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The Quantum Hall Effect Michael Richardson In 1985, Klaus von Klitzing was awarded the Nobel Prize for his discovery of the quantized Hall effect. The quantum Hall effect has provided an amazingly accurate method for calibrating resistance. Some of the successful explanations of the effect are summarized in the following. I. INTRODUCTION The classical Hall effect was discovered in 1879 by Edwin Hall. He correctly speculated that the charge carriers in a wire would be deflected to one side of the wire if a magnetic field were set up perpendicular to the flow of the current. In other words, the magnetic field led to a transverse voltage in the wire. This transverse voltage could be measured experimentally and was seen to increase as the magnetic field increased. It is possible to demonstrate the same effect in a two dimensional electron gas. The electrons are trapped at the interface between a metal and a semiconductor. Then a current is established in one direction and a magnetic field is applied perpendicular to the electron gas. In 1980, von Klitzing did experiments to test the Hall effect on a two dimensional electron gas at very low temperature and high magnetic field. He found that the Hall conductivity (current divided by the transverse voltage) had plateaus of constant conductivity as the magnetic field was varied. More astonishingly, the values of the conductivity at these steps were integer multiples of e2/h that were measured to an accuracy of one part in 10 million. Ever since, theorists have attempted to explain the phenomenon. A fairly simple explanation involving Landau levels can be made for an ideal system. Yet more complicated explanations are needed to explain real systems. II. A SIMPLE APPROACH If the electron gas is subjected to a strong magnetic field, then the electron orbitals split into Landau levels. The number of orbitals in each level is 𝐷 = (2𝜋𝑒𝐵/ℏ𝑐)(𝐿/2𝜋)2 = 𝑒𝐵𝐿2 /ℎ𝑐. If the voltage is adjusted so that the Fermi level of the electron gas falls right between two Landau levels, then the total number of electrons in the electron gas is given by 𝑁 = 𝑠𝐷 One can rearrange these equations to determine that 𝜎 = 𝑠𝑒 2 /ℎ. No elastic collisions are possible from one state to another state in the same Landau level because all the states in the level are filled. Because the magnetic field is strong and the temperature is low, ℏ𝜔𝑐 ≫ 𝑘𝐵 𝑇. Therefore, the electrons are unlikely to absorb the necessary energy to move to the next Landau level from a phonon. As a result, the conductivity remains quantized while the Fermi energy is between the Landau levels. III. LAUGHLIN’S EXPLANATION According to the above explanation, the occurrence of the quantum Hall effect would not be expected for systems with partially filled Landau levels. Yet it does occur in such systems. Robert Laughlin used the principle of gauge invariance to explain the quantization of the conductance. He imagined a 2D electron system bent into a cylinder so that the current moves in a loop. The surface of the cylinder experiences a strong magnetic field that is everywhere normal to the cylinder. Now consider a magnetic flux through the cylinder. Any change in this flux will create an electromotive force that will cause the charges to move around the loop and, due to the magnetic field, deflect to one side of the cylinder. If the change in the magnetic flux is equal to an integral multiple of the elementary quantum of magnetic flux ℎ𝑐/𝑒, then the Hamiltonian describing the system will remain unchanged. This is because, according to the Aharonov-Bohm principle, the system is gauge invariant under flux changes of integral multiples of ℎ𝑐/𝑒. To calculate the Hall conductance, one divides the transverse current by the Hall (transverse) voltage. 𝜎 = 𝐼/𝑉𝐻 The current is found by using the electromagnetic relationships 𝜕𝑈 𝜕𝜑 = −𝑉𝐻 𝐼 = 𝐼𝑐 , 𝜕𝑡 𝜕𝑡 𝐼=𝑐 𝛿𝑈 𝛿𝜑 The energy 𝛿𝑈 is equal to 𝑁𝑒𝑉𝐻 , where N is the number of electrons that are moved to the other side of the cylinder due to the emf from the change in flux. The change in flux is the elementary quantum of magnetic flux. So 𝐼=𝑐 𝑁𝑒𝑉𝐻 ℎ𝑐/𝑒 Given that the Hall conductance is 𝜎 = 𝐼/𝑉𝐻 , the above equation yields 𝑒2 𝜎=𝑁 ℎ Because the system returns to its original state after the flux change, one assumes that if the system were to undergo the same flux change again, then the same number of electrons would be transferred. Yet quantum mechanically, two systems in the same state do not have to produce exactly the same results under the same measurement. Therefore, something more must ensure that the average number of electrons transferred is also an integer. 1 2𝜋 𝐾𝑑𝐴 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑆 where the integral is over a surface with no boundaries and K is the local curvature of the surface. Shiing-shen Chern generalized this formula to the geometry of eigenstates that can be parameterized by variables over a toroidal surface. The “adiabatic curvature” is found by taking the limit of the Berry phase mismatch divided by the loop as the loop size goes to zero. This formula shows that as long as the adiabatic curvature is the same for each cycle of increasing magnetic flux, the same integer will be produced every time. This information, combined with Laughlin’s explanation demonstrates why the Hall conductance is quantized in integer multiples of 𝑒 2 /ℎ. V. CONCLUSION The incredible precision with which one may measure the value of 𝑒 2 /ℎ has allowed for a new standard for resistance (ℎ/𝑒 2 ). The quantum Hall effect also allows a precise calculation of the fine structure constant. The theory behind the effect is still being researched by many scientists. IV. TOPOLOGICAL QUANTUM NUMBERS After the system undergoes the cycle of changing its flux by ℎ𝑐/𝑒, it returns to the same state. Its wave function is the same as before, except that the phase of the wave function may be different. This accumulated phase is known as the Berry phase. When it is compared to the accumulation of phase by vectors undergoing parallel transport on a curved surface, the Hall conductance can be compared to the curvature of the surface. In other words, just as the curvature of a surface is linked to the accumulation of phase in parallel transport, so is the Hall conductance linked to the Berry phase that arises under the adiabatic cycle of the wave function. Doing so allows a formula from topology to be used in order to explain the integral quantum Hall effect. The formula, by Gauss and Charles Bonnet, is References C. Kittel, Introduction to Solid State Physics. 498-503 (2005) J. Avron, D. Osadchy, R. Seiler, Physics Today. 56, 8 (2003) http://en.wikipedia.org/wiki/Quantum_Hall_effect http://nobelprize.org/nobel_prizes/physics/ laureates/1985/ Left-handed Materials Josh Holt March 5, 2008 1 Theoretical Introduction In the 1960’s, Victor Veselago investigated the optical repercussions of a hypothetical medium with simultaneously negative and real electric permittivity ² and magnetic permeability µ. 11 At a flat interface of his conjectured material with “normal” material, the direction of wave propagation is opposite to the Poynting vector of energy propagation; thus electromagnetic waves obey the left-hand, instead of the right-hand, rule. Veselago coined such media exhibiting this effect left-handed materials (LHMs). One must then choose the negative root q of the index of refraction given by n = ± ²µ/²0 µ0 (where the naught subscript indicate respective free-space quantities). Although the ensuing consequences of a negative refractive index, including negative refraction, reverse Doppler shift and reversal of Cherenkov radiation, are peculiar, they do not violate any fundamental physical laws. For example, negative refraction allows creation of a so-called a “Veselago lens,” a flat, parallel slab of LHM inside a right-handed material (RHM) with the condition that both media have the same isotropic refractive index and the same impedance. Rays from a point source impinging on a Veselago lens would be refocused to a point on the opposite side of the material. Interest in LHMs irrupted after the work by Pendry which argued that the Veselago lens is a “perfect lens” in the sense that it gives a perfect image of the point source. 5 His statement is based upon the observation that the evanescent waves of a form exp(iky y − κx) that usually decay in the near-field region are amplified by the LHM. Pendry claimed that the amplified evanescent waves restore a perfect image in both the near-field and far-field regions, but he did not present a solution in coordinate space. That the cherished diffraction limit could be violated did not set well with many physicists. A real-space solution was offered by Ziolkowski and Heyman, 12 revealing that Pendry’s solution diverges exponentially at each point of a 3D domain near the focus, just where the fields of the evenescent waves increase due to amplification. Simultaneous attacks against so-called “superlensing” followed from many groups. 2,8,3 Eventually the debate settled until it was generally agreed that superlensing can be found in the near-field regime while resolution in the far-field remains on the order of wavelength. 6 Even so, implications of negative refraction and the possibility of beating the diffraction limit could not be ignored. The theoretical controversy amplified the need to reveal new experimental evidence of negative refraction. 1 Left-handed Materials 2 Solid State II Experimental Evidence Since large magnetic response, in general, and a negative permeability at optical frequencies, in particular, do not occur in natural materials, metamaterials have been contrived to facilitate ² < 0 and µ < 0 simultaneously. The earliest probe for negative refraction used a two-dimensional array of split-ring resonators (which couple to and alter the local magnetic field) to produce negative µ over a particular frequency region and wire elements to produce negative ², showing microwave transmission at a positive from normal angle using a teflon prism and transmission at a negative angle using a manufactured LHM metamaterial prism. 10 An immediate debate questioning the veracity of the results arose 1 but, ultimately, the effects of negative refraction were repeatably verified. Permeability, permittivity, and refractive index are bulk, effective medium properties. Although metamaterial consist of discrete scattering elements, it may be approximated as an effective medium for wavelengths that are larger than the unit cell size. This approximation is analogous to the effect of the periodic Bloch potential on band electrons in condensed matter. The advantage in these metamaterials, of course, is that they can be scaled to any particular wavelength of light, whereas the dispersion relation of any solid-state electronic device is effectively pinned to a fixed lattice spacing. Photonic crystals and structures have naturally followed to implement negative refraction within optical frequencies. It was shown that a dielectric photonic crystal made of non-magnetic materials can behave as a LHM with negative ² and µ if it has a negative group velocity in the vicinity of the Γ-point of the second Brillouin zone both theoretically 7 and experimentally. 4,9 However, the exsistance of surface plasmons in photonic crystals make interpretation of experiments and simulations difficult. 3 Conclusions Physical understanding since Veselago’s initial theory has bloomed into applications probably beyond what he originally imagined (see, for example, Ref. 12). Applications founded on negative refraction include beam steerers, modulators, band-pass filters, and lenses permitting subwavelength point source focusing. The primary players which have stakes in LHM technology include the telecom industry (which relies on the ultrafast switching of the optical 1.5 µm wavelength for high-throughput communication), data storage and imaging technology (who are continually trying to work around the diffraction limit). The non-intuitive effects of LHMs have driven research in metamaterial and led to their much improved understanding. References [1] N. Garcia and M. Nieto-Vesperinas. Is there an experimental verif ication of a negative index of refraction yet? Opt. Lett., 27:885–887, 2002. Left-handed Materials Solid State II [2] N. Garcia and M. Nieto-Vesperinas. Left-handed materials do not make a perfect lens. Phys. Rev. Lett., 88:207403, 2002. [3] F. D. M. Haldane. Electromagnetic surface modes at interfaces with negative refractive index make a ”not-quite-perfect” lens. cond-mat/0206420, 2002. [4] P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar. Negative refraction and left-handed electromagnetism in microwave photonic crystals. Phys. Rev. Lett., 92:127401, 2004. [5] J. B. Pendry. Negative refraction make a perfect lens. Phys. Rev. Lett., 85:3966–3969, 2000. [6] V. A. Podolskiv and E. E. Narimanov. Near-sighted superlens. Opt. Lett., 30:75–77, 2005. [7] A. L. Pokrovsky and A. L. Efros. Sign of refractive index and group velocity in lefthanded media. Solid State Comm., 124:283–287, 2002. [8] A. L. Pokrovsky and A. L. Efros. Diffraction theory and focusing of light by a slab of left-handed material. Physica B, 338:333–337, 2003. [9] Vladimir M. Shalaev, Wenshan Cai, Uday K. Chettiar, Hsiao-Kuan Yuan, Andrey K. Sarychev, Vladimir P . Drachev, , and Alexander V. Kildishev. Negative index of refraction in optical metamaterials. Opt. Lett., 30:3356–3358, 2005. [10] R. A. Shelby, D. R. Smith, and S. Schultz. Experimental verification of a negative index of refraction. Science, 292:77–79, 2001. [11] V. G. Veselago. Properties of materials having simultaneously negative values of the dielectric(²) and magnetic (µ) susceptibilities. Sov. Phys.-Solid State, 85:3966–3969, 1967. [12] R. W. Ziolkowski and E. Heyman. Wave propagation in media having negative permittivity and permeability. Phys. Rev. E, 64:056625, 2001. 1 Graphene: Carbon, Now in Stunning 2D Jon Paul Johnson, University of Utah, Physics 5520 If thin is in, graphene has a bright future as the thinnest stable material known. If you were to remove a single atomic layer from graphite, one of the more traditionally-built forms of carbon, you would have graphene— a sheet of carbon atoms in a honeycomb array with the thickness of a single carbon atom. The discovery of stable graphene sheets in 2004 came as a surprise1, since it was assumed that it was energetically unfavorable for carbon to be in this configuration and not rolled up in a carbon nanotube, a fullerene, or some other more stable three-dimensional folded shape. Now this intriguing material with its predicted bizarre electrical properties is accessible, which has attracted the attention of electrical engineers looking for a replacement for silicon transistor channels as the limits of that material are expected to be reached within a few decades. Graphene also provides an unlikely sandbox to study quantum effects because its electronic structure yields charge carriers that are massless fermions. among the forest of thicker graphite flakes. The thin flakes have a slightly different appearance from the substrate, and the contrast is maximized for certain SiO2 thicknesses. Much of the interest in graphene comes from the electrical dispersion relation of electron waves in a honeycomb lattice, which at the Fermi level (also known as the neutrality point in graphene) is gapless and linear. See Figure 1. This had been predicted long before stable graphene was discovered2,3. Sp2hybridized bonds connect the carbon atoms in the lattice, with the last electron per carbon almost completely delocalized. Since effective mass depends on the curvature of the dispersion relation, its linearity at the Fermi level implies charge carriers with zero effective mass, and these carriers are quasiparticles that behave as relativistic particles with a speed that is lower than the normal speed of light by about a factor of 300. This enhances some quantum electrodynamic effects, which depend on the Graphene was first discovered by a group at the University of Manchester which, in an effort to study thin sheets of graphite, presumably got more than they bargained for when they discovered the presence of individual flakes of graphene in their samples. Sample preparation was (and is) rather more hunter-gatherer than agricultural in nature: “micromechanical cleavage” of a piece of graphite involves either peeling layers off of it with adhesive tape or drawing with it on a solid SiO2 surface. Then a search is performed with an optical microscope to find the few thin graphene flakes scattered Figure 1. The dispersion relation for graphene. 1 2 electrons and holes used to carry charge in today’s devices, but first there must be advances made in epitaxial growth of this intriguing material6. Without a way to create graphene layers compatible with the parallel processing techniques used by the semiconductor industry, graphene is likely to remain a condensed matter research tool— though a very interesting and, as of now, unique one. inverse of the speed of the particles involved, to the point that they are measurable even at room temperature. Specifically, the quantum Hall effect has been observed in graphene at room temperature. The electron transport properties graphene make it a candidate as a conductive channel in electronic devices. Graphene is conductive without doping, and its carriers exhibit very high mobility and even ballistic transport on small enough scales at room temperature. The carrier velocity (~106 m/s) and the fact that an applied electric field in either direction creates charge carriers could make graphene useful for ultra fast-switching field effect transistors. 1. A. K. Geim & K. S. Novoselov, The rise of graphene. Nature Materials 6, 183-191 (2007). 2. P. R. Wallace, The band theory of graphite, Phys. Rev. 71, 622-634 (1947). One drawback for the use of graphene in a more traditional transistor design is actually the lack of a gap at the neutrality point, which creates charge carriers for conduction regardless of whether or not an electric field is applied to the sheet. One way round this is to artificially create a semiconducting gap by introducing quantum confinement effects on a sheet of graphene. Limiting one of the two remaining spatial dimensions by creating ribbons of graphene4 could open a gap that would make graphene suitable as an alternative to silicon as the channel material in field effect transistors. Another way a semiconducting gap could be manufactured is by using the lattice mismatch between graphene and the crystal used as a seed to grow graphene epitaxially (for example, 5). 3. J. C. Slonczewski & P. R. Weiss, Band structure of graphite. Phys. Rev. 109, 272279 (1958). 4. L. Brey & H. A. Fertig, Electronic states of graphene nanoribbons. Phys. Rev. B 73, 235411 (2006). 5. S.Y. Zhou et al., Substrate-induced band gap opening in epitaxial graphene. Nature Materials 6, 770-775 (2007). 6. A. K. Geim & A. H. MacDonald, Graphene: Exploring carbon flatland. Physics Today 60, 35-41 (2007). The discovery of graphene nicely rounds out the collection of available forms of atomically thin carbon, taking its place with 0D fullerenes and 1D nanotubes. Graphene has the potential to make waves in electronic device design by swapping massless wavelike quasiparticles for the massive 2 March 5, 2008 Living Green With Thermoelectricity Ben Mangum Department of Physics, University of Utah Thermoelectric devices capable of converting heat directly into electricity and thermoelectric coolers, which do not contain any moving parts or ozone depleting gases may become a major player in reducing carbon footprints. While thermoelectric devices have traditionally been plagued by low efficiencies (ZT < 1), recent technological advances allowed for the fabrication of new nanostructured materials with much higher figures of merit (ZT ∼ 3). Such advances are demonstrated in materials that maintain an high electric conductivity and low thermal conductivity by minimizing phononic contributions. By introducing inclusions at various length scales, phonon contributions can be minimized. The development of high-efficiency thermoelectric devices will play an increasingly larger role in society as environmental costs of current technologies are considered more carefully. I. (a) INTRODUCTION Heat Source For almost two centuries now, scientists have possessed both the understanding and the ability to make generators capable of converting heat directly to electricity and refrigerators with no moving parts to wear out. Thomas Seebeck discovered in 1823 that two dissimilar metals joined together with a temperature gradient across the junction was capable of deflecting a compass needle. With later clarification from Ørsted, this discovery meant that such a simple device was capable of generating ‘thermoelectricity’ [1]. Working independently in 1834 Jean Peltier discovered that applying a voltage across such a junction was capable of establishing a temperature gradient across the junction [4]. Lord Kelvin (William Thompson) described the heating or cooling of a current-carrying conductor with a temperature gradient in 1851, showing that as some metals exhibit cooling while others produce heat as current flows from high to low potential with an applied temperature gradient. These reversible processes (the Seebeck effect, the Peltier effect, and the Thompson effect) are collectively known as the thermoelectric effect [1]. By 1838 Heinrich Lenz had demonstrated the use of a thermoelectric (TE) device; by placing a drop of water between a bismuth wire and an antimony wire and applying a current, the water would freeze and then melt by reversing the direction of the current [3]. In the 1950’s a renewed interest in the thermoelectric effect arose as a consequence of the discovery that doped semiconductors were much exhibited a large thermoelectric effect, thus much of the early semiconductor research was in pursuit of thermoelectric refrigerators [3]. Despite the fact that these thermoelectric technologies are now centuries old, there has been very little, or at least very slow, progress in putting TE devices to practical use. Currently, due to low efficiencies TE devices are used for niche markets such as laser diode cooling, and power generation for space bound devices like the Voyager deep space probe [2]. However, with a new arsenal of tools and techniques, coupled with the green awakening sweeping the globe, thermoelectric technologies are ripe for more investigation and implementation into everyday living. N (b) Cooled Surface N P Cool Side P Dissipated Heat FIG. 1: Schematic of thermoelectric (TE) devices. Panel (a) shows a TE generator; as heat flows from a heat source to a heat sink, heat carriers move along the temperature gradient. Utilizing both n-doped and p-doped semiconductors for the junctions the heat carriers are also charge carriers and establish a potential difference across the device capable of driving a load. This is an example of a Seebeck generator. Panel (b) shows a Peltier cooler, and is essentially the converse of the effect just described; an applied voltage will preferentially move heat carriers to one side of the device establishing a temperature gradient. II. THEORY A schematic of both a TE cooler and a TE generator are shown in Fig. 1. When n-doped and p-doped semiconductors are joined to both a heat source and a heat sink by metal contacts, thermal carriers (electrons or holes) move from hot to cold. As seen in Fig. 1(a) this motion of thermal, but also charge, carriers creates a potential difference and thus a current in the material is created as indicated by the arrows in the figure. In like manner a Peltier (TE) cooler is shown in Fig. 1(b). An applied voltage drives the motion of charge carriers as indicated by the arrows, while this net flow of carriers from one side of the device to the other establishes a thermal gradient. The efficiency of any device is thermodynamically limited to a maximum efficiency (Carnot efficiency) of Tc /(Th − Tc ) for refrigerators. Modern compressor based 2 refrigeration systems can reach 30-90% of Carnot efficiency, thus if TE devices could reach 30% Carnot efficiency, they could compete with modern compressor based refrigeration technologies. The efficiency of thermoelectric devices is often characterized by a dimensionless figure of merit ZT, where Z has units of ◦ C −1 and T is the average temperature. Z is a function of the thermal conductivity (κ), the electrical coductivity (σ), and the Seebeck coefficient (S): Z= S2σ κ (1) The Seebeck coefficient (S) is simply a measure of a material’s ability to establish a voltage difference in response to a given temperature difference: S = ∆V /∆T . An examination of metals and degenerate semiconductors yields [2]: S= 8π 2 kb2 ∗ π 2/3 m T 3e~2 3n (2) where n is the carrier concentration, and m∗ is the effective mass of the carrier [2]. It must also be borne in mind that both the electrical conductivity and the thermal conductivity are also functions of carrier concentration; σ also being a function of the electric charge (e) and mobility (µ), σ = neµ. The thermal conductivity has contributions from the motion of electrons and holes, but is also influenced by phonons in the lattice, thus κ = κe +κl , with κe = neµLT , where L is the Lorentz factor. Noting that σ and κ are directly proportional to the carrier concentration n, but that S ∝ n−2/3 the optimum carrier concentration can then be found. This sensitive balance of carrier concentration effects, indicate then that metals, although having high electrical conductivity make poor thermoelectric materials due to a low Seebeck coefficient and high thermal conductivity. On the other hand, insulators, while having large Seebeck coefficients and small electrical contributions to thermal conductivity, have low electrical conductivities also making them poor TE materials [3]. The optimum balance between Seebeck coefficient and conductivities is typically struck with highly doped semiconductors: n ∼ 1019 cm−3 [2]. III. RECENT ADVANCEMENTS Optimizing the concentration of charge carriers in a material is now really the task at hand for determin- [1] http://en.wikipedia.org/wiki/Thermoelectric effect [2] G.J. Snyder, and E.S. Toberer, “Complex thermoelectric materials,” Nature Mater. 7, 105–114 (2008). [3] G. Mahan, B. Sales, J. Sharp, “Thermoelectric Materials: New Approaches to an Old Problem,” Phys. Today, March ing the efficiency of a material for potential use as a TE device, however, to make use of quantum mechanical predictions, firstly the electronic band structure of the material must be known [4]. The inability to predict crystal structures is the major bottleneck in predicting which new materials might be suitable for TE devices. Recent progress in x-ray diffraction techniques, the ability to carefully grow new materials through the use of nano-wires and thin film deposition, along with modern advances in computing technologies all increase the likelihood of finding new and better TE materials [4]. Theoretical predictions suggest that the TE efficiencies can be improved by quantum confinement of the electron charge carriers. In a quantum confined structure, high confinement and low dimensionality leads to narrow electronic energy bands. This equates to high effective masses and thus large Seebeck coefficients [2]. Being able to create a ‘phonon glass’ while maintaining an ‘electron crystal’ is seen as another key to creating high ZT materials. The thermal conductivity contribution due to phonons can be reduced by increased phonon scattering accomplished by a variety of methods such as scattering phonons within the unit cell by creating point defects or alloying, as well as scattering phonons at interfaces through the use of multiphase composites with nanometer scale structuring [2]. To date, the most efficient TE materials are made from superlattice nanowires with a ZT of 2.5 - 3 [5]. IV. CONCLUSIONS Despite relatively low efficiencies of even modern thermoelectric devices, great improvements have been made in recent years. Several years ago the most efficient devices had a figure of merit only as high as ZT ∼ 1, whereas today nanostructured materials are rated as high as ZT ∼ 3. It is predicted that the heretofore elusive barrier of ZT > 4 will soon be broken, thus allowing thermoelectric coolers to compete directly with current compressor based refrigeration technologies. The optimization of carrier concentrations is certainly vital to creating high ZT devices, but the next major advances will likely come from finding ways to maintain an ’electronic crystal’ while simultaneously being a ’phonon glass.’ In an era where the environmental cost of any technology or device is become ever more scrutinized, thermoelectric power generation is also poised to move out of its niche market as ever more efficient TE materials are made available. 1997 pp. 42–47 [4] F.J. DiSalvo, “Thermoelectric Cooling and Power Generation,” Science, 285, 703–706 (1999). [5] K. Walter, https://www.llnl.gov/str/May07/Williamson.html The Quantum Cascade Laser February 27, 2008 Nick Borys Introduction The quantum cascade laser (QCL) is a novel semiconductor laser device which allows for the creation of laser sources in a broad wavelength range of approximately 1µm up through 30µm (near-infrared to terahertz radiation). This wavelength range is exceptionally important in chemical detection due to the majority of vibrational and rotational energy levels of molecules within this energy range. Consequently, QCLs have direct applications in environmental analysis, trace detection, and defense and security technology. Further, QCLs are the first reliable and low-cost laser source in this wavelength ranges since material stability and processing complications make the regime hard to access with conventional semiconductor diode lasers [1]. energy small DOS Although the QCL is based off of semiconductor materials, it is fundamentally different from conventional diode lasers. A diode laser relies upon the recombination of an electron in the conduction band and a hole in the valance band within the semiconductorbased active region. Consequently, the wavelength of a diode laser is defined and limited by the material dependent bandgap energy. On the other hand, a QCL is based on a layering of semiconductor materials such that layers of neighboring energy wells and barriers are created (Figure 1). The physical thickness of the wells and barriers define the energy levels of the laser. So, rather than relying on the material properties alone, the wavelength of the QCL can be tuned simply by modifying the thickness of the layers in the active region consequently opening up the broad wavelength regime of the QCL [1]. distance g large DOS n=3 g n=3 n=2 n=1 n=2 Injector Active Region n=1 Figure 1 Energy diagram of a QCL Technical Description of the Quantum Cascade Laser The QCL was first demonstrated in 1994 by a group at Bell Labs that consisted of Federico Capasso, Claire Gmachi, Deborah Sivco, Alfred Cho, Jerome Faist, Carol Sirtori, and Albert Hutchinson [2]. As mentioned above, it is principally different than a semiconductor diode laser in that the QCL operates with electrons cascading through a series of potential wells that make up the conduction band. Since the QCL only involves conduction electrons, and not valance band holes, it is referred to as a unipolar laser [1]. The QCL consists of two fundamental regions (Figure 1): the injector and the active region. Electrons are injected from the injector ground state (dashed line labeled “g” in Figure 1) into the n=3 state of the active region. The injected electrons then radiatively relax into the n=2 state and quickly, further relax into the n=1 state. From the n=1 state, the electrons can then tunnel into another injector region and leave the active region, thus completing the fundamental light emission process of the QCL [1, 2]. The ground-state of the injector region combined with the three energy levels in the active region essentially comprises a four-level laser system. Electrons can efficiently tunnel through the injector region into the n=3 state of the active layer. The scattering rate of n=3 state to the n=2 state is quite large (several picoseconds). However, the scattering rate from the n=2 state to the n=1 state is an order of magnitude smaller (~0.3 ps). This arrangement of scattering rates accomplished by engineering the well and barrier widths such that there is minimal wavefunction overlap of the n=3 and n=2 states and maximal overlap between the n=2 and n=1 states. The net consequence of the scattering rates is that the n=3 state remains populated by electrons from the injector ground state while the n=2 state is quickly depopulated resulting in population inversion between the two states – a requirement for laser emission [1, 2]. Additionally, the injector region is engineered so that it promotes population inversion between the two states in the active region. The wells and barriers of the injector region are designed such that the ground-state energy is very close to the n=3 energy level of the active region. However, this equality is only for electrons being injected. Notice, in Figure 1, the n=3 energy level of the active region falls in an energy range where the injector has a large density of states. On the right-hand side, however, the n=3 energy level falls in an energy range of the injector region that has a low density of states. This engineered configuration helps prohibit electron tunneling into the subsequent injector region directly from the n=3 state thus making it more likely the injected electrons will radiatively relax into the n=2 state [1, 2]. As illustrated in Figure 1, a typical QCL consists of several active regions separated by injectors, referred to as stages. As of 1999, a typical QCL has a 25-75 stages. Consequently, one single electron in a QCL produces up to N photons for a device that has N stages as compared to a diode laser where an injected electron will only produce a single photon (i.e., once an electron combines with a hole in a diode laser, it will not produce any additional photons). This, combined with the ability to support larger device currents, leads to QCLs outperforming diode lasers by up to factors of 1000 [1]. Finally, the full QCL device has the numerous semiconductor layers comprising the stages clad with semiconducting material of a lower refractive index so that the radiation is guided through the device. In more complicated commercial laser setups this cladding is also grated such that single mode can be significantly amplified over another [1, 2]. Summary Since its invention 1994, the QCL has quickly found a considerable amount of practical applications. A primary reason for this is the large wavelength range QCLs can be engineered for. This wavelength range, combined with the relative ease of production such that QCLs can be produced in large volumes, opens up laser spectroscopy solutions for a large amount of industries that cannot otherwise afford expensive laser systems [3]. As an example, QCLs are poised to provide a significantly better solution for the large gas-sensing market [4]. Furthermore, the QCL is also finding use for theoretical studies. In 2007, a paper was published in Nature that documented using QCLs to directly probe the phase of laser emission. This method is claimed to open up the ability to probe the dynamics of the photons as they traverse the laser cavity and should find possible use in studying laser dynamics [5]. So in addition to a significant impact in several industries, the QCL has what appears to be a promising future in scientific fields as well. References [1] F Capasso, C Gmachl, D L Sivco, A Y Cho, Physics World, 27-33 (June 1999). [2] J Faist, F Capasso, D L Sivco, C Sirtori, A L Hutchinson, A Cho, Science 264, 553-556 (1994). [3] I Howieson, Laser Focus World 41, (2005). [4] E Normand, Laser Focus World 43, (2007). [5] D Citrin, Nature 449, 669-670 (2007).