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

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

3. (nanometers)  “Nanostructures”
Modern growth techniques (starting in the 1980’s), especially MBE & MOCVD, make 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
(nanometers)  “Nanostructures”
Superlattices = “2 dimensional” structures Quantum Wells = “2 dimensional” structures Quantum Wires = “1 dimensional” structures Quantum Dots = “0 dimensional” structures!!
Clearly, it is not only the electronic properties of materials which 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!

4. Quantum Confinement in Nanostructures: Overview
Electrons Confined in 1 Direction: Quantum Wells (thin films):  Electrons can easily move in Dimensions!
Electrons Confined in 2 Directions: Quantum Wires:  Electrons can easily move in Dimension!
Electrons Confined in 3 Directions: Quantum Dots:  Electrons can easily move in Dimensions!
Each further confinement direction changes a continuous k component to a discrete component characterized by a quantum number n. kx ky nz
1 Dimensional
Quantization! kx nz ny
2 Dimensional
Quantization! ny nz nx
3 Dimensional
Quantization!

5. km  (/a) Em  (km)2/(2mo) Em  ()2/(2moa2)
PHYSICS: Back to the bandstructure chapter:
Consider the 1st Brillouin Zone for the infinite crystal. The maximum wavevectors are of the order
km  (/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 Brillouin Zone edge. That is, they are of the form:
E  (k)2/(2mo)
So, the energy at the Brillouin Zone edge has the form:
Em  (km)2/(2mo) or
Em  ()2/(2moa2)

6. PHYSICS
SUPERLATTICES  Alternating layers of material. Periodic, with periodicity L (layer thickness). Let kz = 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 Brillouin Zone is much smaller than that for an infinite crystal. The maximum wavevectors are of the order: ks  (/L)
 At the BZ edge in the z direction, the energy has the form:
Es  ()2/(2moL2) + E2(k)
E2(k) = the 2 dimensional energy for k in the x,y plane.
Note that: ()2/(2moL2) << ()2/(2moa2)

7. Primary Qualitative Effects of Quantum Confinement
Consider electrons confined along 1 direction (say, z) to a layer of width L:
Energies
The energy bands are quantized (instead of continuous) in kz & shifted upward. So kz is 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) & shifted:
En = (n)2/(2m*L2)
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.

8. Quantum Confinement Terminology
Quantum Well  QW
= A single layer of material A (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 A layer are independent of those in other A layers.
Superlattice  SL
= Alternating layers of materials A & B with similar layer thicknesses.

9. Brief Elementary Quantum Mechanics & Solid State Physics Review
Quantum Mechanics of a Free Electron:
The energies are continuous: E = (k)2/(2mo) (1d, 2d, or 3d)
The 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 :
The 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*)
The wavefunctions are Bloch Functions = traveling waves:
Ψnk(r) = eikr unk(r); unk(r) = unk(r+R)

10. Some Basic Physics Density of states (DoS) Structure
in 3D:
Structure
Degree of Confinement
Bulk Material 0D
Quantum Well 1D 1
Quantum Wire 2D
Quantum Dot 3D
d(E)

11. 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 = 0 at x = 0 & x = L (V  there)
Energies:
En = (n)2/(2moL2), n = 1,2,3
Wavefunctions:
ψn(x) = (2/L)½sin(nx/L) (a standing wave!)
Qualitative Effects of Quantum Confinement:
Energies are quantized & ψ changes from a traveling wave to a standing wave.

12. Real Quantum Structures aren’t this simple!!
In 3Dimensions…
For the 3D infinite potential well: R
Real Quantum Structures aren’t this 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 are too simple!

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

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

15. The 1d (finite) rectangular potential well A brief math summary!
Define: α2  (2moε)/(ħ2); β2  [2mo(ε - Vo)]/(ħ2)
The Schrödinger Equation becomes:
(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/dx are continuous SO:

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

17. Even eigenfunction solutions (a finite number):
Circle, ξ2 + η2 = ρ2, crosses η = ξ tan(ξ) Vo o o b

18. Odd eigenfunction solutions:
Circle, ξ2 + η2 = ρ2, crosses η = -ξ cot(ξ)
|E2| < |E1| Vo b o o b

20. Quantum Confinement in Nanostructures
Confined in:
1 Direction: Quantum well (thin film)
Two-dimensional electrons
2 Directions: Quantum wire
One-dimensional electrons
3 Directions: Quantum dot
Zero-dimensional electrons
Each confinement direction converts a continuous k in a discrete quantum number n. kx ky nz kx nz ny ny nz nx

21. Quantization in a Thin CrystalE
/a
/d
EFermi
EVacuum
Photoemission
Inverse Photoemission
Electron Scattering k
= zone boundary
Quantization in a Thin Crystal
An energy band with continuous k is quantized into N discrete points kn in a thin film with N atomic layers. d
n = 2d / n
kn = 2 / n = n  /d
N atomic layers with the spacing a = d/n
N quantized states with kn ≈ n  /d ( n = 1,…,N )

22. Quantization in Thin Graphite Films
Lect. 7b, Slide 11
1 layer = graphene
2 layers
EVacuum
EFermi
3 layers
Photoemission
/d k
/a
 layers = graphite
4 layers
N atomic layers with spacing a = d/n :
 N quantized states with kn ≈ N  /d

23. becoming continuous for N  
Quantum Well States
in Thin Films
becoming continuous for N  
discrete for small N
Paggel et al.
Science 283, 1709 (1999)

24. Counting Quantum Well States
Periodic Fermi level crossing of quantum well states with increasing thickness n
Number of monolayers N

25. Quantum Well Oscillations in Electron Interferometers
Fabry-Perot interferometer model: Interfaces act like mirrors for electrons. Since electrons have so short wavelengths, the interfaces need to be atomically precise. n 1 2 3 4 5 6
Himpsel
Science 283, 1655 (1999)
Kawakami et al.
Nature 398, 132 (1999)

26. The Important Electrons in a Metal
Energy  EFermi
Energy Spread  3.5 kBT
Transport (conductivity, magnetoresistance, screening length, ...)
Width of the Fermi function:
FWHM  3.5 kBT
Phase transitions (superconductivity, magnetism, ...)
Superconducting gap:
Eg  3.5 kBTc (Tc= critical temperature)

27. Energy Bands of Ferromagnets
Calculation Photoemission data Ni
Energy Relative to EF [eV]
k|| along [011] [Å-1 ]
States near the Fermi level cause the energy splitting between majority and minority spin bands in a ferromagnet (red and green).

28. Quantum Well States and Magnetic Coupling
The magnetic coupling between layers plays a key role in giant magnetoresistance (GMR), the Nobel prize winning technology used for reading heads of hard disks. This coupling oscillates in sync with the density of states at the Fermi level.
(Qiu, et al.
PR B ‘92)

29. Spin-Polarized Quantum Well States
Magnetic interfaces reflect the two spins differently, causing a spin polarization.
Minority spins discrete,
Majority spins continuous

30. Giant Magnetoresistance and Spin - Dependent Scattering
Parallel Spin Filters  Resistance Low
Opposing Spin Filters  Resistance High
Filtering mechanisms
Interface: Spin-dependent Reflectivity  Quantum Well States
Bulk: Spin-dependent Mean Free Path  Magnetic “Doping”

31. Giant Magnetoresistance (GMR):
Magnetoelectronics
Spin currents instead of charge currents
Magnetoresistance = Change of the resistance in a magnetic field
Giant Magnetoresistance (GMR):
(Metal spacer, here Cu)
Tunnel Magnetoresistance (TMR):
(Insulating spacer, MgO)

33. ELEC 7970 Special Topics on Nanoscale Science and Technology
Quantum Wells, Nanowires, and Nanodots
Summer 2003
Y. Tzeng
ECE
Auburn University

34. Quantum confinement Trap particles and restrict their motion
Quantum confinement produces new material behavior/phenomena
“Engineer confinement”- control for specific applications
Structures
Quantum dots (0-D) only confined states, and no freely moving ones
Nanowires (1-D) particles travel only along the wire
Quantum wells (2-D) confines particles within a thin layer
(Scientific American)

35. Figure 11: Energy-band profile of a structure containing three quantum wells, showing the confined states in each well. The structure consists of GaAs wells of thickness 11, 8, and 5 nm in Al0.4  Ga0.6  As barrier layers. The gaps in the lines indicating the confined state energies show the locations of nodes of the corresponding wavefunctions.
Quantum well heterostructures are key components of many optoelectronic devices, because they can increase the strength of electro-optical interactions by confining the carriers to small regions. They are also used to confine electrons in 2-D conduction sheets where electron scattering by impurities is minimized to achieve high electron mobility and therefore high speed electronic operation.

38. February 2003 The Industrial Physicist Magazine
Quantum Dots for Sale
                     Nearly 20 years after their discovery, semiconductor quantum dots are emerging as a bona fide industry with a few start-up companies poised to introduce products this year. Initially targeted at biotechnology applications, such as biological reagents and cellular imaging, quantum dots are being eyed by producers for eventual use in light-emitting diodes (LEDs), lasers, and telecommunication devices such as optical amplifiers and waveguides. The strong commercial interest has renewed fundamental research and directed it to achieving better control of quantum dot self-assembly in hopes of one day using these unique materials for quantum computing.
Semiconductor quantum dots combine many of the properties of atoms, such as discrete energy spectra, with the capability of being easily embedded in solid-state systems. "Everywhere you see semiconductors used today, you could use semiconducting quantum dots," says Clint Ballinger, chief executive officer of Evident Technologies, a small start-up company based in Troy, New York...

39. Quantum Dots for Sale The Industrial Physicist reports that quantum dots are emerging as a bona fide industry.
Emission Peak[nm]
535±10
560±10
585±10
610±10
640±10
Typical FWHM [nm]
<30
<40
1st Exciton Peak [nm - nominal]
522
547
572
597
627
Crystal Diameter [nm - nominal]
2.8
3.4
4.0
4.7
5.6
Part Number (4ml) SG-CdSe-Na-TOL
Part Number (8ml) SG-CdSe-Na-TOL
Evident Nanocrystals Evident's nanocrystals can be separated from the solvent to form self-assembled thin films or combined with polymers and cast into films for use in solid-state device applications. Evident's semiconductor nanocrystals can be coupled to secondary molecules including proteins or nucleic acids for biological assays or other applications.

40. EviArray Capitalizing on the distinctive properties of EviDots™, we have devised a unique and patented microarray assembly. The EviArray™ is fabricated with nanocrystal tagged oligonucleotide
probes that are also attached to a fixed substrate in such a way that the nanocrystals can only fluoresce when the DNA probe couples with the corresponding target genetic sequence.

41. EviDots - Semiconductor nanocrystals EviFluors- Biologically functionalized EviDots EviProbes- Oligonucleotides with EviDots EviArrays- EviProbe-based assay system
Optical Transistor- All optical 1 picosecond performance Telecommunications- Optical Switching based on EviDots Energy and Lighting- Tunable bandgap semiconductor

42. Why nanowires?
“They represent the smallest dimension for efficient transport of electrons and excitons, and thus will be used as interconnects and critical devices in nanoelectronics and nano-optoelectronics.” (CM Lieber, Harvard)
General attributes & desired properties
Diameter – 10s of nanometers
Single crystal formation -- common crystallographic orientation along the nanowire axis
Minimal defects within wire
Minimal irregularities within nanowire arrays

43. Nanowire fabrication Challenging! Template assistance
Electrochemical deposition
Ensures fabrication of electrically continuous wires since only takes place on conductive surfaces
Applicable to a wide range of materials
High pressure injection
Limited to elements and heterogeneously-melting compounds with low melting points
Does not ensure continuous wires
Does not work well for diameters < nm
CVD
Laser assisted techniques

44. Magnetic nanowires Important for storage device applications
Cobalt, gold, copper and cobalt-copper nanowire arrays have been fabricated
Electrochemical deposition is prevalent fabrication technique
<20 nm diameter nanowire arrays have been fabricated
Significantly higher storage densities – improve traditional magnetic (superparramagnetic) storage limit – improved storage capacity
Making the bits perpendicular to the surface is important for fitting a lot of information in a small area. The bits in disk drive media today are parallel to the, I’ve most common methods of fabricating magnetic nanowires.   Electrochemical deposition into a porous media is a prevalent technique.   The porous media may be porous silica, alumina, polycarbonate membranes and even polyaniline nanotubules.  Advanced lithographic techniques (electron- beam) are also used.  Nanowire diameters as low as ~10 nm have been reported and wire length may range from ~1um to >20um depending on the wire’s diameter. In the case of the 14nm Cobolt wires that I showed in my presentation, they use a heated mixture of polystyrene and polymethylmethacrylate on a silicon substrate.  The PMMA then forms 14nm cylinders.  They use an electric field to control the orientation of the cylinders.  After exposure to UV, the PMMA cylinders break down and a pore array results.  Finally, electrodeposition is used to grow the wires. With regard to your questions from Friday, I’ve looked through the literature to figure out what are the most common methods of fabricating magnetic nanowires.   Electrochemical deposition into a porous media is a prevalent technique.   The porous media may be porous silica, alumina, polycarbonate membranes and even polyaniline nanotubules.  Advanced lithographic techniques (electron- beam) are also used.  Nanowire diameters as low as ~10 nm have been reported and wire length may range from ~1um to >20um depending on the wire’s diameter. In the case of the 14nm Cobolt wires that I showed in my presentation, they use a heated mixture of polystyrene and polymethylmethacrylate on a silicon substrate.  The PMMA then forms 14nm cylinders.  They use an electric field to control the orientation of the cylinders.  After exposure to UV, the PMMA cylinders break down and a pore array results.  Finally, electrodeposition is used to grow the wires.
Cobalt nanowires on Si substrate
(UMass Amherst, 2000)

45. Silicon nanowire CVD growth techniques
With Fe/SiO2 gel template (Liu et al, 2001)
Mixture of 10 sccm SiH4 & 100 sccm helium, 5000C, 360 Torr and deposition time of 2h
Straight wires w/ diameter ~ 20nm and length ~ 1mm
With Au-Pd islands (Liu et al, 2001)
Mixture of 10 sccm SiH4 & 100 sccm helium, 8000C, 150 Torr and deposition time of 1h
Amorphous Si nanowires
Decreasing catalyst size seems to improve nanowire alignment
Bifurcation is common
30-40 nm diameter and length ~ 2mm

46. Template assisted nanowire growth
Create a template for nanowires to grow within
Based on aluminum’s unique property of self organized pore arrays as a result of anodization to form alumina (Al2O3)
Very high aspect ratios may be achieved
Pore diameter and pore packing densities are a function of acid strength and voltage in anodization step
Pore filling – nanowire formation via various physical and chemical deposition methods

47. Al2O3 template preparation
Anodization of aluminum
Start with uniform layer of ~1mm Al
Al serves as the anode, Pt may serve as the cathode, and 0.3M oxalic acid is the electrolytic solution
Low temperature process (2-50C)
40V is applied
Anodization time is a function of sample size and distance between anode and cathode
Key Attributes of the process (per M. Sander)
Pore ordering increases with template thickness – pores are more ordered on bottom of template
Process always results in nearly uniform diameter pore, but not always ordered pore arrangement
Aspect ratios are reduced when process is performed when in contact with substrate (template is ~0.3-3 mm thick)

48. The alumina (Al2O3) template
(T. Sands/ HEMI group
alumina template
Si substrate
100nm
(M. Sander)

49. Electrochemical deposition
Works well with thermoelectric materials and metals
Process allows to remove/dissolve oxide barrier layer so that pores are in contact with substrate
Filling rates of up to 90% have been achieved
Bi2Te3 nanowire
unfilled pore
alumina template
(T. Sands/ HEMI group

50. Template-assisted, Au nucleated Si nanowires
Gold evaporated (Au nanodots) into thin ~200nm alumina template on silicon substrate
Ideally reaction with silane will yield desired results
Need to identify equipment that will support this process – contamination, temp and press issues
Additional concerns include Au thickness, Au on alumina surface, template intact vs removed
Au dots Au
100nm
1µm
(M. Sander)
template (top)

51. Nanometer gap between metallic electrodes
Before breaking
SET with a 5nm CdSe nanocrystal
After breaking
Electromigration caused by electrical current flowing through a gold nanowire yields two stable metallic electrodes separated by about 1nm with high efficiency. The gold nanowire was fabricated by electron-beam lithography and shadow evaporation.

52. Quantum and localization of nanowire conductance
Nanoscale size exhibits the following properties different from those found in the bulk:
quantized conductance in point contacts and narrow channels whose characteristics (transverse) dimensions approach the electronic wave length
Localization phenomena in low dimensional systems
Mechanical properties characterized by a reduced propensity for creation and propagation of dislocations in small metallic samples.
Conductance of nanowires depend on
the length,
lateral dimensions,
state and degree of disorder and
elongation mechanism of the wire.

53. Short nanowire “Long” nanowire
Conductance during elongation of short wires exhibits periodic quantization steps with characteristic dips, correlating with the order-disorder states of layers of atoms in the wire.
The resistance of “long” wires, as long as A exhibits localization characterization with ln R(L) ~ L2

54. Electron localization
At low temperatures, the resistivity of a metal is dominated by the elastic scattering of electrons by impurities in the system. If we treat the electrons as classical particles, we would expect their trajectories to resemble random walks after many collisions, i.e., their motion is diffusive when observed over length scales much greater than the mean free path. This diffusion becomes slower with increasing disorder, and can be measured directly as a decrease in the electrical conductance.
When the scattering is so frequent that the distance travelled by the electron between collisions is comparable to its wavelength, quantum interference becomes important. Quantum interference between different scattering paths has a drastic effect on electronic motion: the electron wavefunctions are localized inside the sample so that the system becomes an insulator. This mechanism (Anderson localization) is quite different from that of a band insulator for which the absence of conduction is due to the lack of any electronic states at the Fermi level.

55. Molecular nanowire with negative differential resistance at room temperature

56. Resistivity of ErSi2 Nanowires on Silicon
ErSi2 nanowires on a clean surface of Si(001).
Resistance of nanowire vs its length.
ErSi2 nanowire self-assembled along a <110> axis of the Si(001) substrate, having sizes of 1-5nm, 1-2nm and <1000nm, in width, height, and length, respectively. The resistance per unit length is 1.2M/nm along the ErSi2 nanowire. The resistivity is around 1cm, which is 4 orders of magnitude larger than that for known resistivity of bulk ErSi2, i.e., 35 cm. One of the reasons may be due to an elastically-elongated lattice spacing along the ErSi2 nanowire as a result of lattice mismatch between the ErSi2 and Si(001) substrate.

57. Last stages of the contact breakage during the formation of nanocontacts.
Conductance current during the breakage of a nanocontact. Voltage difference between electrodes is 90.4 mV
Electronic conductance through nanometer-sized systems is quantized when its constriction varies, being the quantum of conductance, Go=2 e2/h, where e is the electron charge and h is the Planck constant, due to the change of the number of electronic levels in the constriction.
The contact of two gold wire can form a small contact resulting in a relative low number of eigenstates through which the electronic ballistic transport takes place.

58. Setup for conductance quantization studies in liquid metals
Setup for conductance quantization studies in liquid metals. A micrometric screw is used to control the tip displacement.
Evolution of the current and conductance at the first stages of the formation of a liquid metal contact. The contact forms between a copper wire and (a) mercury (at RT) and (b) liquid tin (at 300C). The applied bias voltage between tip and the metallic liquid reservoir is 90.4 mV.

59. Conductance transitions due to mechanical instabilities for gold nanocontacts in UHV at RT: (a) between 0 and 1 quantum channel. (b) between 0 and 2 quantum channels.
Conductance transitions due to mechanical instabilities for gold nanocontacts in UHV at RT: Transition from nine to five and to seven quantum channels.