranged periodically, with periodicity L (= layer thickness). Let k z = wavevector perpendicular to the layers. In a superlattice, the potential V has 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. The maximumn wavevectors are of the order: k s  (  /L)  At the BZ edge in the z direction, the energy is E s  (   ) 2 /(2m o L 2 ) + E 2 (k) E 2 (k) = the 2 dimensional energy for k in the x,y plane. Note that: (   ) 2 /(2m o L 2 ) << (   ) 2 /(2m o a 2 )

14. Consider electrons confined along one direction (say, z) to a layer of width L: Energies The Energy Bands are Quantized (instead of continuous) in k z & shifted upward. So k z is quantized: k z = k n = [(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: E n = (n   ) 2 /(2m*L 2 ), Energies are quantized! Also, the wavefunctions are 2 dimensional Bloch functions (traveling waves) for k in the x,y plane & standing waves in the z direction. Primary Qualitative Effects of Quantum Confinement

15. 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: E gA < E gB 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 A layer are independent of those in other A layers. Superlattice  SL  Alternating layers of materials A & B with similar layer thicknesses. Quantum Confinement Terminology

16. Quantum Mechanics of a Free Electron: –The energies are continuous: E = (  k) 2 /(2m o ) (1d, 2d, or 3d) –Wavefunctions are Traveling Waves: ψ k (x) = A e ikx (1d), ψ k (r) = A e ik  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 = E nk –At the bottom of the conduction band or the top of the valence band, in the effective mass approximation, the bands can be written: E nk  (  k) 2 /(2m*) –Wavefunctions = Bloch Functions = Traveling Waves: Ψ nk (r) = e ik  r u nk (r); u nk (r) = u nk (r+R) Brief Elementary Quantum Mechanics & Solid State Physics Review

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

18. StructureConfinement Dimen sions (dN/dE) Bulk Material0D Quantum Well1D1 Quantum Wire2D Quantum Dot3D  (E)

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

20. 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, in real structures, there are many electrons & also holes! Also, there is often an effective mass mismatch at the boundaries. That is the boundary conditions we’ve used may be too simple!

21. QM Review: The 1d (Finite) Potential Well In most QM texts!!  Analogous to a Quantum Well We want to solve the Schrödinger Equation for: [-{ħ 2 /(2m o )}(d 2 /dx 2 ) + V]ψ = εψ (ε  E) V = 0, -(b/2) < x < (b/2); V = V o otherwise We want the bound states: ε < V o

22. Solve the Schrödinger Equation: [-{ħ 2 /(2m o )}(d 2 /dx 2 ) + V]ψ = εψ (ε  E) V = 0, -(b/2) < x < (b/2) V = V o otherwise The Bound States are in Region II Region II: ψ(x) is oscillatory Regions I & III: ψ(x) is decaying -(½)b (½)b VoVo V = 0

23. The 1d (Finite) Rectangular Potential Well A brief math summary! Define: α 2  (2m o ε)/(ħ 2 ); β 2  [2m o (ε - V o )]/(ħ 2 ) The Schrödinger Equation becomes: (d 2 /dx 2 ) ψ + α 2 ψ = 0, -(½)b < x < (½)b (d 2 /dx 2 ) ψ - β 2 ψ = 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/dx are continuous SO:

24. Algebra (2 pages!) leads to: (ε/V o ) = (ħ 2 α 2 )/(2m o V o ) ε, α, β are related to each other by transcendental equations. For example: tan(αb) = (2αβ)/(α 2 - β 2 ) Solve graphically or numerically. Get: Discrete Energy Levels in the well (a finite number of finite well levels!)

25. Even Eigenfunction solutions (a finite number): Circle, ξ 2 + η 2 = ρ 2, crosses η = ξ tan(ξ) VoVo b

26. b Odd Eigenfunction solutions (a finite number): Circle, ξ 2 + η 2 = ρ 2, crosses η = - ξ = cot(ξ) VoVo

27. Quantum Confinement Discussion Review 1. For Confinement in 1 Direction  Electrons can easily move in 2 Dimensions!  1 Dimensional Quantization! “2 Dimensional Structure” Quantum Wells (thin films) kxkx kyky nznz 2. For Confinement in 2 Directions  Electrons can easily move in 1 Dimension!  2 Dimensional Quantization! “1 Dimensional Structure” Quantum Wires kxkx nznz nyny 3. For Confinement in 3 Directions  Electrons can easily move in 0 Dimensions! 3 Dimensional Quantization! “0 Dimensional Structure” Quantum Dots nyny nznz nxnx

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