Quantum Confinement Quantum Confinement Overview History In 1970

Quantum Confinement

Quantum Confinement: Overview • 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.

• 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!!

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!

Quantum Confinement: Overview 1. Thin Films & Quantum Wells • Electrons are Confined in 1 Direction: They can easily have Motion in 2 Dimensions! There is 1 Dimensional Quantization! ky kx nz

2. Quantum Wires • Electrons are Confined in 2 Directions: They can easily have Motion in 1 Dimension! There is 2 Dimensional Quantization! ny kx nz

3. Quantum Dots • Electrons are Confined in 3 Directions: They can easily have Motion in 0 Dimensions! There is 3 Dimensional Quantization! Each further confinement direction nz ny nx changes a continuous k component to a discrete component characterized by a quantum number n.

• PHYSICS: Back to the bandstructure chapter: • Consider the 1 st 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/(2 mo) • So, the energy at the Brillouin Zone edge has the form: Em ( km)2/(2 mo) Or Em ( )2/(2 moa 2)

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 max. wavevectors are of the order: ks ( /L) At the BZ edge in the z direction, the energy has the form: Es ( )2/(2 mo. L 2) + E 2(k) = the 2 dimensional energy for k in the x, y plane. Note that: ( )2/(2 mo. L 2) << ( )2/(2 moa 2)

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/(2 m*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.

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: Eg. A < Eg. B 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.

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

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

QM Review: The 1 d (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/(2 mo)](d 2 ψ/dx 2) = Eψ • Boundary Conditions: ψ = 0 at x = 0 & x = L (V there) • Energies: En = ( n )2/(2 mo. L 2), 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.

In 3 Dimensions… • For the 3 D 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!

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

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

The 1 d (finite) rectangular potential well A brief math summary! Define: α 2 (2 moε)/(ħ 2); β 2 [2 mo(ε - Vo)]/(ħ 2) The Schrödinger Equation becomes: (d 2/dx 2) ψ + α 2ψ = 0, -(½)b < x < (½)b (d 2/dx 2) ψ - β 2ψ = 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:

• Algebra (2 pages!) leads to: (ε/Vo) = (ħ 2α 2)/(2 mo. Vo) 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!)

• Even eigenfunction solutions (a finite number): Circle, ξ 2 + η 2 = ρ2, crosses η = ξ tan(ξ) Vo o o b

• Odd eigenfunction solutions: Circle, ξ 2 + η 2 = ρ2, crosses η = -ξ cot(ξ) Vo b o o b |E 2| < |E 1|

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

Quantization in a Thin Crystal E An energy band with continuous k quantized into N discrete points kn thin film with N atomic layers. is in a Electron Scattering EVacuum Inverse Photoemission EFermi Photoemission 0 /d k /a d = zone boundary N atomic layers with the spacing a = d/n n = 2 d / n N quantized states with kn ≈ n /d ( n = 1, …, N ) kn = 2 / n = n /d

Quantization in Thin Graphite Films E Lect. 7 b, Slide 11 1 layer = graphene 2 layers EVacuum EFermi 3 layers Photoemission 0 /d k /a 4 layers N atomic layers with spacing a = d/n : N quantized states with kn ≈ N /d layers = graphite

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

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

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 Himpsel Science 283, 1655 (1999) 2 3 4 5 Kawakami et al. Nature 398, 132 (1999) 6

The Important Electrons in a Metal Energy EFermi Energy Spread 3. 5 k. BT Transport (conductivity, magnetoresistance, screening length, . . . ) Width of the Fermi function: FWHM 3. 5 k. BT Phase transitions (superconductivity, magnetism, . . . ) Superconducting gap: Eg 3. 5 k. BTc (Tc= critical temperature)

Energy Bands of Ferromagnets Calculation Photoemission data Energy Relative to EF [e. V] Ni 0. 7 0. 9 k|| along [011] 1. 1 [Å-1 ] States near the Fermi level cause the energy splitting between majority and minority spin bands in a ferromagnet (red and green).

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)

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

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”

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, Mg. O)

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

Quantum confinement Ø Trap particles and restrict their motion Ø Quantum confinement produces new material behavior/phenomena Ø “Engineer confinement”- control for specific applications Ø Structures ØQuantum dots (0 -D) only confined states, and no freely moving ones ØNanowires (1 -D) particles travel only along the wire ØQuantum wells (2 -D) confines particles within a thin layer http: //www. me. berkeley. edu/nti/englander 1. ppt http: //phys. educ. ksu. edu/vqm/index. html (Scientific American)

Figure 11: Energy-band profile of a structure containing three quantum wells, showing the confined states in each well. The structure consists of Ga. As wells of thickness 11, 8, and 5 nm in Al 0. 4 Ga 0. 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 electrooptical 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. http: //www. utdallas. edu/~frensley/technical/hetphys/node 11. html#SECTION 000500000000 http: //www. utdallas. edu/~frensley/technical/hetphys. html

http: //www. eps 12. kfki. hu/files/Woggon. EPSp. pdf

http: //www. evidenttech. com/pdf/wp_biothreat. pdf http: //www. evidenttech. com/why_nano. php

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 selfassembly 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. . . http: //www. evidenttech. com/news. php

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 <30 <40 1 st 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 (4 ml) SG-Cd. Se-Na-TOL 05 -535 -04 05 -560 -04 05 -585 -04 05 -610 -04 05 -640 -04 Part Number (8 ml) SG-Cd. Se-Na-TOL 05 -535 -08 05 -560 -08 05 -585 -08 05 -610 -08 05 -640 -08 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. http: //www. evidenttech. com/why_nano/docs. php http: //www. evidenttech. com/index. php

Evi. Array Capitalizing on the distinctive properties of Evi. Dots™, we have devised a unique and patented microarray assembly. The Evi. Array™ 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. http: //www. evidenttech. com/why_nano/docs. php

Evi. Dots - Semiconductor nanocrystals Evi. Fluors- Biologically functionalized Evi. Dots Evi. Probes- Oligonucleotides with Evi. Dots Evi. Arrays- Evi. Probe-based assay system Optical Transistor- All optical 1 picosecond performance Telecommunications- Optical Switching based on Evi. Dots Energy and Lighting- Tunable bandgap semiconductor

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 Ø Diameter – 10 s of nanometers Ø Single crystal formation -- common crystallographic orientation along the nanowire axis Ø Minimal defects within wire Ø Minimal irregularities within nanowire arrays http: //www. me. berkeley. edu/nti/englander 1. ppt

Nanowire fabrication Ø Challenging! Ø Template assistance Ø Electrochemical deposition Ø Ensures fabrication of electrically continuous wires since only takes place on conductive surfaces Ø Applicable to a wide range of materials Ø High pressure injection Ø Limited to elements and heterogeneously-melting compounds with low melting points Ø Does not ensure continuous wires Ø Does not work well for diameters < 30 -40 nm Ø CVD Ø Laser assisted techniques http: //www. me. berkeley. edu/nti/englander 1. ppt

Magnetic nanowires Ø Important for storage device applications Ø Cobalt, gold, copper and cobalt-copper nanowire arrays have been fabricated Ø Electrochemical deposition is prevalent fabrication technique Ø <20 nm diameter nanowire arrays have been fabricated Cobalt nanowires on Si substrate (UMass Amherst, 2000) http: //www. me. berkeley. edu/nti/englander 1. ppt

Silicon nanowire CVD growth techniques Ø With Fe/Si. O 2 gel template (Liu et al, 2001) Ø Mixture of 10 sccm Si. H 4 & 100 sccm helium, 5000 C, 360 Torr and deposition time of 2 h Ø Straight wires w/ diameter ~ 20 nm and length ~ 1 mm Ø With Au-Pd islands (Liu et al, 2001) Ø Mixture of 10 sccm Si. H 4 & 100 sccm helium, 8000 C, 150 Torr and deposition time of 1 h Ø Amorphous Si nanowires Ø Decreasing catalyst size seems to improve nanowire alignment Ø Bifurcation is common Ø 30 -40 nm diameter and length ~ 2 mm http: //www. me. berkeley. edu/nti/englander 1. ppt

Template assisted nanowire growth Ø Create a template for nanowires to grow within Ø Based on aluminum’s unique property of self organized pore arrays as a result of anodization to form alumina (Al 2 O 3) Ø Very high aspect ratios may be achieved Ø Pore diameter and pore packing densities are a function of acid strength and voltage in anodization step Ø Pore filling – nanowire formation via various physical and chemical deposition methods http: //www. me. berkeley. edu/nti/englander 1. ppt

Al 2 O 3 template preparation Ø Anodization of aluminum Ø Start with uniform layer of ~1 mm Al Ø Al serves as the anode, Pt may serve as the cathode, and 0. 3 M oxalic acid is the electrolytic solution Ø Low temperature process (2 -50 C) Ø 40 V is applied Ø Anodization time is a function of sample size and distance between anode and cathode Ø Key Attributes of the process (per M. Sander) Ø Pore ordering increases with template thickness – pores are more ordered on bottom of template Ø Process always results in nearly uniform diameter pore, but not always ordered pore arrangement Ø Aspect ratios are reduced when process is performed when in contact with substrate (template is ~0. 3 -3 mm thick) http: //www. me. berkeley. edu/nti/englander 1. ppt

The alumina (Al 2 O 3) template (T. Sands/ HEMI group http: //www. mse. berkeley. edu/groups/Sands/HEMI/nano. TE. html) alumina template Si substrate 100 nm (M. Sander) http: //www. me. berkeley. edu/nti/englander 1. ppt

Electrochemical deposition Ø Works well with thermoelectric materials and metals Ø Process allows to remove/dissolve oxide barrier layer so that pores are in contact with substrate Ø Filling rates of up to 90% have been achieved Bi 2 Te 3 nanowire unfilled pore alumina template http: //www. me. berkeley. edu/nti/englander 1. ppt (T. Sands/ HEMI group http: //www. mse. berkeley. edu/groups/Sands/HEMI/nano. TE. html)

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

Nanometer gap between metallic electrodes Before breaking After breaking SET with a 5 nm Cd. Se nanocrystal Electromigration caused by electrical current flowing through a gold nanowire yields two stable metallic electrodes separated by about 1 nm with high efficiency. The gold nanowire was fabricated by electron-beam lithography and shadow evaporation. http: //www. lassp. cornell. edu/lassp_data/mceuen/homepage/Publications/EMPaper. pdf

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. http: //dochost. rz. hu-berlin. de/conferences/conf 1/PDF/Pascual. pdf

Short nanowire Conductance during elongation of short wires exhibits periodic quantization steps with characteristic dips, correlating with the orderdisorder states of layers of atoms in the wire. “Long” nanowire The resistance of “long” wires, as long as 100 -400 A exhibits localization characterization with ln R(L) ~ L 2 http: //dochost. rz. hu-berlin. de/conferences/conf 1/PDF/Pascual. pdf

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. http: //www. cmth. ph. ic. ac. uk/derek/research/loc. html

Molecular nanowire with negative differential resistance at room temperature http: //research. chem. psu. edu/mallouk/articles/b 203047 k. pdf

Resistivity of Er. Si 2 Nanowires on Silicon Er. Si 2 nanowires on a clean surface of Si(001). Resistance of nanowire vs its length. Er. Si 2 nanowire self-assembled along a <110> axis of the Si(001) substrate, having sizes of 1 -5 nm, 1 -2 nm and <1000 nm, in width, height, and length, respectively. The resistance per unit length is 1. 2 M� /nm along the Er. Si 2 nanowire. The resistivity is around 1�� cm, which is 4 orders of magnitude larger than that for known resistivity of bulk Er. Si 2, i. e. , 35 �� cm. One of the reasons may be due to an elasticallyelongated lattice spacing along the Er. Si 2 nanowire as a result of lattice mismatch between the Er. Si 2 and Si(001) substrate. http: //www. riken. go. jp/lab-www/surf-inter/tanaka/gyouseki/ICSTM 01. pdf

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 m. V Electronic conductance through nanometer-sized systems is quantized when its constriction varies, being the quantum of conductance, Go=2 e 2/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. http: //physics. arizona. edu/~stafford/costa-kraemer. pdf

Setup for conductance quantization studies in liquid metals. A micrometric screw is used to control the tip displacement. http: //physics. arizona. edu/~stafford/costa-kraemer. pdf 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 300 C). The applied bias voltage between tip and the metallic liquid reservoir is 90. 4 m. V.

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. http: //physics. arizona. edu/~stafford/costa-kraemer. pdf Conductance transitions due to mechanical instabilities for gold nanocontacts in UHV at RT: Transition from nine to five and to seven quantum channels.