Build-up of interference patterns with single electrons

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Eur. J. Phys. 34 (2013) 511–517 doi:10.1088/0143-0807/34/3/511

Build-up of interference patterns withsingle electrons

Giorgio Matteucci1,5, Michele Pezzi1, Giulio Pozzi1,Gian Luigi Alberghi1,2, Filippo Giorgi1,2,Alessandro Gabrielli1,2, Nicola Semprini Cesari1,2,Mauro Villa1,2, Antonio Zoccoli1,2, Stefano Frabboni3

and Gian Carlo Gazzadi4

1 Department of Physics and Astronomy, University of Bologna, V/le B Pichat, 6/2,I-40127 Bologna, Italy2 INFN-Bologna, V/le B Pichat, 6/2, I 40127 Bologna, Italy3 Department of Physics, University of Modena and Reggio Emilia, Via G Campi 213/a,I-41125 Modena, Italy4 CNR-Institute of Nanoscience-S3, Via G Campi 213/a, I-41125 Modena, Italy

E-mail: giorgio.matteucci@unibo.it

Received 19 December 2012, in final form 13 January 2013Published 4 March 2013Online at stacks.iop.org/EJP/34/511

AbstractA conventional transmission electron microscope, equipped with a fastrecording system able to measure the electron arrival time and the positionof single electrons, is used to show the build-up of interference patterns. Twoexperiments are presented. The first is the electron version of the Grimaldi andYoung experiments performed with light, where single electrons strike on anopaque thin wire. Interference fringes are observed in the geometrical shadowof the wire and diffraction effects are clearly displayed at the wire edges.The second, original experiment reports the build-up of two-slit interferencepatterns with single electrons.

(Some figures may appear in colour only in the online journal)

1. Introduction

It is well known that Grimaldi observed that light travelling near an opaque obstacle or throughvery narrow apertures produces patterns of bright and dark fringes on a white screen. Grimaldireferred to this unexpected observation as diffraction of light (see [1], proposition II). About150 years later, Young adopted and improved the same experimental arrangements devised by

5 Author to whom any correspondence should be addressed.

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Grimaldi and explained the fringes as the result of the superposition of two portions of lightwaves passing respectively along both sides of an opaque obstacle or through the celebratedtwo-hole setup.

In particular, the two-slit device has been widely adopted to discuss the foundations ofquantum mechanics [2] and to demonstrate the wave behaviour of material particles (see[3, 4] and references therein). According to Feynman, the observation of the build-up of aninterference pattern with single electrons puts ‘you up against the paradoxes and mysteriesand peculiarities of nature one hundred per cent’ [5].

The build-up of the interference pattern with single electrons has been demonstrated byreplacing the two-slit interferometry device with an electrostatic biprism, i.e. a thin chargedwire whose field acts on electrons in a similar way to that in which a Fresnel biprism actson light [6–8]. The observation of the cumulative effect of many electrons is so enthrallingthat the readers of Physics World selected this experiment as the most beautiful in physics[9, 10].

To reveal single electron events, the final viewing screen of our conventional transmissionelectron microscope (TEM) was replaced by a CMOS detector developed for experiments infuture colliders [11]. This device, which was also very sensitive to electrons, was connectedto a fast recording system. With improved accuracy on previous recording devices, it hasbeen possible to observe not only the build-up of high statistic electron interference patterns,but also to measure, for the first time, the time distribution of the arrival of single electrons[12, 13].

This paper aims to present the single electron counterparts of the famous experimentscarried out by Grimaldi and Young to investigate the nature of light. In the same wayas Grimaldi and Young illuminated an opaque obstacle with a beam of light, consideredat that time as a homogeneous stream of corpuscles, a beam of single electrons isused here to illuminate a metal wire which is an impenetrable barrier for electrons.The effect of this opaque obstacle on the stream of particles is observed on our specialdetector placed behind the wire. As expected, an interference pattern is revealed in thegeometrical shadow of the wire while Fresnel diffraction fringes are clearly detected atthe obstacle rims. It must be emphasized that diffraction and interference experiments havebeen performed using, respectively, a slit, a circular hole, two-slit, two-hole, multiple-slitdevices, etc. For diffraction from one hole and interference from two holes see [14, 15]and references therein, while for a comprehensive review on electron interferometry see[7, 16].

Recently, using nanotechnology tools, two- and multiple-slit setups have been produced tocarry out electron interference experiments [17, 18]. Taking advantage of this method, a two-slit device was produced and used to carry out interference experiments with single electrons.With respect to the experiments with an electrostatic biprism, the present arrangement hasthe following advantages: (1) electrons travel through the slits in an electromagnetic field freeregion and (2) the images are Fraunhofer patterns, the interpretation of which is much simplerthan for images obtained with the electrostatic biprism.

The observation of the build-up of interference patterns with successively travellingelectrons helps students of quantum physics understand the wave properties of an individualparticle. The usefulness and relevance to physics education is stressed by Feynman: ‘Thequantum behaviour of atomic objects (electrons, protons, neutrons, photons, and so on) is thesame for all, they are all ‘particle waves’, or whatever you want to call them. So what we learnabout the properties of electrons (which we shall use for our examples) will apply also to all‘particles’, including photons of light’ [2].

Build-up of interference patterns with single electrons 513

165 µs

Figure 1. Pictorial view of the stack of frames recorded in a typical experiment to display the finalinterference pattern.

2. Experimental set-up

The sensor used to detect electrons is a custom silicon chip named Apsel4D. It consists ofa matrix of 128 × 32-pixel monolithic active pixel sensors developed in 0.13 μm CMOStechnology and equipped with a fast digital readout [11]. Each square pixel (side-50 μm)provides a hit/not-hit information tagged with a time stamp label. The sensor chip wasoriginally designed for applications of vertex detectors of the next generation to particlephysics experiments. These chips were designed with efficiency (>90%), reliability, cost ofmaterials and high particle rates in mind. To satisfy these requirements, the Apsel4D chip hasbeen developed as a very fast and efficient application specific integrated circuit. The pixelsdo not provide information on the detector charge, thus minimizing the output data stream.

Figure 1 represents schematically the build-up of images as operated by our system.Starting from the bottom, the first frame shows a one-hit event. Two successive empty framesare shown, while the fourth frame reports a further one-hit event. The frame at the top offigure 1 shows the final pattern resulting from the superposition of a large number of recordedframes. With this procedure it is possible to observe, step by step, the build-up of an image byselecting the recorded non-empty frames.

Interference experiments were carried out with a Philips EM400T TEM equipped witha standard thermionic source (hair-pin filament), operating at 60 kV accelerating voltage (deBroglie wavelength λ = 4.9 pm). The condenser lenses of the microscope are excited at theirmaximum strength to obtain the lateral coherence conditions needed for the observation ofinterference effects.

In the first experiment we present, the diffracting obstacle is a platinum wire (0.6 μmdiameter) connected to the ground [1]. The wire, which is an impenetrable barrier for theincoming electrons, is inserted at the standard specimen level. The wire produces a divisionof the electron beam into two parts which pass through its right- and left-hand sides. Theobjective lens of the microscope is switched off and the diffraction mode is used to select anobservation plane at a distance of a few centimetres below the wire itself [1]. The projectorlens system conjugates this observation plane with the detector plane.

In the second experiment, a two-slit device replaced the thin wire. A gold layer ∼250 nmthick was deposited by flash evaporation on a commercial copper grid coated with a carbonfilm. The two slits were produced by milling the gold-carbon layers using the focusing ion

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Figure 2. Fresnel diffraction pattern of the wire obtained by summing up many frames

beam technique. The width of each slit was about 95 nm and their separation and length 430and 1550 nm, respectively. Due to the resulting small diffraction angle conditions (of the orderof 10−5 rad), Fraunhofer interference patterns are recorded once the electron optical set-upis arranged in the so-called low-angle diffraction mode. Accordingly, the condenser lensesare excited at their maximum strength to obtain the required lateral coherence conditions andthe objective lens is weakly excited to project the interference pattern onto the intermediateimage plane named the ‘selected area aperture plane’. The remaining lenses of the electronmicroscope provide an enlargement of this pattern onto the detector. Therefore the electronmicroscope works as a diffraction camera whose length can be increased up to several hundredmetres.

3. Experimental results

In this section we will first present the experimental results in which the diffracting obstacle isa thin conducting wire. The electron beam current which strikes on the wire is strongly reducedand the electron optical conditions are arranged so that, at least in principle, one electron ata time is going through the microscope column. Under these conditions the performance ofthe detector to display the hit of single electrons and the time-stamp label has been tested.Secondly, a quantitative analysis of the detector capabilities, together with clear evidence ofthe build-up of an interference pattern, are fully reported for the two-slit experiment.

3.1. Build-up of diffraction and interference patterns from an opaque obstacle

Figure 2 shows the magnified image of the electron distribution, formed on a plane placedat a distance of 22 mm from a wire (0.6 μm diameter). This picture is obtained with theoverlapping of 131 000 frames. Faint interference fringes are barely detected in the centralregion representing the ‘geometrical shadow’ of the wire, while diffraction fringes are seenmore clearly at the wire rims outside the geometrical shadow. As discussed in detail in [1],it is impossible to give a classical explanation of how electrons, passing on both sides of thewire, form patterns of bright and dark fringes either in the geometrical shadow of the wire oroutside it.

By integrating the signal along the wire-axis direction of figure 2, it is possible to averagethe noise and obtain the graph of figure 3(a). The faint interference fringes in the shadowregion, resulting from the overlapping of the two portions of the wave passing on the right-and left-hand sides of the wire, are now clearly visible, as well as the modulation of thediffraction at the two rims of the wire itself. An accurate calibration of the experimental set-upis very troublesome in these non-standard electron–optical conditions. Therefore, simulationsof the experimental image were carried out by calculating the Fresnel–Kirchhoff integral usinga plane wave illumination and, in particular, for different wire-observation plane distances.

Build-up of interference patterns with single electrons 515

(a)

(b)

Figure 3. (a) Integrated intensity along the wire axis of figure 2; (b) theoretical Fresnel imagecalculated in a plane at a distance of 22 mm. Plane wave illumination is assumed.

Figure 3(b) shows the result obtained for a distance of 22 mm. The agreement with theexperimental pattern is extraordinary.

3.2. Build-up of two-slit interference fringes

In our experiment, the electron beam current was adjusted so that in a typical run 130 000hit pixels were recorded in about 7 min, that is an electron detection rate of approximately310 electrons s−1. The electronic unit control records 6060 frames per second, i.e. a single frameevery 165 μs. It turns out that the detector detects, approximately, one hit every 20 frames.

Figure 4(a) reports the multiplicity distribution of frames where multiplicity is the numberof hit pixels in each frame. As can be seen in the histogram, the fraction of frames with electronmultiplicity higher than one is at the per cent level. An outstanding feature of the detector isits ability to record the arrival time of particles. From the measurement of the time intervalwhich separates two adjacent non-empty frames, the distribution of the relative arrival timeof detected electrons is obtained (figure 4(b)). It turns out that the average time intervalis about 3.2 ms. The distance from the source to the detector is approximately 1.2 m andelectron velocity is about 134 000 km s−1; therefore, the time of flight of electrons through the

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Figure 4. (a) Multiplicity distribution of frames. The bin zero, that is not plotted and countsthe empty frames, is of the order of 2.4 × 106. (b) Distribution of time intervals between twoconsecutive non-empty events.

Figure 5. Pictorial view of the build-up of the interference pattern with 1, 2, 10, 100, 1000, 94 000electrons, respectively.

microscope is about 9 ns. As a consequence, each electron is completely read out before thenext one is emitted by the source.

The coherence relation among single electrons becomes evident when the collected framesare added up to display the cumulative effect. The step wise building up of the interferencepattern is shown in figure 5, where the integrated results over 1, 2, 10, 100, 1000, 94 000non-empty frames, respectively, are reported. The first three pictures report 1, 2 and 10 brightspots representing the hit on the detector of single electrons.

It can be ascertained that the superposition of the effects of the first 1000 electrons makes itpossible to detect unambiguously a typical two-slit interference pattern. The final image showsthe interference maxima whose intensity is modulated by the diffraction envelope of both slits.The colour scale representing the entries is different in the six plots. The spacing of theinterference fringes is as expected and consistent with the value calculated by the knowledgeof the calibrated slits-detector distance, in accordance with previous similar experiments [12].

Build-up of interference patterns with single electrons 517

4. Conclusions

A new detector, developed for high-energy physics experiments and a standard electronmicroscope were used to display the gradual build-up of diffraction from a thin wire and two-slit interference patterns with individual electrons travelling through the microscope column.We believe that both our original experiments provide a stimulating approach to show studentsthe wave–particle duality of electrons and in particular the build-up of interference patterns asa result of the coherent superposition of a large number of single ‘independent’ particles.

These results are a convincing demonstration of one of the most intriguing features ofquantum mechanics and thus are of overwhelming educational and cultural interest.

Acknowledgment

We wish to thank R Berti for technical assistance.

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