Direct observation of conduction electron beam transmissionthrough a Bi intercrystalline boundary

M. V. TsoiGrenoble High Magnetic Field Laboratory, Max-Planck-Institut fu¨r Festkorperforschung and Centre National de la Recherche Scientific,

Boıte Postale 166, F-38042 Grenoble Cedex 9, Franceand Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow Region, Russia

A. Bohm and M. PrimkeGrenoble High Magnetic Field Laboratory, Max-Planck-Institut fu¨r Festkorperforschung and Centre National de la Recherche Scientific,

Boıte Postale 166, F-38042 Grenoble Cedex 9, France

V. S. TsoiInstitute of Solid State Physics RAS, 142432 Chernogolovka, Moscow Region, Russia

P. WyderGrenoble High Magnetic Field Laboratory, Max-Planck-Institut fu¨r Festkorperforschung and Centre National de la Recherche Scientific,

Boıte Postale 166, F-38042 Grenoble Cedex 9, France~Received 2 October 1997!

A technique for studying conduction electron/interface interactions is used to study reflection and refractionof an electron beam at a Bi intercrystalline boundary. The technique involves scanning a small laser spot overthe surface of a bicrystal, exciting conduction electrons locally, and using a magnetic field to focus excitedelectrons onto a fixed point contact~an electron collector!, after the electrons have been reflected from, ortransmitted through, the intercrystalline boundary. Refraction of conduction electrons, high bicrystal interfacetransparency, and a correlated electron transmission through the interface have all been observed.@S0163-1829~97!50648-5#

Reflection of conduction electrons from the surfaces ofconductors has been studied for a long time using a widevariety of techniques.1 Studies of reflection from boundariesbetween two conductors are much rarer.2 We know of onlyone previous direct study of transmission—Sharvin andSharvin used a twinning plane in Al parallel to the crystalslab surface to produce a radio frequency size effect~Gant-makher effect!, which required for its existence strong cor-related transmission of electrons through the interface.3

In this paper, we describe the use of transverse electronfocusing~TEF! ~Refs. 4 and 5! to study transmission of con-duction electrons through, and reflection from, an intercrys-talline boundary in a Bi bicrystal. By injecting electrons intoa conductor, using a magnetic field to focus these conductionelectrons on the surface of the crystal, and by detecting thesefocused electrons using a point contact, it is possible to fol-low the trajectories of the electrons inside the crystal fromthe outside. We first describe the technique and our sample,then present our results and analysis, and then a summaryand conclusions.

We describe the basic principles of the technique for aconductor with a cylindrical Fermi surface~FS!, both be-cause this is the case for Bi having the electron FS close to acylindrical one~for details of the FS of Bi, see Ref. 6!, andbecause it simplifies the analysis. The electrons of interestare only those on the FS. For a metal with a cylindrical FS ina magnetic fieldH, the electron orbits in real space are sim-ply circles with the electron velocity always directed perpen-dicular to the cylinder axis, independent of the direction of

H. Therefore in real space an electron in a given crystalremains in a given plane perpendicular to the cylinder axis,independent of whether it is scattered or not, so long as thescattering returns it to the same cylindrical FS. The value ofH and its direction simply change the cyclotron radiusRc ofthe electron orbit according to the equation

Rc~H,w!5Rc~H,w50!/cosw5pFc

eH

1

cosw, ~1!

wherepF is the Fermi momentum,c the light velocity,e theelectron charge, andw the angle betweenH and the cylinderaxis.

We describe the motion of an electron through a bicrystalfor a pedagogically simple case as illustrated in Fig. 1. Thebicrystal sample surface is in thexy plane. An intercrystal-line boundary is a mirror-symmetry plane between crystal Awith its FS cylinder along thex axis and crystal B with its FScylinder along they axis. We choose an appliedH in theboundary plane so that cyclotron radii@from Eq. ~1!# areequal in both crystals.

The construction in Fig. 1~b! describes inp space the lawsof reflection from the sample surface~pxpy plane! and trans-mission through the intercrystalline boundary~pzH plane!for an electron in stateA. In the construction, we coincidedthe origins0 ~zero! of the Brillouin zones of both adjacentcrystals. We treat the electron/interface scattering as elasticand for simplicity assume conservation of an electron mo-mentum component along an interfacept . For the electron

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reflection, elastic scattering impels the electron to stay on FS,and conservation ofpt impels it to stay on lineAB—line ofthe constantpt—perpendicular to the surface. Therefore, theelectron goes from stateA to stateB. The analysis of anelectron momentum change under transmission through aninterface is not trivial. The basic principles of conductionelectron transmission were formulated by Pippard.7 After theelectron transmission from crystal A into crystal B the elec-tron state can be determined similarly to the reflection case.In our case, the electron goes fromA to C at the intersectionof the constantpt , line AC ~perpendicular to thepzpy plane!with the FS of crystal B.

In Fig. 1~a! trajectories of electrons injected through the

emitter E are depicted in real space. In crystal A, injectedelectrons move in thezy plane. We assume that the injectionis uniform in direction in thezy plane. An injected electronwill execute a part of the circle and reproduce it after thespecular reflection from the sample surface. It is crucial thatin a Fermi system even for electrons injected isotropically inan angle, the density of electron flux incident onto a surfaceis singular for points at distances 2Rc , 4Rc , etc., away fromE, i.e., where electrons emitted perpendicular to the surfacereturn to it. At these points caustics—regions of high densityof electron trajectories@visible in Fig. 1~a!#—intersect thesample surface. In Fig. 1~a!, trajectories of the electrons in-jected perpendicular to the surface are shown in thick lines.If we scan an electron collectorC over the sample surface, itwill collect injected electrons incident onto the surface at thecollector location, and register a focusing peak~spike! onlyat distances 2Rc @point A in Fig. 1~a!#, 4Rc , etc., fromE onthey axis. For perfectly specular reflection from the surface,the heights of the peaks after 0, 1, 2, etc., reflections withinthe same crystal will be all the same. If the probability ofspecular reflection isq,1, then the ratio of the heights ofadjacent peaks will beq. These problems are treated in detailin regular TEF.4,5,8

Consider now what happens when crystal A has an inter-crystalline boundary with crystal B. As illustrated in Fig.1~a!, transmitted electrons are ‘‘refracted’’ at the boundary,initially moving in theyz plane, but then in thexz plane. Incrystal B under the specified conditions, focusing will be atpoints B, C, etc., where caustics intersect the sample sur-face. The heights and locations of the peaks give informationabout the transmission probability and correlation.

If H is made so large that 2Rc becomes less than thediameterd of the electron emitter, then the series of ballis-tically produced focusing peaks collapse together to giveskipping paths where an apparently continuous electron fluxsimply propagates along the sample surface in directions per-pendicular to the axes of the electron cylinders in each crys-tal @skipping pathE-O-C in Fig. 1~a!#. We called this casedrift flux to distinguish it from theballistic flux when2Rc@d as illustrated in Fig. 1~a!. Figures 1~c!–1~e! showschematically the paths that would be followed by electronstransmitted or reflected from the boundary, now assigningthree different electron cylinders to each crystal, to illustratewhat is actually expected in Bi, where the FS contains threenoncollinear electron cylinders. A mobile collector scannedover the entire surface allows direct visualization of the driftflux both before it is incident upon the boundary and after itis reflected and refracted, for chosenH and boundary orien-tations.

In the actual experiments, a laser beam focused to a smallspot on the surface of a Bi bicrystal~emitterE! creates elec-tronic excitations~electrons! ~Ref. 9! that are focused by auniform and constant magnetic fieldH onto a point-contactcollectorC ~the technical setup is described in Ref. 10!. Pre-vious TEF measurements11 proved the similarity of an ohmicmicrocontact emitter with an electric current passing throughit and a small laser spot emitter. In the actual situation, thelaser spot (d'15mm) is scanned over the sample surface,inducing an electron-hole flux due to thermoelectric effectsand producing a voltage atC ~Ref. 12! that varies with theposition ofE. For ease of exposition, we describe the alter-

FIG. 1. Dynamics of conduction electrons in a bicrystal in realspace and inp space~b!. ~a! Electron focusing by a magnetic fieldand electron refraction at an intercrystalline boundary~zH plane!.The electron injection through the emitterE is uniform in directionin the zy plane. Skipping electron trajectories are drawn up to thethird specular reflection from the sample surface~xy plane!. Visiblecaustics~regions of high density of electron trajectories! cross theyandx axes at pointsA, B, C—points of electron ‘‘focusing.’’ Tra-jectories of the ‘‘focused’’ electrons, injected perpendicular to thesurface, are shown in thick lines. LineE-O-C shows the refractingskipping path.~b! The laws of reflection from a sample surface~pxpy plane! and transmission through an intercrystalline boundary~pzH plane! are illustrated for an electron in stateA. Two FS cyl-inders are for two parts of the bicrystal, respectively. Electron tra-jectories inp space—cross sections of the FS with a plane perpen-dicular to H—are shown in thick lines. When reflecting from thesurface the electron goes from stateA to stateB. When transmittingthrough the boundary it goes from stateA to stateC. For details,see the text.~c!–~e! Schematic drawing of the skipping paths~1–3and 18– 38 in adjacent crystals, respectively! propagating along thesample surface~arrows indicate the direction of propagation for agiven orientation of the magnetic fieldH!. The three paths for eachcrystal are due to the three ‘‘cylinders’’ of the Bi Fermi surface~seetext!. ~c! No interface~all paths, 1–3 and 18– 38, are shown comingfrom one emitter!. ~d! and ~e! illustrate the possibilities of electrontransmission through a boundary@paths 1→18 in ~d! and 2→28,2→18 in ~e!# and electron reflection from the boundary@path 1→3in ~d!# for two different boundary orientationsAA8 andBB8.

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native situation whereE is held fixed andC scanned—thepatterns should be essentially the same in the two cases.

The sample studied was a Bi bicrystal accidentally ob-tained from growth in a polished quartz demountable form.The common ballistic electron mean-free path in the twocrystals was about 0.2 mm (T51.2 K). TheC3 axes of thetwo crystals were parallel to each other and inclined by 12°from the normal to the sample surface. The interface wasparallel to the axis of one of the electron cylinders in crystalA. Crystal B was rotated relative to A around theC3 axis byabout 30°. The intercrystalline boundary at the surface wasmade visible by etching the surface. Electron reflection fromsuch an etched surface in Bi is expected to be diffuse with ahigh probability of electron-electron intervalley scattering,13

i.e., under reflection from such a surface, an electron on onecylinder can be scattered to another cylinder, thereby chang-ing its plane of motion within the same crystal.

With this background, we now turn to our data.Drift Flux. Preliminary measurements were made to

check the interface location and to determine the planes ofelectron motion~skipping paths! in both crystals, to allow adirection ofH to be chosen that would yield good visualiza-tion of the electron paths. Figure 2~a! shows a typical experi-mental two-dimensional~2D! gray-scale pattern forH largeenough to produce a drift flux. Lighter color indicates higherflux intensity. The emitter was located above the scannedarea at about (150,250). Visible are the incident electronflux coming from the top center of the picture; the boundarydesignated by a white line; and the refracted electron fluxexiting from the picture at the bottom, left of center. Theangle of refraction is;30°, in good agreement with therelative crystallographic orientation of the two crystals. Thefluxes in both crystals damp and broaden as they propagate,behaviors that we attribute to electron relaxation and inter-valley transitions, exacerbated by the surface damage pro-duced by the etching used to make the boundary visible.

However, the similar intensities of the fluxes just on the twosides of the boundary indicate that any intensity change atthe boundary is small and that the transparency of the bound-ary for the drift flux is close to unity. Separate experimentsto better examine the reflected flux confirmed that the re-flected intensity was only a few percent.

Ballistic Flux. As a reference for our boundary refractionand reflection studies with ballistic flux, we first examinedthe ballistic flux TEF pattern for our bicrystal far from theintercrystalline boundary. Figure 2~b! shows that we couldresolve three gradually broadening and weakening focusingpeaks generated from an emitter located outside of the pic-ture at about (420,2110). For the TEF boundary studies,Hwas chosen to give a cyclotron radius of about the distanceof the emitter from the boundary. Once the skipping paths inthe two crystals were known,E was set at a distance of;100mm from the boundary@i.e., at about (130,250) inFig. 2~c!# in crystal A, andC was scanned over the area ofinterest around the boundary to determine the direction ofHand its precise value to give the most intense focusing peaksdue to the transmitted electrons. In the resulting geometry,the ballistic flux was incident nearly normal to the boundary.Figure 2~c! shows the presence of two focusing peaks afterrefraction at the boundary~white line!—a fairly well-definedpeak at about~25,135! for one electron cylinder in crystal B,and a less well-defined one at about~170,110! for anotherelectron cylinder. Two peaks originated in splitting at theboundary of a single electron beam for one cylinder in crys-tal A into two electron beams for two different cylinders incrystal B. The presence of these two peaks requires corre-lated transmission of electrons through the boundary. Thefact that the intensities of the refracted peaks are much lowerthan those of the unrefracted peaks in Fig. 2~b!, shows thatthe probability of correlated transmission is low (,0.1). Amore quantitative determination of this transmission prob-

FIG. 2. Density distributions of injected through-the-emitter electrons over the Bi bicrystal surface in a magnetic fieldH. ~a! 2Dgray-scale density plot of the drift flux refracting at the intercrystalline boundary~indicated as a white line!. Lighter color indicates higherflux intensity ~H550 0e in, approximately, thex axis direction!. For (x,y), the flux enters from approximately~200, 0! and exits atapproximately~150,400!, ~b, c! 2D gray-scale density plots of the ballistic flux~TEF!. ~b! Without the boundary~H510 Oe in, approxi-mately, thex axis direction!. Three ordinary TEF peaks are visible centered at, roughly,~430,10!, ~440,120!, and ~450,240!. The emitterposition is approximately~420,2110! out of the scanned area.~c! Transmission through the boundary~indicated as a white line; H56 Oe in, approximately, thex axis direction!. Two broad, weak TEF peaks after electron transmission through the boundary are visible,one at approximately~25,135! and another at approximately~170,110!. The emitter position is approximately~130,250! out of the scannedarea. The white region within~90–130,0–50! is due to the enhanced contrast needed to make the two weak peaks visible.

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ability is complex and will require a detailed theoreticalanalysis that does not yet exist.

To summarize, we have applied a technique, involvingTEF with light-induced electron excitation, for the study ofconduction electron transmission through~with refraction!,and reflection from, an intercrystalline boundary in Bi forbothdrift andballistic fluxes. For thedrift flux, the observeduncorrelated transmission probability was nearly unity andreflection was only a few percent. For theballistic flux, incontrast, the correlated transmission probability was small

(,0.1). Therefore the high intercrystalline electron transpar-ency originates in uncorrelated electron transmission. Notethat the technique can be used for the analysis of an interfacestructure in situ probing the interface with conductionelectrons.5

Useful discussions with Professor J. Bass are gratefullyacknowledged. This work was partially supported by GrantNo. 95-02-05202 from RFFR and Grant No. 020/3 from SC-STP.

1D. Schumacher,Surface Scattering Experiments with ConductionElectrons, Springer Tracts in Modern Physics Vol. 128~Springer, Berlin, 1993!.

2V. S. Tsoi, N. P. Tsoi, and S. E. Yakovlev, Zh. Eksp. Teor. Fiz.95, 921 ~1989! @Sov. Phys. JETP68, 530 ~1989!#.

3Yu. V. Sharvin and D. Yu. Sharvin, Zh. Eksp. Teor. Fiz.77, 2153~1979! @Sov. Phys. JETP50, 1033~1979!#.

4V. S. Tsoi, Zh. Eksp. Teor. Fiz. Pis’ma Red.19, 114 ~1974!@JETP Lett.19, 70 ~1974!#.

5V. S. Tsoi, J. Bass, and P. Wyder, Adv. Phys.41, 365 ~1992!.6V. S. Edel’man, Adv. Phys.25, 555 ~1976!.7A. B. Pippard,The Dynamics of Conduction Electrons~Gordon

and Breach, New York, 1965!.8Yu. A. Kolesnichenko, Fiz. Nizk. Temp.18, 1059 ~1992!

@Low Temp. Phys.18, 741 ~1992!#.9The laser beam also creates hole excitations~holes! that we

neglect as having a very short mean-free path in Bi comparedwith an electron mean-free path. I. F. Sveklo and V. S. Tsoi,Zh. Eksp. Teor. Fiz. 103, 1705 ~1993! @JETP 76, 839~1993!#.

10J. Heil, A. Bohm, M. Primke, and P. Wyder, Rev. Sci. Instrum.67, 307 ~1996!.

11J. Heil, M. Primke, K. U. Wurz, and P. Wyder, Phys. Rev. Lett.74, 146 ~1995!.

12The voltage~current! across the collector is proportional to theelectron flux incident onto it.

13S. I. Bozhko, I. F. Sveklo, and V. S. Tsoi, Fiz. Nizk. Temp.15,710 ~1989! @Sov. J. Low Temp. Phys.15, 397 ~1989!#.

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