PPPPPP*PP,PP andPoster Session, Thursday, June 17Theme F686 - N1123Topological Analysis of the Integer Quantum Hall Effect12,341UAyl<strong>in</strong> YildizUPP, Afif SiddikiPPDeniz EksiP Ismail SokmenP1 Department of Physics, Dokuz Eylul University, Izmir 35160, Turkey2PDepartment of Physics, Istanbul University, Istanbul 34134, TurkeyPDepartment of Physics, Harvard University, Cambridge MA 02138, USA4PDepartment of Physics, Trakya University, Edirne 22030, Turkey3Abstract-We discuss the role of topology on the <strong>in</strong>teger quantum Hall effect (IQHE). The characteristics of the edges as well as the bulkstates <strong>in</strong> a Hall bar with two identical square gates <strong>in</strong> the <strong>in</strong>terior and a Corb<strong>in</strong>o disc geometries with and without disorder have been analyzed<strong>in</strong> detail. The current distribution obta<strong>in</strong>ed from the L<strong>in</strong>ear Response Theory (LRT) is presented for all systems.Vortices <strong>in</strong> Type II superconductors, Aharonov-Bohmeffect and many other examples put forward the important roleof topology <strong>in</strong> condensed matter physics. S<strong>in</strong>ce the discoveryof the quantum Hall effect [1], important topological<strong>in</strong>vestigations have been performed [2-5].The famous gauge <strong>in</strong>variance argument of Laughl<strong>in</strong> [2] isfundamental to the phenomena. Laughl<strong>in</strong>'s argument, focussedon a closed cyl<strong>in</strong>der with a strong magnetic field normal to itssurface, is threaded by a time dependent magnetic flux. Eachtime the flux was <strong>in</strong>creased by one flux quantum, an electronwas argued to be adiabatically transferred from the <strong>in</strong>side tothe outside edge of the cyl<strong>in</strong>der. This resulted <strong>in</strong> a flow ofelectrical current proportional to the electro-motive driv<strong>in</strong>gforce, with a precisely quantized coeffcient of proportionality -the Hall conductance [2]. The Laughl<strong>in</strong>'s argument waselaborated on by Halper<strong>in</strong> to a Corb<strong>in</strong>o disc - a hollow discshaped sample [3]. With<strong>in</strong> a Corb<strong>in</strong>o-disc geometry, thequantized Hall conductance could be understood as a discretetransfer of electrons between the <strong>in</strong>ner and outer edges, onefor each magnetic flux quantum thread<strong>in</strong>g the bore of the disc.The importance of the edge state was first demonstrated bythis work.The equivalence of Hall bar sample enclosed on itself underthe periodic boundary conditions as well as gauge <strong>in</strong>varianceand Corb<strong>in</strong>o disc topologies is one of the <strong>in</strong>trigu<strong>in</strong>g researchtopics. In this study we def<strong>in</strong>e two geometries - a Hall bar withidentical square gates <strong>in</strong> the <strong>in</strong>terior region and a Corb<strong>in</strong>o discas an annular 2DEG system surround<strong>in</strong>g a metallic contact[i.e., an electron reservoir (electrode)] and surrounded <strong>in</strong> turnby a second metallic contact as shown <strong>in</strong> Figure 1.The Hall bar geometry <strong>in</strong>volve a physical edge which connectsthe prob<strong>in</strong>g contacts. A unique feature of Corb<strong>in</strong>o disc is that,unlike other two-dimensional semiconductor devices, theirboundaries consist entirely of metallic contacts and no edgeconnects the contacts. This different sample configuration ledto the observation of completely different effects.The characteristics of the <strong>in</strong>compressible strips (IS) havebeen analyzed for both geometries with and without disorder.We apply the Thomas-Fermi Approximation (TFA) assum<strong>in</strong>gthe electrostatic quantities vary slowly <strong>in</strong> the quantummechanical scale such as magnetic length [7]. We obta<strong>in</strong>edcircular ISs <strong>in</strong> Corb<strong>in</strong>o disc near <strong>in</strong>ner and outer annulus atT=5K temperature under the <strong>in</strong>fluence of B=8T, whichobviously confirm the edge picture for Hall conductancestressed by Halper<strong>in</strong>. Under T=4K, B=8.5T conditions theobta<strong>in</strong>ed edge states <strong>in</strong> the clean Hall bar sample are parallelto the straight l<strong>in</strong>e connect<strong>in</strong>g source and dra<strong>in</strong>. On the otherhand, the circular edge states <strong>in</strong> Corb<strong>in</strong>o geometry cut thecorrespond<strong>in</strong>g l<strong>in</strong>e perpendicularly. With <strong>in</strong>creas<strong>in</strong>g magneticfield the existence of ISs <strong>in</strong> both systems is described by thebulk states. This picture is valid for a small magnetic fieldrange.In the presence of a small fixed current the system is locally<strong>in</strong> thermal equilibrium. In this case the l<strong>in</strong>earity between thecurrent density and the electric field is almost conserved thatthe L<strong>in</strong>ear Response theory (LRT) is valid [8]. The Hallcurrent <strong>in</strong> Hall bar sample is carried by edge states for B=7.5Tand T=6K conditions, while with <strong>in</strong>creas<strong>in</strong>g the magnetic fieldup to 8T current penetrates the ISs around the gates.Our calculations show that the existence of disorderbroadens the ISs and the bulk picture is ascendent <strong>in</strong> bothsystems <strong>in</strong>dependent of the geometry.In summary, we analyze the edge or bulk states realizationsof the IQHE <strong>in</strong> a Hall bar with two identical square gates <strong>in</strong>the <strong>in</strong>terior and a Corb<strong>in</strong>o disc geometries with and withoutdisorder. We conclude that the edge and bulk states areactually related and the topological characters of the Hall barand Corb<strong>in</strong>o disc systems are quite different.*Correspond<strong>in</strong>g author: HTayl<strong>in</strong>.yildiz@deu.edu.trTFigure 1. (a) Hall bar geometry with identical square gates <strong>in</strong> the<strong>in</strong>terior region (red squares). (b) Corb<strong>in</strong>o disc geometry. The blueregions <strong>in</strong>dicate the two-dimensional electron gas, the red areas aresource (S) and dra<strong>in</strong> (D) elements and the yellow squares denotes themetallic contacts <strong>in</strong> both samples.[1] K. Von Klitz<strong>in</strong>g, G. Dorda and M. Pepper, Phys. Rev. Lett. 45,494 (1980).[2] R. B. Laughl<strong>in</strong>, Phys. Rev. B 23(10), 5632 (1981).[3] B.I. Halper<strong>in</strong>, Phys. Rev. B 25, 2185 (1982).[4] G. Kirczenow, J.Phys.: Condens. Matter. 6, L583-L588 (1994).[5] E. Yahel, A. Plalevski, and H. Shtrikman, Superlattices andMicrostructures 22(4), 537 (1997).[7] A. Siddiki, R. R. Gerhardts, Phys. Rev. B 70 (2004) 195335.[8] K. Güven, R. R. Gerhardts, Phys. Rev. B 67 (2003) 115327.6th Nanoscience and Nanotechnology Conference, zmir, 2010 661
Poster Session, Thursday, June 17Theme F686 - N1123Fully Differential High Voltage Amplifier Design for Stick-slip Nanoposition<strong>in</strong>gNazmi Burak Budanur 1* , Devrim Yılmaz Aksın 1 , Oğuzhan Gürlü 21 Electronics & Communications Department, Istanbul Technical University, Istanbul 34469, Turkey2 Physics Department, Istanbul Technical University, Istanbul 34469, TurkeyAbstract – A fully differential high voltage amplifier to drive stick-slip piezoelectric actuators is designed. The amplifier consists of a fullydifferential amplifier and a common mode amplifier as ICs, and a power boost<strong>in</strong>g stage with discrete components. By this design approach weachieve slew rates of 300 V/μs on high capacitive loads of 10 nF.The motivation of this work is to build a high resolutionnanopositioner to be used <strong>in</strong> the sample position<strong>in</strong>g stage of ascann<strong>in</strong>g tunnel<strong>in</strong>g microscope (STM). Due to the stick-slipmotion pr<strong>in</strong>ciple, a high voltage (with an amplitude ofapproximately 300V) ramp signal with very high slew rates isneeded to drive piezoelectric ceramics with several hundredthsof grams of load on them. Period of the ramp signal should be<strong>in</strong> the range of 0.1s <strong>in</strong> order to make the motion <strong>in</strong> reasonabletime scales. Additionally, a high voltage and fast controlelectronic can be applied <strong>in</strong> other systems that require nanoposition<strong>in</strong>g by means of piezo electric positioners.There are s<strong>in</strong>gle ended examples of HV amplifiers <strong>in</strong>literature with an operational amplifier <strong>in</strong> the <strong>in</strong>put stagefollowed by class-AB power boost<strong>in</strong>g output stages <strong>in</strong> whichthe output common mode is determ<strong>in</strong>ed by a simple negativefeedback [1], [2]. Our design, shown <strong>in</strong> Figure 1, has threema<strong>in</strong> blocks: Input fully differential amplifier, class-AB poweramplifier and common mode feedback amplifier. Inputamplifier and the common mode amplifier are designed <strong>in</strong>0.35 micron CMOS technology with 3.3V sources, and thepower amplifier is build with discrete components with a155V DC source. S<strong>in</strong>ce the output common mode level is tobe determ<strong>in</strong>ed at the half of the high voltage DC source,common mode sens<strong>in</strong>g circuit divides the output voltage to asuitable level.We design and simulate our circuit on Cadence VirtuosoSpectre with SPICE models of discrete components and theAMS 0.35μm libraries. The power stage will be realized withdiscrete components whereas the low power sections will berealized through AMS. AC simulation result of the differentialloop is shown <strong>in</strong> Figure 2. As it is clear from the figure, theDC ga<strong>in</strong> of the amplifier is 90dB, its ga<strong>in</strong> band width productis 370kHz and its phase marg<strong>in</strong> is 83 degree.Figure 2. AC Simulation result.In conclusion, a fully differential amplifier design with a300V/μs slew rate is done to drive stick-slip piezoelectricnanopositioners. This work is a collaborative project of ITUVLSI and nano scale surface science labs.* budanur@itu.edu.tr[1] Colclough, M. S., 2000, A Fast high-voltageamplifier for driv<strong>in</strong>g piezoelectric positioners, Review ofScientific Instruments, vol.71 pp. 4323-4324[2] Wang, D. H., Zhu, W., Yang, Q., D<strong>in</strong>g, W.M., 2009,A High-voltage and High-power Amplifier for Driv<strong>in</strong>gPiezoelectric Stack Actuators, Journal of Intelligent MaterialSystems and Structures, Vol. 20 pp. 1987-2001Figure 1. HV Amplifier Design BlocksOne of the ma<strong>in</strong> advantages of the fully differentialapproach is the ability to use relatively low voltage devices.S<strong>in</strong>ce high voltage BJT and MOS transistors have largegeometries, parasitic capacitors of these devices determ<strong>in</strong>e thefrequency behavior of the whole circuit. It is possible to obta<strong>in</strong>a differential voltage on the load, approximately double of theDC source.6th Nanoscience and Nanotechnology Conference, zmir, 2010 662
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