PHYS 3313 – Section 001 Lecture #21

PHYS 3313 – Section 001Lecture #21 Monday, Nov. 26, 2012 Dr. Jaehoon Yu • Isotope Effect • Cooper Pair • Application of Superconductivity • Semi-Conductor • Nano-technology PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Announcements • Your presentations are in classes on Dec. 3 and Dec. 5 • All presentation ppt files must be sent to me by 8pm Sunday, Dec. 2 • Final exam is 11am – 1:30pm, Monday, Dec. 10 • Reading assignments • CH10.1, 10.3 and 10.4 • Colloquium this week PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Superconductivity Isotope effect: • M is the mass of the particular superconducting isotope. Tc is a bit higher for lighter isotopes. • Mercury with M=119.5u, Tc=4.815K • Mercury isotope with M=203.4u, Tc=4.146K • It indicates that the lattice ions are important in the superconducting stateunlike classical conduction where zero resistivity can only come from non-interaction between electrons and the lattice Bardeen-Cooper-Schrieffer theory (electron-phonon interaction): • Electrons form Cooper pairswhich propagate throughout the lattice. • Propagation is without resistance because the electrons move in resonance with the lattice vibrations (phonons). PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Superconductivity How is it possible for two electrons to form a coherent pair? Consider the crude model. Each of the two electrons experiences a net attraction toward the nearest positive ion. Relatively stable electron pairs can be formed. The two fermions combine to form a boson. Then the collection of these bosons condense to form the superconducting state. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Superconductivity • Considering just one of the two electrons, the propagation wave that is created by lattice deformation due to the Coulomb attraction between the electron and ions is associated with phonon transmission, and the electron-phonon resonance allows the electron to move without resistance. • The complete BCS theory predicts other observed phenomena. • An isotope effect with an exponent very close to 0.5. • It gives a critical field. • Predicts that metals with higher resistivity in room temperature are better superconductors PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Superconductivity Quantized!! Quantum Fluxoid A conducting electron in the first excited state with the energy gap Egwrtthe ground state no longer moves with 0 resistance. This means that Eg is the energy needed to break a Cooper pair apart Eg(0) ≈ 3.54kTc at T = 0. Magnetic flux through a superconducting ring PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

The Search for a Higher Tc • Keeping materials at extremely low temperatures is very expensive and requires cumbersome insulation techniques. • Making liquid He is very hard PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

The Search for a Higher Tc The copper oxide superconductors fall into a category of ceramics. Most ceramic materials are not easy to mold into convenient shapes. There is a regular variation of Tc with n. Tc of thallium-copper oxide with n = 3 PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

The Search for a Higher Tc Higher values of n correspond to more stacked layers of copper and oxygen. thallium-based superconductor PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Superconducting Fullerenes • Another class of exotic superconductors is based on the organic molecule C60. • Although pure C60 is not superconducting, the addition of certain other elements can make it so. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Applications of Superconductivity Josephson junctions: • The superconductor / insulator / superconductor layer constitutions. • In the absence of any applied magnetic or electric field, a DC current will flow across the junction (DC Josephson effect). • Junction oscillates with frequency when a voltage is applied (AC Josephson effect). • They are used in devices known as SQUIDs. SQUIDs are useful in measuring very small amounts of magnetic flux. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Applications of Superconductivity • In an electrodynamic (EDS) system, magnets on the guideway repel the car to lift it. • In an electromagnetic (EMS) system, magnets attached to the bottom of the car lie below the guideway and are attracted upward toward the guideway to lift the car. Maglev: Magnetic levitation of trains PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Generation and Transmission of Electricity • Significant energy savings if the heavy iron cores used today could be replaced by lighter superconducting magnets. • Expensive transformers would no longer have to be used to step up voltage for transmission and down again for use. • Energy loss rate for transformers is • MRI obtains clear pictures of the body’s soft tissues, allowing them to detect tumors and other disorders of the brain, muscles, organs, and connective tissues. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

CHAPTER 11Semiconductor Theory and Devices • 11.1 Band Theory of Solids • 11.2 Semiconductor Theory • 11.3 Semiconductor Devices • 11.4 Nanotechnology It is evident that many years of research by a great many people, both before and after the discovery of the transistor effect, has been required to bring our knowledge of semiconductors to its present development. We were fortunate to be involved at a particularly opportune time and to add another small step in the control of Nature for the benefit of mankind. - John Bardeen, 1956 Nobel lecture

Categories of Solids • There are three categories of solids, based on their conducting properties: • conductors • semiconductors • insulators • The electrical conductivity at room temperature is quite different for each of these three kinds of solids • Metals and alloys have the highest conductivities • followed by semiconductors • and then by insulators

Semiconductor Conduction • The free-electron model does not apply to semiconductors and insulators, since these materials simply lack enough free electrons to conduct in a free-electron mode. • There is a different conduction mechanism for semiconductors than for normal conductors.

Band Theory of Solids • In order to account for decreasing resistivitywith increasing temperature as well as other properties of semiconductors, a new theory known as the band theory is introduced. • The essential feature of the band theory is that the allowed energy states for electrons are nearly continuous over certain ranges, called energy bands, with forbidden energy gaps between the bands. • Consider initially the known wave functions of two hydrogen atoms far enough apart so that they do not interact.

Interaction of the wave functions occurs as the atoms get closer: An atom in the symmetric state has a nonzero probability of being halfway between the two atoms, while an electron in the antisymmetric state has a zero probability of being at that location. In the symmetric case the binding energy is slightly stronger resulting in a lower energy state. Thus there is a splitting of all possible energy levels (1s, 2s, and so on) When more atoms are added (as in a real solid), there is a further splitting of energy levels. With a large number of atoms, the levels are split into nearly continuous energy bands, with each band consisting of a number of closely spaced energy levels. Band Theory of Solids Symmetric Antisymmetric

An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. Kronig and Penney assumed that an electron experiences an infinite one-dimensional array of finite potential wells in 1931 Each potential well models attraction to an atom in the lattice, so the size of the wells must correspond roughly to the lattice spacing. Kronig-Penney Model

Since the electrons are not free their energies are less than the height V0 of each of the potentials, but the electron is essentially free in the gap 0 < x <a, where it has a wave function of the form where the wave number k is given by the usual relation: In the region between a < x < a + b the electron can tunnel through and the wave function loses its oscillatory solution and becomes exponential: Kronig-Penney Model

Matching solutions at the boundary, Kronig and Penney find Here K is another wave number. The left-hand side is limited to values between +1 and −1 for all values of K. Plotting this it is observed there exist restricted (shaded) forbidden zones for solutions. Kronig-Penney Model

Important differences between the Kronig-Penney model and the single potential well • For an infinite lattice the allowed energies within each band are continuous rather than discrete. In a real crystal the lattice is not infinite, but even if chains are thousands of atoms long, the allowed energies are nearly continuous. • In a real three-dimensional crystal it is appropriate to speak of a wave vector . The allowed ranges for constitute what are referred to in solid state theory as Brillouin zones. • In a real crystal the potential function is more complicated than the Kronig-Penney squares. Thus, the energy gaps are by no means uniform in size. The gap sizes may be changed by the introduction of impurities or imperfections of the lattice. • These facts concerning the energy gaps are of paramount importance in understanding the electronic behavior of semiconductors.

Band Theory and Conductivity • Band theory helps us understand what makes a conductor, insulator, or semiconductor. • Good conductors like copper can be understood using the free electron. • It is also possible to make a conductor using a material with its highest band filled, in which case no electron in that band can be considered free. • If this filled band overlaps with the next higher band, however (so that effectively there is no gap between these two bands) then an applied electric field can make an electron from the filled band jump to the higher level. • This allows conduction to take place, although typically with slightly higher resistance than in normal metals. Such materials are known as semimetals.

Valence and Conduction Bands • The band structures of insulators and semiconductors resemble each other qualitatively. Normally there exists in both insulators and semiconductors a filled energy band (referred to as the valence band) separated from the next higher band (referred to as the conduction band) by an energy gap. • If this gap is at least several electron volts, the material is an insulator. It is too difficult for an applied field to overcome that large an energy gap, and thermal excitations lack the energy to promote sufficient numbers of electrons to the conduction band. • For energy gaps smaller than about 1 electron volt, it is possible for enough electrons to be excited thermally into the conduction band, so that an applied electric field can produce a modest current. The result is a semiconductor.

Semiconductor Theory • At T = 0 we expect all of the atoms in a solid to be in the ground state. The distribution of electrons (fermions) at the various energy levels is governed by the Fermi-Dirac distribution of Equation: • β = (kT)−1 and EF is the Fermi energy. • When the temperature is increased from T = 0, more and more atoms are found in excited states. • The increased number of electrons in excited states explains the temperature dependence of the resistivity of semiconductors. Only those electrons that have jumped from the valence band to the conduction band are available to participate in the conduction process in a semiconductor. More and more electrons are found in the conduction band as the temperature is increased, and the resistivity of the semiconductor therefore decreases.

Some Observations • Although it is not possible to use the Fermi-Dirac factor to derive an exact expression for the resistivity of a semiconductor as a function of temperature, some observations follow: • The energy E in the exponential factor makes it clear why the band gap is so crucial. An increase in the band gap by a factor of 10 (say from 1 eV to 10 eV) will, for a given temperature, increase the value of exp(βE) by a factor of exp(9βE). • This generally makes the factor FFD so small that the material has to be an insulator. • Based on this analysis, the resistance of a semiconductor is expected to decrease exponentially with increasing temperature. • This is approximately true—although not exactly, because the function FFD is not a simple exponential, and because the band gap does vary somewhat with temperature.

Holes and Intrinsic Semiconductors • When electrons move into the conduction band, they leave behind vacancies in the valence band. These vacancies are called holes. Because holes represent the absence of negative charges, it is useful to think of them as positive charges. • Whereas the electrons move in a direction opposite to the applied electric field, the holes move in the direction of the electric field. • A semiconductor in which there is a balance between the number of electrons in the conduction band and the number of holes in the valence band is called an intrinsic semiconductor. • Examples of intrinsic semiconductors include pure carbon and germanium.

Impurity Semiconductor • It is possible to fine-tune a semiconductor’s properties by adding a small amount of another material, called a dopant, to the semiconductor creating what is a called an impurity semiconductor. • As an example, silicon has four electrons in its outermost shell (this corresponds to the valence band) and arsenic has five. • Thus while four of arsenic’s outer-shell electrons participate in covalent bonding with its nearest neighbors (just as another silicon atom would), the fifth electron is very weakly bound. • It takes only about 0.05 eV to move this extra electron into the conduction band. • The effect is that adding only a small amount of arsenic to silicon greatly increases the electrical conductivity.

n-type Semiconductor • The addition of arsenic to silicon creates what is known as an n-type semiconductor (n for negative), because it is the electrons close to the conduction band that will eventually carry electrical current. • The new arsenic energy levels just below the conduction band are called donor levels because an electron there is easily donated to the conduction band.

Acceptor Levels • Consider what happens when indium is added to silicon. • Indium has one less electron in its outer shell than silicon. The result is one extra hole per indium atom. The existence of these holes creates extra energy levels just above the valence band, because it takes relatively little energy to move another electron into a hole • Those new indium levels are called acceptor levels because they can easily accept an electron from the valence band. Again, the result is an increased flow of current (or, equivalently, lower electrical resistance) as the electrons move to fill holes under an applied electric field • It is always easier to think in terms of the flow of positive charges (holes) in the direction of the applied field, so we call this a p-type semiconductor (p for positive). • In addition to intrinsic and impurity semiconductors, there are many compound semiconductors, which consist of equal numbers of two kinds of atoms.

Thermoelectric Effect • When there is a temperature gradient in a thermoelectric material, an electric field appears. • This happens in a pure metal since we can assume the system acts as a gas of free electrons. • As in an ideal gas, the density of free electrons is greater at the colder end of the wire, and therefore the electrical potential should be higher at the warmer end and lower at the colder end. • The free-electron model is not valid for semiconductors; nevertheless, the conducting properties of a semiconductor are temperature dependent, as we have seen, and therefore it is reasonable to believe that semiconductors should exhibit a thermoelectric effect.

Thermoelectric Effect • In one dimension the induced electric field E in a semiconductor is proportional to the temperature gradient, so that where Q is called the thermoelectric power. • The direction of the induced field depends on whether the semiconductor is p-type or n-type, so the thermoelectric effect can be used to determine the extent to which n- or p-type carriers dominate in a complex system. • This thermoelectric effect is sometimes called the Seebeck effect.

The Thomson, Peltier Effects & Thermocouple • Thompson Effect: In a normal conductor, heat is generated at the rate of I2R. But a temperature gradient across the conductor causes additional heat to be generated. Here heat is generated if current flows toward the higher temperature and absorbed if toward the lower. • Peltiereffect occurs when heat is generated at a junction between two conductors as current passes through the junction. • Thermocouple: An important application of the Seebeck thermoelectric effect is in thermometry. The thermoelectric power of a given conductor varies as a function of temperature, and the variation can be quite different for two different conductors.This difference makes possible the operation of a thermocouple.

Semiconductor Devices pn-junction Diodes: p-type and n-type semiconductors are joined together The principal characteristic of a pn-junction diode is that it allows current to flow easily in one direction but hardly at all in the other direction. We call these situations forward bias and reverse bias, respectively.

Zener Diodes • The Zener diode is made to operate under reverse bias once a sufficiently high voltage has been reached. The I-V curve of a Zener diode is shown in Figure 11.15. Notice that under reverse bias and low voltage the current assumes a low negative value, just as in a normal pn-junction diode. But when a sufficiently large reverse bias voltage is reached, the current increases at a very high rate. A Zener diode reference circuit. A typical I-V curve for a Zener diode.

Light Emitting Diodes (LED) • Another important kind of diode is the light-emitting diode (LED). Whenever an electron makes a transition from the conduction band to the valence band (effectively recombining the electron and hole) there is a release of energy in the form of a photon (Figure 11.17). In some materials the energy levels are spaced so that the photon is in the visible part of the spectrum. In that case, the continuous flow of current through the LED results in a continuous stream of nearly monochromatic light. Figure 11.17: Schematic of an LED. A photon is released as an electron falls from the conduction band to the valence band. The band gap may be large enough that the photon will be in the visible portion of the spectrum.

Photovoltaic Cells • An exciting application closely related to the LED is the solar cell, also known as the photovoltaic cell. Simply put, a solar cell takes incoming light energy and turns it into electrical energy. A good way to think of the solar cell is to consider the LED in reverse (Figure 11.18). A pn-junction diode can absorb a photon of solar radiation by having an electron make a transition from the valence band to the conduction band. In doing so, both a conducting electron and a hole have been created. If a circuit is connected to the pn junction, the holes and electrons will move so as to create an electric current, with positive current flowing from the p side to the n side. Even though the efficiency of most solar cells is low, their widespread use could potentially generate significant amounts of electricity. Remember that the “solar constant” (the energy per unit area of solar radiation reaching the Earth) is over 1400 W/m2, and more than half of this makes it through the atmosphere to the Earth’s surface. There has been tremendous progress in recent years toward making solar cells more efficient. Figure 11.18: (a) Schematic of a photovoltaic cell. Note the similarity to Figure 11.17. (b) A schematic showing more of the working parts of a real photovoltaic cell. From H. M. Hubbard, Science 244, 297-303 (21 April 1989).

Transistors • Another use of semiconductor technology is in the fabrication of transistors, devices that amplify voltages or currents in many kinds of circuits. The first transistor was developed in 1948 by John Bardeen, William Shockley, and Walter Brattain (Nobel Prize, 1956). As an example we consider an npn-junction transistor, which consists of a thin layer of p-type semiconductor sandwiched between two n-type semiconductors. The three terminals (one on each semiconducting material) are known as the collector, emitter, and base. A good way of thinking of the operation of the npn-junction transistor is to think of two pn-junction diodes back to back. Figure 11.22: (a) In the npn transistor, the base is a p-type material, and the emitter and collector are n-type. (b) The two-diode model of the npn transistor. (c) The npn transistor symbol used in circuit diagrams. (d) The pnp transistor symbol used in circuit diagrams.

Semiconductor Lasers • Like the gas lasers described, semiconductor lasers operate using population inversion—an artificially high number of electrons in excited states • In a semiconductor laser, the band gap determines the energy difference between the excited state and the ground state • Semiconductor lasers use injection pumping, where a large forward current is passed through a diode creating electron-hole pairs, with electrons in the conduction band and holes in the valence band. • A photon is emitted when an electron falls back to the valence band to recombine with the hole. • Since their development, semiconductor lasers have been used in a number of applications, most notably in fiber-optics communication. • One advantage of using semiconductor lasers in this application is their small size with dimensions typically on the order of 10−4 m. • Being solid-state devices, they are more robust than gas-filled tubes.

Integrated Circuits • The most important use of all these semiconductor devices today is not in discrete components, but rather in integrated circuits commonly called chips. • Some integrated circuits contain a million or more components such as resistors, capacitors, transistors, and logic switches. • Two benefits: miniaturization and processing speed

Moore’s Law and Computing Power Figure 11.29: Moore’s law, showing the progress in computing power over a 30-year span, illustrated here with Intel chip names. The Pentium 4 contains over 50 million transistors. Courtesy of Intel Corporation. Graph from http://www.intel.com/research/silicon/mooreslaw.htm.

Nanotechnology & Carbon Nanotubes • Nanotechnology is generally defined as the scientific study and manufacture of materials on a submicron scale. • These scales range from single atoms on the order of 0.1 nm up to 1 micron (1000 nm). • This technology has applications in engineering, chemistry, and the life sciences and, as such, is interdisciplinary. • In 1991, following the discovery of C60buckminsterfullerenes, or “buckyballs,” Japanese physicist SumioIijima discovered a new geometric arrangement of pure carbon into large molecules. • In this arrangement, known as a carbon nanotube, hexagonal arrays of carbon atoms lie along a cylindrical tube instead of a spherical ball.

Structure of a Carbon Nanotube • There is virtually no limit to the length of the tube. FromChris Ewels/www.ewels.info • leads to two types of nanotubes. A single-walled nanotube has just the single shell of hexagons as shown. • In a multi-walled nanotube, multiple layers are nested like the rings in a tree trunk. • Single-walled nanotubes tend to have fewer defects, and they are therefore stronger structurally but they are also more expensive and difficult to make.

Applications of Nanotubes • Because of their strength they are used as structural reinforcements in the manufacture of composite materials • (batteries in cell-phones use nanotubes in this way) • Nanotubes have very high electrical and thermal conductivities, and as such lead to high current densities in high-temperature superconductors. • One problem in the development of truly small-scale electronic devices is that the connecting wires in any circuit need to be as small as possible, so that they do not overwhelm the nanoscale components they connect. • In addition to the nanotubes already described, semiconductor wires (for example indium phosphide) have been fabricated with diameters as small as 5 nm. • These nanowires have been shown useful in connecting nanoscale transistors and memory circuits. These are referred to as nanotransistors

Graphene • A new material called graphene was ﬁrst isolated in 2004. Graphene is a single layer of hexagonal carbon, essentially the way a single plane of atoms appears in common graphite. • A. Geim and K. Novoselov received the 2010 Nobel Prize in Physics for “ground-breaking experiments.” Pure graphene conducts electrons much faster than other materials at room temperature. • Graphene transistors may one day result in faster computing.

Graphene Figure 11.33 Schematic diagram of graphene-based transistor developed at the University of Manchester. The passage of a single electron from source to drain registers 1 bit of information—a 0 or 1 in binary code.

Quantum Dots • Quantum dots are nanostructures made of semiconductor materials. • They are typically only a few nm across, containing up to 1000 atoms. • Each contains an electron-hole pair ]confined within the dot’s boundaries, (somewhat analogous to a particle conﬁned to a potential well discussed in Chapter 6. • Properties result from the fact that the band gap varies over a wide range and can be controlled precisely by manipulating the quantum dot’s size and shape. • They can be made with band gaps that are nearly continuous throughout the visible light range (1.8 to 3.1 eV) and beyond.

Nanotechnology and the Life Sciences • The complex molecules needed for the variety of life on Earth are themselves examples of nanoscale design. • Examples of unusual materials designed for specific purposes include the molecules that make up claws, feathers, and even tooth enamel.

Information Science • It’s possible that current photolithographic techniques for making computer chips could be extended into the hard-UV or soft x-ray range, with wavelengths on the order of 1 nm, to fabricate silicon-based chips on that scale • In the 1990s physicists learned that it is possible to take advantage of quantum effects to store and process information more efficiently than a traditional computer. To date, such quantum computers have been built in prototype but not mass-produced.

PHYS 3313 – Section 001 Lecture #21