Beenakker, C.W.J.

Foxon, C.T.

Harris, J.J.

Houten, H. van

Kouwenhoven, L.P.

Mooij, J.E.

Wees, B.J. van

Williamson, J.G.

Leiden University Scholary Publications

19th International Conference onTHE PHYS/CS OFS E MIC ONDUCTORSVOLUME 1Warsaw, Po l andAugust 75-75, 7355EDITOR: W. ZAWADZKIInstitute of PhysicsPolish Academy of Sciences39QUANTUM BALLISTIC ELECTRON TRANSPORT IN ACONSTRICTED TWO-DIMENSIONAL ELECTRON GASB.J. van Wees*, H. van HoutenW, C.W.J. Beenakker*, L.P. Kouwenhoven*,J.G. Williamson* , J.E. Mooij*, C.T. Foxon+ and J.J. Harris+* Department of Applied Physics, Delft University of TechnologyP.O. Box 5046, 2600 GA Delft, The Netherlands# Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands+Philips Research Laboratories, Redhill, United KingdomWe present an experimental and theoretical study of electron transport in constrictedgeometries, defmed in a high mobility two-dimensional electron gas (2DEG). In zeromagnetic field, the conductance of single point contacts, defined by a lateral depletiontechnique, changes in quantized Steps of 2e2/h, when the width is varied. Thisquantization, which persists in a magnetic field, is shown to result from the ballistictransport through the point contact, in which one-dimensional subbands are formed.Electron focusing has been observed in a double point contact geometry, showing ballisticand phase coherent transport along the boundary of a 2DEG. A description of thefocusing is given in terms of a non-local voltage measuremenLINTRODUCTIONElectron transport in low dimensional Systems has been studied predominantly in thediffusive regime, where the elastic mean free path IQ is smaller than the dimensions of theSystem. In this diffusive regime, quantum effects in the conductance at low temperaturesmay manifest themselves äs localization, the constructive interference of back scatteredelectron waves, or äs aperiodic oscillations äs a function of magnetic field, known äsuniversal conductance fluctuations. The observation of these effects is directly related tothe fact that phase coherence is not destroyed by elastic scattering, and mayextend on ascale far beyond the mean free path le between impurity scattering.We have studied electron transport in the ballistic regime, where electrons are scattered(or reflected) at the boundaries of the conductor only. This has been achieved by definingsubmicron geometries in the two-dimensional electron gas of high-mobility (1£=8.5μιτι)GaAs/AlGaAs hetero structures. The two-dimensionality of the electron transport inhetero structures makes these Systems ideal starting points for the study of electrontransport in constricted geometries. Confmement in two directions can be obtained byfabricating narrow wires in a 2DEG, either with etching techniques1, or by lateraldepletion, using a split-gate on top of the hetero structure2. The large Fermi wave length40λρ (typically 40 nm) makes ihese Systems attractive for the study of quantum transportBecause of the lack of impurity scattering in ballistic Systems, localization and universalconductance fluctuations are suppressed. The quantum size effects, arising from thelateral confinement of electrons, are therefore preferably studied in a ballistic System.We give a survey of our experimental and theoretical study of electron transpon innarrow and short constrictions, through which quantum ballistic transport occurs. With adouble point contact geometry, the coherent ballistic electron transport along the 2DEGboundary has been investigatedQUANTUM BALLISTIC TRANSPORT IN SINGLE POINT CONTACTS.The inset of Fig. l shows a schematic layout of the samples used to study electrontransport in narrow and short constrictions. By means of electron beam and opticallithography a split gate is fabricated on top of the hetero structure, having a width of 250nm between the gate electrodes. The point contact is defined by depleting the electron gasunderneath the gate. At Vg=-0.6V the electron gas undemeath the gate is fully depletedand a constriction with W«250 nm is formed. A further reduction of the gate voltagenarrows the constriction until it is fully pinched off at Vg=-2.2V. Ohmic contacts,between which the resistance can be measured, are attached to the wide regions.Fig. l Conductancequantization in zeromagnetic field. Theinset shows aschematic layout ofthe sample. Thedepletion regionsaround the gates areindicated.-1.0-2.0 -1.8 -1.6 -1.4GATE VOLTAGE (V)-1.2In the experiments the resistance between the ohmic contacts is measured äs a functionof gate voltage, both in the absence and presence of a magnetic field. The result for zeromagnetic field3 is given in Fig.l. Similar results have been obtained independently by41Wharam et alA The conductance of the point contact is shown, obtained from themeasurement after subtraction of a constant series resistance. A sequence of plateaux isobserved, where the conductance is quantized in mulriples of 2e2/h.To interpret these results, we model the constriction region äs a narrow channel, inwhich the electrons are confined by an electrostatic potential eV(x)= 1/2 mcüo2 x2 + eV0.As a result of this confinement one-dimensional subbands are formed with subbandSeparation ncuo. The narrow channel may now be envisaged äs an electron wave guidewith dispersion relation:• + eV0 (lThe modes, or l D subbands, in which the electron waves propagate are indexed with n(=1,2,3,..) and ky denotes the wave vector along the channel. In Fig. 2 the subbandoccupation is shown for two different values of the gate voltage.Fig.2 Subband occupation for two different gate voltages. A decrease in gate voltagereduces the number of occupied subbands.In the wide 2DEG regions the electron states are occupied to the Fermi level EFB(=12.5 meV). A reduction of the gate voltage increases both Cu0) which is a measure of thelateral confinement, äs well äs eV0, the electrostatic energy in the constriction. As shownin Fig. 2, both result in a reduction of the number of occupied subbands Nc.By applying a voltage V over the constriction, the right and left-going states arepopulated to different electrochemical potentials μι and μ2· As is clear from fig. 2, the netcurrent I results from the energy interval εΥ=μι-μ2, in which only right-going states areoccupied. The conductance can now be evaluated:42Ι ι Nc μι ]Gc = γ = γ Σ l 5-eNn(E)vn(E)dE (2.μ2Essential for the quantization is the energy and subband index independence of theproduct of spin-degenerate l D density of states Nn(E)= 2/π (dEn/dky)'1 and the velocityVn(E) = l/h dEn/dky , which gives the result:r Nc 2e2 2e2 _ , EF K „G c = Σ -ü- = -r-Int(— - + ) (3n=l n nIn the classical limit Nc»l, this equation describes a Sharvin contact resistance5. Eq. 3can also be viewed äs a direct consequence of the quantum mechanical. Landauerformula6, applied to the case of a perfect conductor. The resulting finite conductance wasfirst identified äs a quantum contact resistance by Imry^. Eq. 3 predicts a contactresistance of 2e2/h per occupied quantum channel. The conductance increases stepwisewhenever the Fermi level EF, controlled by the gate voltage, reaches a new l D subband.It must be noted, however, that Eq. 3 has been derived for an infinitely long channel.The actual constrictions are not only narrow but also short (The length L may be estimatedfrom the geometry of the depletion regions, shown in Fig. 1). Also Eq. 3 only holds if noelectron states with negative velocity are occupied in the energy ränge μι-μ2, whichrequires the absence of back scattering in or near the constriction. Although impurityscattering may probably be neglected (le»W,le»L), quantum mechanical reflection ofelectron waves may occur äs a result of the relatively abrupt widening at the ends of theconstriction. We surmise that the transition regions in between the quantized plateaux maybe explained by the partial reflection of electron waves.In a perpendicular magnetic field hybrid magneto-electric subbands are formed äs aresult of both electric and magnetic confmement. Because of the translational invariancealong the narrow channel, the electron transport may still be envisaged äs propagation ofelectron waves, which now have a different dispersion8 :En = (n - j)hcu + ~^~^ eVo . with τη"=τη — J ' and ω = Λ/ ωο +(ln~)2 ^4The magnetic fields increases the subband Separation which is already present in anarrow channel. Despite the different dispersion relations the relation Nn(E) vn(E) = 4/hstill holds in a magnetic field and the conductance is:G c = Σ Γ = - I n t ( . + ) (5n=l n n43Eqs. 4 and 5 show a gradual transition between the quantization in zero field,determined by CUQ, and the quantization in high magnetic fields, determined by Cuc=eB/m.Experimentally this is observed in Fig. 3, which shows the conductance of a pointcontact, obtained from the measured resistance for several values of the magnetic field,after subtraction of a gate voltage independent series resistance^.12Fig. 3 Conductancequantization in a magneticfield. A magnetic fieldincreases the subbandSeparation, which leads tobroadening of the plateaux.-2 -1.8 -1.6 -1.4 -1.2 -lGATE VOLTAGE (V)As in the zero field case, a sequence of quantized plateaux is observed. The effect ofthe magnetic field is to reduce the number of plateaux in a given gate voltage interval.This clearly shows magnetic depopulation of subbands. At high fields plateaux at oddmultiples of e2/h are beginning to be resolved äs a result of the spin-splitting of the l Dsubbands in the constriction. Spin-splitting in a parallel magnetic field has been srudiedby Wh'aram et al.4COHERENT ELECTRON FOCUSINGIn the previous sections the electron motion was restricted laterally by the electric fieldof the split-gate. However, äs we will discuss below, electrons can also be confined tothe boundary of the 2DEG by the application of a magnetic field. Classically, electronspropagate along the boundary in skipping orbits (see inset Fig. 5) with cyclotron radiuslc= mvF/(eB), with multiple specular reflections at the 2DEG boundary. As a result of the44translational invariance in the direction along the boundary, the quantum mechanicaltransport may be treated äs the propagation of electron waves along the 2DEG boundary.The quantization of the periodic motion perpendicular to the boundary, associated with theskipping orbits, leads to discrete modes in which these waves can propagate. Thesemodes, or magnetic edges states, also play a role in the (quantum) Hall effect in narrowwires^O. Experiments have been perforrned^, in which electrons are injected into the2DEG by means of an injector point contact, and are collected in a second point contact,with Separation L=3jim, after deflection by the magnetic field (see Fig.4)Fig. 4 Experimental setup for the electronfocusing experiment. Agate on top of a Hall bardefines the injector andcollector point contacts.The inset shows thedouble point contacts.F ig .5 E lec t ronfocusing spectra atseveral temperatures. At4 K focusing peaks atthe classical positions(arrows) are seen, atlower temperatures afine structure isresolved.-0.4 -0.2 0 0.2 0.4 0.6B (T)45The classical propagation in skipping orbits gives rise to electron focusing. Peaks areobserved in the collector voltage for fields Bf = n 2mvF/(eL), (indicated by arrows inFig.5) where an integer number n of cyclotron diameters fits in between the pointcontacts. At 4 K up to 8 peaks are observed, which illustrates the high degree ofspecularity of the reflections at the 2DEG boundary. At low temperatures fine structuredevelops in the collector signal. This can be understood by the coherent exitation of anumber of edges states by the injector. At the collector point contact interference12 occurs,depending on the relative phases of the waves. The observation of this interference showsthe phase-coherent propagation of electron waves along the 2DEG boundary.The electron focusing experiment may be described äs a non-local voltagemeasurement. In a three-terminal setup, shown in Fig.5, the collector signal for reversefields is a measure of. the longitudinal resistance, whereas for positive fields a Hallresistance RXy is superimposed. Alternatively the experiment has been performed äs afour-terminal Hall measurement (Hg.6).Fig. 6 Electronfocusing in afour-terminal Hallgeometry.-0.5-0.3 -0.2 -0.1 0 0.1B (T)For reverse fields the classical Hall resistance Rxy= B/(ne) is observed, for positive fieldsthe electron focusing gives rise to a modulation around the average Hall voltage13.CONCLUSIONSWe have observed a number of new phenomena associated with quantum ballistictransport. The possibility to make constrictions which resemble an electron waveguide,by means of lateral confinement from a split-gate, has led to the observation ofconductance quantization in the absence of a magnetic field. The propagation of electronwaves along a 2DEG boundary has been studied with an electron focusing experiment.46The results show that quantum transport can be controlled on a microscopic scale, whichleads us to expect more fascinating developments in this new field.We would like to thank M.E.I. Broekaan, P.H.M. van Loosdrecht, C.E. Timmering andL.W. Lander for their contributions to this work. We acknowledge the Delft Centre forSubmicrontechnology for their contribution towards the sample fabrication and thestichting "FOM" for financial supporLW Temporary address: Philips Laboratories Briarcliff, New York 10510.REFERENCES1. G.Timp, A.M. ChangJP. Mankiewich, R. Behringer, J.E. Cunningham, T.Y. Changand R.E. Howard, Phys. Rev. Lett. 59. 732 (1987); M.L. Roukes, A. Scherer, S.J.Allen Jr., H.G. Craighead, R.M. Ruthen, E.D. Beebe and J.P. Harbison, Phys.Rev.Lett. 5_i, 3011 (1987); H. van Houten, B.J. van Wees, M.GJ. Heijman, and J.P.Andre, Appl. Phys. Lett. 49., 1781 (1986).2. T.J. Thomton, M. Pepper, H. Ahmed, D. Andrews, and GJ. Davies, Phys. Rev. |Lett._5_6_, 1198 (1986); H.Z. Zheng, H.P. Wei, and D.C. Tsui, Phys. Rev. B M, 5635 l(1986).3. B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P.Kouwenhoven, D. van der Marel and C.T. Foxon, Phys. Rev. Lett. 60. 848 (1988).4. D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost,D.G. Hasko, D.C. Peacock, D.A. Ritchie and G.A.C. Jones, J.Phys. C: Solid StatePhys. 2_L L209 (1988).5. Y.V. Sharvin, Sov. Phys. JETP 2L 655 (1965).6. R. Landauer, Z. Phys. B 6JL 217 (1987); M. Buttiker, Phys. Rev. Lett. 5JZ., 1761(1986).7. Y. Imry in Directions in Condensed Matter Physics, Vol l, ed. G. Grinstein and G.Mazenko, World Scientific, Singapore, 102, (1986).8. K.F. Berggren and D.J. Newson, Semicond. Sei. Technol. l, 327 (1986); S.B.Kaplan and A.C. Warren, Phys. Rev. B.3A 1346 (1986).9. B.J. van Wees, L.P. Kouwenhoven, H. van Houten, C.W.J. Beenakker, J.E. Mooij,C.T. Foxon, and J.J. Harris, submitted to Phys. Rev. B.10. C.W.J. Beenakker and H.van Houten, Phys. Rev. Lett. £Q_, 2406 (1988).11. H. van Houten, B.J. van Wees, J.E. Mooij, C.W.J. Beenakker, J.G. Williamsonand C.T. Foxon, Europhys. Lett. 5_, 721 (1988).12. C.W.J Beenakker, H. van Houten and B.J. van Wees, submitted to Europhys. Lett.13. H. van Houten, C.W.J. Beenakker, J.G. Williamson, M.E.I. Broekaart, P.H.M.van Loosdrecht, B.J. van Wees, J.E. Mooij, C.T. Foxon and J.J. Harris, to bepublished.

Quantum ballistic electron transport in a constricted two-dimensional electron gas

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Quantum ballistic electron transport in a constricted two-dimensional electron gas

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