XXXIV International School on the Physics of Semiconducting ...

Imaging Coherent Electron FlowBrian LeRoyDelft University of TechnologyHarvard UniversityJune 10, 2005www.mb.tn.tudelft.nlleroy@mb.tn.tudelft.nl

3.1.2 Transmitir N IN vezes cada um dos símbolos da constelação.É preciso conhecer a mensagem transmitida no receptor, o quetorna esta fase supervisionada. Uma vez recebidos os símbolos, éfeita uma média aritmética entre as amostras correspondentes acada um deles, o que nos dá uma estimativa inicial de cada umdos símbolos da constelação distorcida. Algumasparticularizações deste passo serão comentadas na seção 4.1.3.1.3 As estimativas obtidas no passo anterior são então usadascomo condição inicial para a rede neural. Com isto, os neurôniosjá estarão identificados com cada um dos símbolos transmitidos.3.2. Treinamento:Esta fase serve para que seja refinada, se constatada talnecessidade, a estimativa inicial obtida na fase anterior, antes quea rede comece a operar.3.2.1. Separar um conjunto de N T vetores y(k) recebidos para otreinamento e ajuste da rede neural.3.2.2. Para cada vetor y(k), obter o índice do neurônio vencedor,através de:i = arg min ( || w j (k) – y(k) || ) (11)jonde w j (k) é o vetor do j-ésimo neurônio no instante k.3.2.3. Uma vez obtido o índice i, utilizar a seguinte fórmula deatualização dos pesos do i-ésimo neurônio:w i (k+1) = w i (k) + α[y(k) – w i (k)] (12)onde α é o passo de adaptação.3.2.4. Se julgar-se necessário, embaralhar os N T padrões e voltara 3.2.1.3.3. Operação:Após as duas fases anteriores, considera-se que a rede está apta aoperar como decisor ótimo (de acordo com o critério ML). Oprocesso de decisão consiste em se determinar, para cada vetorrecebido y(k), o neurônio que tem o vetor de pesos mais próximodesta entrada. Como os neurônios já foram devidamenteidentificados, o símbolo correspondente ao vencedor é aestimativa de nosso decisor.É muito importante que a rede continue a ser adaptada nosmoldes de 3.2.2 e 3.2.3 para todos os símbolos, para compensareventuais modificações do canal. Isto caracteriza a operação “online”do decisor já devidamente treinado.4. APLICAÇÃO A SISTEMAS 4-PSK E 16-QAM COM VARIAÇÕES DE GANHO EFASEDentre as muitas possíveis escolhas para uma base ortonormal,iremos concentrar-nos neste trabalho em apenas uma: uma baseformada por duas senóides de mesma frequência e defasadas denoventa graus (cosseno e seno). Esta escolha permite-nosabranger uma ampla classe de esquemas de modulação digital.Iremos concentrar nossa análise em dois exemplos significativos:modulações 4-PSK e 16-QAM. No primeiro caso, a informaçãoestá embutida na fase da portadora senoidal. No segundo, ainformação encontra-se na fase e na amplitude da portadora.A componente associada ao termo cossenoidal da base échamada de componente em fase, e o outro termo denomina-secomponente em quadratura. O conjunto de possíveis vetores decomponentes do alfabeto do transmissor denomina-seconstelação.Na exposição a seguir, foi arbitrado o valor da energia de cadasímbolo, sem perda de generalidade, e este valor foi usado noresto do trabalho.4.1. Esquema para a modulação 4-PSKNa modulação 4-PSK, há 4 possibilidades para as componentesde x :X 4-PSK = { [-1; -1], [-1; 1], [1; -1], [1; 1] } (13)Analisando esta constelação, percebe-se que a mesma ésimétrica. Isto permite que exploremos esta característica paraotimizar o algoritmo descrito na seção 3. Mais especificamente, opasso 3.1.2 pode ser alterado. Ao invés de enviarmos os quatrosímbolos desta constelação, basta que enviemos um deles (N INvezes). Feita a média, temos a condição inicial de um dosneurônios. Para obtermos a dos outros, basta imprimir 3deslocamentos de 90 o sucessivos, sendo que o vetor resultante decada deslocamento é considerado como condição inicial de umdos 3 neurônios restantes. A isto denominamos inicializaçãoortogonal da rede, que então estará pronta para a fase 3.2.4.2. Esquema para a modulação 16-QAMNa modulação 16-QAM, há dezesseis possíveis símbolos:X 16-QAM = {[-3; -3], [-3; -1], [-3; 1], [-3; 3], [-1; -3],[-1; -1], [-1; 1], [-1; 3], [1; -3], [1; -1], [1; 1], [1; 3], (14)[3; -3], [3; -1], [3; 1], [3; 3]}Os quadrantes do plano no qual se encontra a constelaçãoexpressa em (14) serão denominados por nós quadrantesmaiores. Dentro de cada um destes, podemos definir outrosquadrantes, de modo a ser ter cada um dos quatro símbolos daconstelação, simetricamente distribuídos, em um dos chamadosquadrantes menores. Portanto, cada símbolo pode sercaracterizado pela indicação dos seus quadrantes maior e menor.A princípio, pode-se propor um decisor baseado em 16 neurônios(um por símbolo), de forma análoga ao caso anterior. Podemos,porém, explorar as características de simetria da constelação 16-QAM para reduzir este número.Neste artigo, propomos um esquema baseado em quatroneurônios para obter uma aproximação do decisor ótimo.Novamente, utilizamos a idéia de inicialização ortogonal,descrita na seção 4.1, sendo transmitido apenas o símbolo demaior potência de um dos quadrantes. Isto permite a realizaçãodo passo 3.1.2 na melhor SNR possível, tornando o algoritmomais robusto ao efeito do ruído.

Two-dimensional electron gasQuantum Point Contact GateOhmic ContactGaAs/AlGaAs HeterostructureTwo-Dimensional Electron Gas (2DEG)2DEG located 57 nm below surface

EF2DEG propertiesFree electrons in 2D with a reduced mass2 2kF1= = m v*2m2 ** 2F= 0.067meLow (tunable) electron density and long Fermi wavelength2π2π11 2λ = = 39nm n = = 4.2×10 / cmFk FHigh mobility and long elastic mean free pathµ = 1.0 x 10 6 cm 2 */V s = vmµ e=11µmλF2FmGood review article for basics about 2DEG and quantum point contactsBeenakker and van Houten, Solid State Physics 44 1 (1991) (cond-mat/0412664)

Quantum point contactsElectrons flowing through a narrow constrictionOhmic contactElectrostatic gate300nmClassicalQuantumWidth

EF+ δµJn = e ∫ ρn( E)vndEEFCurrent in 1-DCalculate the current carried by a single modeJvelocityDensity of statesnρ ( ) nE2e 2e= δµ = ∂Vπ hdN 1 1= =dE π vDensity of states and velocity cancelIn 1-D each mode carries the same amount of currentF

Quantum point contacts-quantizationWhat is the differential conductance?∞0∂IG ≡ =∂V2ehWhat are the values of Tn?T⎧0= ⎨⎩12∑n FnEn< EFAssume 1-D particle-in-a-box of width LEn= E whenFFinite temperature blurs the conductance plateaus2 ∞df 2eGET ( , ) = ∫ GE ( ,0) dE= ∑ f( En−EF)dE hFEL =i>TnλF2iEn=1

Quantum point contacts-modesConductance vs. Gate Voltage14121086420-1.2 -1.0 -0.8 -0.6Gate Voltage (Volts)T = 1.7 KTunneling Regime 1st Mode 2nd Mode

Quantum point contact-energy levelsConductanceGate Voltage6 66554 4 4332 2 2g = 2e 2 /hg = e 2 /h101g = 2e 2 /hV sd

Quantum point contact-energy levelsPlot of dG/dV gRed areas are plateaus,yellow and blue are steep14121086420-1.2 -1.0 -0.8 -0.6Gate Voltage (Volts)Horizontal distance between edges ofdiamonds gives subband spacing

Outline• Introduction• Measurement technique• Imaging a quantum point contact• Interference fringes• Measuring electron density• Electron optics• Electron-electron interactions• Conclusions

Liquid helium temperature scanningprobe microscope - cantileverAFM CantileverTip∆R cR c= 4 x 10 -7 /ÅR c ≈ 2000 ΩV BiasRR C25µmRRVV out∆R∝V outµ ∆R outcRVcV biasBias

Liquid helium temperature scanning probemicroscope – scan tubeWheatstoneBridgeFeedbackCircuitCross Section+V y1”-V x V z-V y+V xPiezoelectric TubeScan Range 20µm

Liquid helium temperature scanning probemicroscopeHeight50 nm1µm0 nm

Experimental techniqueAFM CantileverQuantum PointContact (QPC) GateTwo-dimensionalElectron Gas (2DEG)GaAs/AlGaAsheterostructuren-type GaAssubstratePerturbation from tip causes scatteringwhich changes the conductance of the QPCImage obtained by measuring conductancethrough the QPC as a function of tip positionHigh flow High scattering High signalNo flow No scattering No signal

Weak ScattererImaging mechanismNo Backscattering=Conductance UnchangedStrong ScattererBackscattering=Conductance Reduced

Effect of tip height10 nm 15 nm 20 nm 25 nm 30 nm 35 nm150nm 150nm 150nm 150nm 150nm 150nm0.140.120.100.080.060.040.020.000204060Height (nm)∆G: 0 e 2 /h 1.6 e 2 /h80100Tip voltage -3.5 VoltsSignal only when voltage ontip depletes the 2DEG

Electrostatic simulationn ⎛sV ⎞ εε0∆ n( ρ ) = V2 ⎜ ⎟+1 + / ⎝Vd⎠ d + εh( ρ d )Induced chargen

Quantum mechanical simulationGatesTip0.040.030.020.010.0020406080100Height (nm)Signal only when tip depletes 2DEG

Importance of depletion regionV tip-3.0V V tip-2.6V V tip-2.2V V tip-1.8V V tip0.0V V tip+3.0V50 nm0.060.040.02-3 -2 -1 0 1 2 3Tip Voltage (V)

Imaging requirementsPerturbation from tip must backscatterelectrons through the QPC.This requires that there is an area depleted ofelectrons beneath the tip.Can be accomplished by either bringing tip closeto surface or increasing the tip voltageImaging only works with a negative tip voltageSize of perturbation is set by distancebetween the tip and the 2DEGApproximately Lorentzian in shape

Outline• Introduction• Measurement technique• Imaging a quantum point contact• Measuring electron density• Interference fringes• Electron optics• Electron-electron interactions• Conclusions

Topinka, LeRoy et al., Science 289, 2323 (2000)Imaging a quantum point contact1st Plateau 2nd Plateau 3rd Plateau∆G:0 e 2 /h -1.7e 2 /h

Coherent fringesConstructive & Destructive BackscatteringFringes spaced by λ F /2Constructive Interference: 2k F L = 2π nDestructive Interference: 2k F L = 2π (n+1/2)

Opening QPCNew Modes of current only appear on plateausPattern remains constant between plateaus

Heating electronsSmall Bias Voltage V bias = 0.2mVG = 2e 2 /h G = 4e 2 /h G = 6e 2 /hLarge Bias Voltage V bias = 3.0 mVG = 2e 2 /hG = 4e 2 /hG = 6e 2 /h∆G: 0 e 2 /h 1.7 e 2 /hLarge bias voltage heats electronsBlurred interference fringes

Extracting a modal patternExperiment TheoryExperimentTheoryDerivation2nd Mode Flow 2nd Plateau Flow 1st Plateau Flow

Modal summaryExperimentTheory1st Mode2nd Mode3rd Mode∆G(percent)|Ψ | 2(arb unit)

Selective mode suppressionNo TipTip Blocking CenterTipTip Blocking SideTip500 nm500 nm500 nm64201.7e 2 /h2.0e 2 /hV g (Volts)Tip selectively modifies transmission coefficient of individual modes

Electron-wave flow through 2DEG“Idealized” 2DEG Potential& Electron WaveActual 2DEG Potential &Electron Wave?2µmOperating Point2µmOperating Point

Coherent electron flow through a 2DEG1µm∆G: 0 e 2 /h 0.4 e 2 /h

Modal patterns far away from QPC1st Mode2nd Mode3rd Mode4th Mode∆G: 0 e 2 /h 0.4 e 2 /h

Simulated imageDoes the presence of the tip influence the image?Simulated electron flowSimulated imageImage the unperturbed electron flow

Unexpectedly high resolution100nmTip Perturbation ~ 120nm FWHMbutFringe Resolution ~ better than 10nmLateral Resolution ~ better than 20nm

Unperturbed Currentand Tip PerturbationHigh spatial resolutionTip Moving Through Current

Quantum and classical simulations0.2 E FPotential, U(x,y)1µm-0.2 E FClassical1µmQuantum Mechanical1µmTopinka, LeRoy et al., Nature 410, 183 (2001)

Formation of CausticsClassical TrajectoriesPy-y Phase SpacePotential Dip

QPC Location shifts by 20 nmMoving QPC location500 nm

Shifting QPCUnequal voltage on QPC gates∆Ge 2 /h0.01500 nm230.4Difference of the two scans1G diff-0.1e 2 /h500 nm500 nm0.0e 2 /h0.1e 2 /h

Mapping 2DEG potentialTip Locations0.5Potential?∆G (e 2 /h)0.40.30.2200 nm0.1-4.0 -3.5 -3.0 -2.5 -2.0Tip Voltage (Volts)

Imaging a QPCImaged electron flow near a quantum point contactModal pattern associated withwavefunctions in QPCInterference fringes spaced by λ F /2Imaged electron flow far from a quantum point contactElectron flow forms narrow branchesBranches are due to caustics caused by dipsin the potentialInterference fringes spaced by λ F /2

Outline• Introduction• Measurement technique• Imaging a quantum point contact• Interference fringes• Measuring electron density• Electron optics• Electron-electron interactions• Conclusions

Interference fringesPresence of interference fringes showsthat electrons are coherentUse interference to create new types of devicesthat rely not only on amplitude of signal but alsothe phase informationWant to understand what causesthe interference fringesCan this interference also be controlled?

Coherent fringesConstructive & Destructive BackscatteringFringes spaced by λ F /2Constructive Interference: 2k F L = 2π nDestructive Interference: 2k F L = 2π (n+1/2)

Finite temperature- thermal smearingFermi energy for free electrons in a 2DEG2 2kF1 * 2EF= = m v*F m* = 0.067me2m2Electrons are not monoenergetic, they have a range ofenergy from the finite temperatureConsider two electrons that differ in energy by 2kTThey will have slightly different wavevectors, k2 2k,E = ,E ± FkT = + −+− *2mAssume kT

Thermal smearing (continued)Now suppose these electrons interfere after a distance LEach of the electrons will have accumulated a phase, kLSo, their phase difference is2kT2∆ kL =v Define the thermal length as when this phase difference is 1LtFLvF= L t ~ 200 nm @ 4.2 K2kTThe thermal length can be better defined by using the spread in energy of theFermi distribution

Thermal smearing (continued)As the length is increased, the range of accumulated phaseswill start to wash out the interference fringesEach electron accumulates a different phase on theroundtrip from the tip to the QPC.Expectation:Fringes decay with distance from QPCNote: The electrons have not lost their coherence itis only being “hidden” by the thermal smearing

Fringe persistence1D model of fringe persistence, calculate transmissionthrough two delta functions as a function of positionFringe amplitude decays with distance

Fringe persistenceFringe Size (%)Distance from QPC (nm)No distance dependence observed

Backscattering from Tip and Impurities...Calculate wave backscattered to the QPC2ikRtip2ikR A itipeAeiΨ= +∑Rtipi RiLook at the amplitude of terms that vary with tip position2Ψ = 2Re∑AA eitipi i tip2 ik ( R −R)RRFinite temperature Range of wavevectorssignaltip2 ∂f = ∫ Ψ dE ∂ Ei

Fringe persistence far from the QPCIntegrand oscillates rapidly with energy except for terms with R i ≈ R tipAA2 2i tip−( Rtip −Ri ) / TsR (tip) = 2∑cos⎡2 kF ( Rtip −Ri) ⎤eRR ⎣⎦i i tipT=2k1/2 Fπ/4mkTTipImpurityAnnular band, width = TCombined backscattering off the tip and the impurities in the annularband around r tip produces interference fringes spaced at λ F /2**S.E.J. Shaw et al, cond-mat/0105354 (2001)Test this theory with an artificial scatterer

Fringe persistence far from the QPCImpurityTipAnnular band, width = thermal lengthCombined backscattering off the tip and the impurities in the annularband around r tip produces interference fringes spaced at λ F /2**S.E.J. Shaw et al, cond-mat/0105354 (2001)Test this theory with an artificial scatterer

Scattering induced fringe enhancementQPC GatesReflectorImpuritiesScan Area∆G(e 2 /h)0.00.05V refl = 0 VV refl = -0.4 VV refl = -0.8 V

ReflectorVoltageFringe strength analysis0.0 Volts-0.4 Volts-0.8 Volts∆G (e 2 /h)FFT (arb. units)0.00 0.05 0 300Reflector enhances interference fringes

Single BounceFringe movementDouble BounceDouble100 nmSingle100 nmMovies range from V refl = -0.5 V to -1.0V in steps of 0.02 VReflecting ArcFringes move twice as fast attwice the distance from the QPCLeRoy et al., PRL 94 126801 (2005)

Single BounceFringe movementDouble BounceAverage Peak Movement0.10 µm/VoltAverage Peak Movement0.23 µm/VoltInterference fringes at twice the radius of the reflecting arcmove twice the speed of the arc

Fringe amplitudeFringes enhancedReflector backscattering suppressedFringes suppressedReflector backscattering enhancedFringe amplitude anti-correlated with reflector conductance

Heating electrons1.7KQPC GatesReflector“4.2K”Scan Area“8.4K”

Temperature dependenceIncreased temperature destroys interference away from reflector

Interference fringesSpaced by half the Fermi wavelength, λ F /2Due to interference between electrons backscattered bythe tip an ones backscattered from impuritiesPersist throughout image because of backscattering fromimpuritiesCan control their location by introducing artificial impuritiesDemonstrated their movementwith a reflecting gateThey become more localized withincreasing temperature

Outline• Introduction• Measurement technique• Imaging a quantum point contact• Interference fringes• Measuring electron density• Electron optics• Electron-electron interactions• Conclusions

Therefore want a local probe of densityMeasuring densityElectron density measured by Shubnikov-de Haas oscillationsgives the average value over the entire sample.Usually a Hall bar of~ 100 micronsns=2e1h ∆( 1 B)For nanoscale devices, important to be able to measure thedensity in the vicinity of the device

Electron densityFree electrons in two dimensions2 2kEF= **m2m=0.067meFermi surface is a circle2 2⎛ k ⎞ kn = 2π⎜ ⎟ =⎝2π⎠ 2πWavelength of interference pattern is related to kUse interference of electron waves

Changing Fermi energyEffect of reducing Fermi Energy from 15 meV to 5 meV500 nmReducing Fermi Energy increases relative strength ofbumps and dips in potential

Effect of reducing Fermi energyBack Gate 0 V Back Gate -1 V Back Gate -3 V Back Gate -5 V∆G: 0 e 2 /h 0.15 e 2 /hReducing Fermi energy, increases fringe spacing andcauses flow to be more diffuse

Measuring local electron densityDensity Decreases, Wavelength IncreasesBack gate -1V Back gate -3V Back gate -5V2DEGWafer ProfileMeasured Density1.12 µm GaAs0.82 µm Al 0.3 Ga 0.7 Asn-type GaAsBackgateParallel platecapacitor model∆ n =11 -20.36 × 10 cm V∆VLeRoy et al., APL 80 4431 (2002)

Is it really local?Calculate spacing between two adjacent fringesPhase accumulated, φNeed phase to change by 2π, must move tip by LL=πkk is wavevector at this locationYes it is local, with resolution ~ fringe spacing

Mapping electron density∆G(e 2 /h)0.00200 nm0.15Average Density, n = 2.95 x 10 11 cm -2Standard Deviation σ = 0.48 x 10 11 cm -2

Electron densityInterference fringes provide a way to measurethe local electron densityspacing =π2nDensity can be controlled by back gate voltagePattern of flow become more diffuse asdensity is decreasedPotential felt by the electrons becomes a largefraction of their energy at low density

Outline• Introduction• Measurement technique• Imaging a quantum point contact• Interference fringes• Measuring electron density• Electron optics• Electron-electron interactions• Conclusions

Electron opticsCreate elements to electrostatically control electronsElectrostatic lensSivan et al., PRB 41 7937 (1990)Image how electrons travel through these types of devices

Electron optics theoryAnalog to Optical Index of Refraction…p Sin( θ ) = p Sin( θ )1 1 2 2n Sin( θ ) = n Sin( θ )1 1 2 2

Electron optics element - prism1µm“Refractive switch for two-dimensional electrons”, J. Spector, H. L. Stormer, K. W.Baldwin, L. N. Pfeiffer, and K. W. West , App Phys Let 56, 2433-2435 (1990)

Electron optics element - ballBall 0.0 VBall -0.5 V∆Ge 2 /h0.00.5Addition of the ball creates elliptical interference fringes

Electron optics element - channel∆Ge 2 /h0.00.75Simulated TrajectoryElectron trajectories are bent by the potential from the gate

Electron opticsImaged electrons traveling through the following devicesElectrostatic prismReflecting ballChannelCan learn about the potential from electrostatic gatesThe images show why it is difficult to make efficientdevices with these type of gatesBranches of current limit the abilityto control the flow

Outline• Introduction• Measurement technique• Imaging a quantum point contact• Interference fringes• Measuring electron density• Electron optics• Electron-electron interactions• Conclusions

InteractionsTo design devices relying on quantum information musthave a way to measure time (distance) over which theelectrons decohere.At low temperatures, electron-electron interactions limitthe distance electrons remain coherent in the 2DEGIn order to create devices relying on coherencemust be able to measure the electron-electronscattering time12∼ ( p−pF) ln ( p−pF)τeeImages can give a direct measure of this scattering time

DC bias techniqueNo Energy LossConductance ReducedEnergy LossConductance Unchanged

Voltage applied across QPC0.0 meV Across QPC∆ge 2 /h0.02.4 meV Across QPC0.4

Images of electron flow2.74 µm from QPC 1.96 µm from QPC 1.20 µm from QPC∆G (e 2 /h)0.00 0.1350 nm 50 nm50 nm0.00 0.35 0.00 0.55E FE FTipQPCMovies range from 0.0 mV to 3.1 mV in steps of 0.15 mV

Measuring electron energyFringe spacing indicates energy of electrons

Measuring e-e scattering lengthElectron-Electron Scattering Time21 E ⎛F∆ ⎞ ⎡ ⎛EF ⎞ ⎛2q⎞TF1⎤= ⎜ ⎟ ⎢ln⎜ ⎟+ ln ⎜ ⎟+⎥τee 4π⎝EF ⎠ ⎣ ⎝ ∆ ⎠ ⎝ pF⎠ 2⎦Valid at T=0, ∆ is excess energyGiuliani and Quinn, PRB 26, 4421 (1982).What is probability that an electron can travel distance L without a collision?2⎡ ⎤ ⎡ ⎤LL⎛∆ ⎞P~ Exp⎢− ⎥ ~ Exp⎢− ⎜ ⎟ ⎥⎢ vτv EF⎣ ee⎥⎦⎢⎣⎝ ⎠ ⎥⎦Signal relies on electrons not colliding should decay like a gaussian…E FAssuming τ ee< τ elE FTipQPC

Differential conductance vs. excess energySignal decays more quickly at longer distances from the QPC0.6 µm0 meV 1 meV 2 meV 3 meV∆g(e 2 /h)0.001.4 µm50 nm0.55∆g(e 2 /h)0.0050 nm0.13

Differential conductance vs. excess energySignal decays more quickly with increasing back gate voltage0 meV 1 meV 2 meV 3 meVBack Gate 0V∆g(e 2 /h)0.0050 nmBack Gate -2V0.3550 nm

Signal Vs. excess energySignal decays more quickly at longer distances from the QPCSignal decays more quickly with increasing back gate voltage

InteractionsImages are sensitive to whether or not an electron haslost energyThis is used to measure the distance over whichelectrons remain coherentThe measurements show that the scattering time dependson the Fermi energy and the excess applied energy.21 E ⎛F∆ ⎞ ⎡ ⎛EF ⎞ ⎛2q⎞TF1⎤= ⎜ ⎟ ⎢ln⎜ ⎟+ ln ⎜ ⎟+⎥τee 4π⎝EF ⎠ ⎣ ⎝ ∆ ⎠ ⎝ pF⎠ 2⎦

ExperimentAcknowledgementsHarvard UniversityTheoryMark TopinkaAnia BleszynskiKathy AidalaRobert WesterveltScot ShawAllison KalbenEric HellerUniversity of California – Santa BarbaraKevin MaranowskiArt Gossard

Image pattern of electronflow from a QPCConclusionsControl interference fringesImage local electron densityImage electron flow throughelectron optics elementsImage electron-electron scattering

Current workCombined scanning tunneling spectroscopy andtransport measurements of carbon nanotubesPhonon generation and absorptionLeRoy et al., Nature 432 374 (2004)

Improved fringe resolutionWiggling Tip VoltageFixed Tip Voltage100nm100nmdI/dVtip (nA/V)1.1 -0.80.0∆G (e 2 /h)0.6Wiggling tip voltage, images the spatial derivative of the flow

Imaging with AC voltage on tipTip Voltage SeriesBias Voltage Series100 nmTip Voltage goes from 0 to -3.5 V100 nmBias Voltage goes from -6.25 to 6.25 mV