Microsoft Word - Aidala_focusing_arXiv.doc
Imaging Magnetic Focusing of Coherent  
Electron Waves 
Katherine E. Aidala,1* Robert E. Parrott,2 Tobias Kramer,2 E.J. Heller,2,3  
R.M. Westervelt,1,2 M.P. Hanson4 and A.C. Gossard4 
1Div. of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 
2Dept. of Physics, Harvard University, Cambridge, MA 02138 
3Dept. of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 
4Dept. of Materials Science, University of California, Santa Barbara, CA 93106 
* Present Address: Dept. of Physics, Mount Holyoke College, South Hadley, MA 01075 
The magnetic focusing of electrons has proven its utility in fundamental studies of 
electron transport.1-4  Here we report the direct imaging of magnetic focusing of 
electron waves, specifically in a two-dimensional electron gas (2DEG).  We see the 
semicircular trajectories of electrons as they bounce along a boundary in the 2DEG, as 
well as fringes showing the coherent nature of the electron waves.  Imaging flow in open 
systems is made possible by a cooled scanning probe microscope.5-17  Remarkable 
agreement between experiment and theory demonstrates our ability to see these 
trajectories and to use this system as an interferometer.  We image branched electron 
flow11 as well as the interference of electron waves.10,11,18  This technique can visualize 
the motion of electron waves between two points in an open system, providing a 
straightforward way to study systems that may be useful for quantum information 
processing and spintronics.19-21 
For nanoscale devices at low temperatures, both the particle and wave aspects of 
electron motion are important.  Electrons can travel ballistically through an unconfined two-
dimensional electron gas (2DEG) while their quantum phase remains coherent.  The 
development of unconfined or open devices in this regime1,22-26 clears the way for new 
applications in electron interferometry, spintronics, and quantum information processing.18-
21,27  The most basic question for open systems is: "What is the pattern of flow for electron 
waves?"  Imaging flow is difficult, because the electrons are often buried inside a 
heterostructure, and because low temperatures and strong magnetic fields are needed to 
observe quantum phenomena.   
A cooled scanning probe microscope (SPM) provides a way to image flow: a 
conducting SPM tip above the sample capacitively couples to the 2DEG below.5-17  Pathways 
for electron waves emerging from a quantum point contact (QPC) were revealed10 and found 
to have a dramatic branched form.11  The branching was shown to be the real-space 
consequence of small-angle scattering by charged donor atoms - the well known mechanism 
that limits the electron mobility.  A charged tip was able to image this flow, because it 
backscattered electrons; some of them follow a time-reversed path back through the QPC, 
measurably reducing its conductance.  Images of flow were obtained by displaying the QPC 
conductance as the tip was scanned in a plane above the sample.10,11  However, this 
backscattering imaging technique cannot be used in a perpendicular magnetic field, because 
the electrons follow curved paths and no longer reverse.  Magnetic fields are important for 
spintronics and quantum information processing,19-21 so it is important to understand electron 
flow with a field present.   
In this letter, we demonstrate a new way to image the flow of electron waves between 
two points in an open system using a cooled SPM, and we use this approach to image 
magnetic focusing of electron waves in a 2DEG.  Magnetic focusing plays an important role 
in the study of ballistic transport.1-4  Images were obtained by using an electron lens formed 
in the 2DEG beneath a charged SPM tip to redirect flow, by throwing a shadow downstream.  
We obtain clear pictures of semicircular bouncing-ball patterns of particle flow that cause 
magnetic focusing.  Branching of the flow is visible.  In addition, we see fringes created by 
the interference of electron waves bouncing off the tip with those travelling directly between 
the two QPCs.  In this way, the cooled SPM acts as a new type of electron interferometer for 
solids.  The observed bouncing-ball patterns of flow and interference fringes are in excellent 
agreement with full quantum simulations that include the tip-induced electron lens and small-
angle scattering, demonstrating that this new imaging technique accurately views the flow of 
electron waves through the device. 
Figure 1 illustrates the device geometry and the imaging technique.  Two QPCs are 
formed in a 2DEG by surface gates (Fig. 1a); a scanning electron micrograph of the device is 
shown in Fig. 1b.  The separation between the QPC centers is L = 2.7 μm.  A 2DEG with 
density 3.8x1011 cm−2 and mobility 500,000 cm2/Vs is located 47 nm beneath the surface of a 
GaAs/AlGaAs heterostructure with the following layers: 5 nm GaAs cap, 20 nm Al0.3Ga0.7As 
barrier, Si delta-doping, then a 22 nm Al0.3Ga0.7As barrier next to the 2DEG in GaAs.  Metal 
gates that define the QPCs were fabricated using e-beam lithography.  The device is mounted 
in the SPM, inside a superconducting solenoid, and cooled to 4.2 K.  A computer controls the 
SPM and records the images.  
Magnetic focusing (Fig. 1d) occurs because electrons leaving one QPC over a range of 
angles circle around and rejoin at the second QPC, when the spacing L is close to the 
diameter of a cyclotron orbit.  For GaAs the electron cyclotron orbit is circular with radius: 
     rc = kF eB ,       (1) 
where e is the electron charge, B is the perpendicular magnetic field, and kF  is the Fermi 
wavevector.  As B is increased, the first focusing peak occurs when L = 2rc .  Additional 
peaks occur at higher fields when L = 2nrc  is an integer multiple of the cyclotron diameter at 
fields: 
     Bn = 2n kF eL .      (2) 
The shape, clarity and spacing of magnetic focusing peaks provide information about ballistic 
flow in the sample material.  The effects of small-angle scattering are shown in the simulation 
of Fig. 1e (and in supplementary material):  magnetic focusing still occurs, but the flow now 
contains branches11 similar to flow in the SPM image in Fig. 1b for B = 0.   
To image electron flow from one point to another, a small movable electron lens is 
created by creating a dip in electron density immediately below the charged SPM tip.  The 
lens deflects electrons (Fig. 1c), throwing a V-shaped shadow downstream.  An image of 
electron flow is obtained by displaying the transmission T between the QPCs as the tip is 
scanned across a plane 10 nm above the surface.  T is measured by recording the voltage 
across the second QPC as a known current is passed through the first.  When electrons are 
ballistically injected into the second QPC from the first, a voltage develops that drives a 
current in the opposite direction to cancel the influx of electrons; this voltage is proportional 
to T. 
Simulations of electron flow that show how the imaging technique works are presented 
in Figs. 1c-f.  A Gaussian ϕ tip = Vo exp − r − rtip( )
2a2( ) is used to model the tip potential in 
the 2DEG, where rtip is the tip position, a is the width, and Vo is the height.  For this paper Vo 
> 0, and the tip creates a dip in electron density.  The relative strength of the tip potential is 
   η =
,        (3) 
where EF is the Fermi energy.  For η <1, the electron gas is partially depleted, and an 
imperfect diverging lens is created with a focal length determined by η.  When η ≥1, the 
2DEG is fully depleted, and electrons can backscatter.  Simulations (Fig. 1c) at B = 0 show 
how a weak lens (η = 0.2) creates a V-shaped shadow downstream by forcing electrons to the 
sides, where they form two caustics in flow along the legs of the V.  Figure 1f shows how we 
image magnetic focusing: when the shadow cast by the lens beneath the tip hits the second 
QPC, T is reduced.  The amount of reduction ΔT is proportional to the original flux, before 
the tip was introduced.  By displaying ΔT vs. tip position rtip as the tip is raster scanned in a 
plane above, an image of the original flow is created.28 
Experimental images of magnetic focusing near the first, second, and third peaks are 
presented in Figs. 2a-c for a weak tip (η ≈ 0.5), and in Figs. 2d-e for a strong tip (η ≈1.0).  
The bouncing semicircular cyclotron orbits imaged here are an experimental visualization of 
the origins of magnetic focusing.  Figure 2a shows a single, semicircular crescent 
characteristic of the first peak.  Branches in flow from small-angle scattering are also visible.  
For Fig. 2b the bounce characteristic of the second peak is clearly seen.  For Fig. 2c, recorded 
near the third focusing peak, it becomes difficult to see distinct bounces, although circular 
features are evident, with radii comparable to rc.  The bouncing-ball orbits begin to form the 
semi-classical equivalent of an edge state.  
With a strongly scattering tip (η ~ 1.0) the SPM images (Figs. 2d-f) have new and 
distinctly finer features, resulting from the interference of electron waves, some traveling 
along new pathways created by scattering at the tip. In the absence of the tip, electrons move 
from one QPC to the other by simultaneously traveling along multiple paths, which add up 
with a particular overall phase. As quantified below, a strongly scattering tip introduces new 
trajectories (and removes some of the old).  These new trajectories interfere with the original 
ones, and create interference fringes as the tip is scanned that can be seen in the experimental 
images. The images in Figs. 2d-f also show bouncing ball orbits similar to their counterparts 
in Figs. 2a-c.  The striking difference is the appearance of narrow fringe-like features.  In 
some locations, a noticeably periodic structure of fringes exists, shown in the blowups Figs. 
2g and 2h.  
We can understand both the classical and quantum behavior by using full, thermally 
averaged quantum simulations of an SPM image including tip scattering.  Figures 3a-c show 
simulations of an image on the first magnetic focusing peak for weak (η = 0.2), moderate 
(η = 0.6) and strong (η =1.2) tips.  These were obtained at 1.7 K using a thermal wavepacket 
calculation (26) with λF = 40 nm, and they include a random background potential from 
donor ions.  For a weak tip in Fig. 3a, the dark area of reduced transmission (ΔT < 0) 
corresponds to a classical cyclotron orbit.  For moderate and strong tips in Figs. 3b and 3c, 
quantum interference fringes created by tip scattering become visible.  The increase in fringe-
like structure as the tip strength is increased is in excellent qualitative agreement with the 
behavior of the experimental SPM images in Figs. 2a-c and Figs. 2d-f.   
To understand the source of contrast in the experimental images, it is very useful to 
compare quantum simulations with classical trajectories.10-11  Figure 3d joins quantum 
simulations from Fig. 3a (beige surface) with classical trajectories (red lines) computed 
without the tip present using ray tracing.  The background potential is shown in blue.  Areas 
with ΔT < 0 are eliminated for ease of comparison.  We see that the original trajectories line 
up with paths of decreased transmission, because the flow is blocked by the tip.  These 
simulations reveal the unusual, and yet very informative way that electron flow is encoded in 
the experimental images.   
The origin of fringing can be understood by a simple semiclassical argument pictured in 
Fig. 4a.  When a new trajectory with phase φ deflects from the tip and reaches the target 
QPC, it contributes an amplitude φiae  which interferes coherently with the background 
amplitude Aoe
iφo  from the nascent trajectories, such that the change in transmission depends 
on the phase difference ΔT ∝ cos(φ − φo) .  This is illustrated in Fig. 4a by the interference of 
a direct path between the QPCs along a single cyclotron orbit, with a path deflected by the 
tip, composed of two cyclotron orbit segments.  The phase of the deflected trajectory   φ = S  
is proportional to the classical action S accumulated along the trajectory.  When the tip is 
moved, S changes, and so does φ .  This leads to a simple equation for the fringe spacing d:  
   d =
 ,      (4) 
where θ is the angle by which the trajectory was deflected by the tip, shown in Fig. 4a. When 
the tip backscatters by θ = π , the fringe spacing is λF /2 as has been seen in previous 
experiments.10,11  When using a weak tip withη <1, the maximum possible scattering angle is 
θ < π , because it cannot backscatter.  This implies that the minimum fringe spacing is 
dmin > λF /2 .  For sufficiently weak tips η ~ 0.1, the fringe spacings are so large that they are 
indistinguishable from classical structures such as branches. 
A direct comparison of fringing between theory (Figs. 4b-d) and experiment (Figs. 4e-
g) shows remarkable agreement.  A series of quantum simulations in a small region (outlined 
in black in Fig. 3c) is compared with SPM images of a comparable area.  The simulations in 
Figs. 4b-d are for η = 0.2, 0.6 and 1.0 respectively, with the corresponding experimental 
images for these tip strengths displayed below in Figs. 4e-g.  The agreement is excellent.  The 
fringe spacing for a strong tip is comparable to λF , to be expected when the electrons are 
scattered by θ ~ π 3.  In this geometry the SPM acts as an interferometer that could be used 
to extract information from the fringes about the momentum and energy of the electrons.  
We have successfully visualized coherent electron transport patterns in a 2DEG 
magnetic focusing experiment and provided theoretical explanations of phenomena not 
previously predicted nor measured.  The complex interaction of focusing, branching, and tip 
scattering has been unraveled, revealing the true nature of electron pathways in a real device. 
References 
1. van Houten, H. et al. Coherent electron focusing with quantum point contacts in a two-
dimensional electron gas.  Phys. Rev. B 39, 8556 (1989). 
2. Sharvin, Y.V., & Fisher, L.M. Observation of focused electron beams in a metal. JETP 
Letters 1, 152 (1965). 
3. Tsoi, V. Focusing of electrons in a metal by a transverse magnetic field. JETP Letters 19, 
70 (1974). 
4. Rokhinson, L.P., Larkina, V., Lyanda-Geller, Y.B., Pfeiffer, L.N., & West, K.W. Spin 
Separation in Cyclotron Motion. Phys. Rev. Lett. 93, 146601 (2004). 
5. Topinka, M.A., Westervelt, R.M., & Heller, E.J.  Imaging Electron Flow Physics Today 
56, 47 (2003), and references therein. 
6. Eriksson, M.A. et al. Cryogenic scanning-probe characterization of semiconductor 
nanostructures. Appl. Phys. Lett. 69, 671-673 (1996). 
7. Tessmer, S.H., Glicofridis, P.I.,  Ashoori. R.C., Levitov, L.S., & Melloch, M.R. Subsurface 
charge accumulation imaging of a quantum Hall liquid. Nature 392, 51 (1998). 
8. Yacoby, A., Hess, H.F., Fulton, T.A., Pfeiffer, L.N., & West, K.W. Electrical imaging of 
the quantum Hall state. Solid State Commun. 111, 1 (1999). 
9. McCormick, K.L. et al. Scanned potential microscopy of edge and bulk currents in the 
quantum Hall regime. Phys. Rev. B 59, 4654-4657 (1999). 
10. Topinka, M.A., et al. Imaging Coherent Flow from a Quantum Point Contact. Science 
289, 2323-2326 (2000). 
11. Topinka, M.A., et al. Coherent Branched Flow in a Two-Dimensional Electron Gas.  
Nature 410, 183 (2001). 
12. Steele, G.A., Ashoori, R.C., Pfeiffer, L.N, & West, K.W. Imaging transport resonances in 
the quantum Hall effect. Phys. Rev. Lett. 95 136804 (2005). 
13. Crook, R., Smith, C.G., Simmons, M.Y., & Ritchie, D.A. Imaging cyclotron orbits and 
scattering sites in a high-mobility two-dimensional electron gas. Phys. Rev. B 62, 5174-5178 
(2000). 
14. Crook, R., et al. Erasable electrostatic lithography for quantum computers. Nature 424, 
751, (2003). 
15. Ihn, T., et al. Local spectroscopy of edge channels in the quantum Hall regime with local 
probe techniques. Physica E 13, 671-674 (2002). 
16. Kicin, S., et al. Local backscattering in the quantum Hall regime. Phys Rev. B 70 205302 
(2004). 
17. Aoki, N., da Cunha, C.R., Akis, R., Ferry, D.K., & Ochiai, Y. Imaging of integer 
quantum Hall edge state in a quantum point contact via scanning gate microscopy. Phys. Rev. 
B 72 155327 (2005). 
18. LeRoy, B.J., et al. Imaging Electron Interferometer. Phys. Rev. Lett. 94, 126801 (2005). 
19. Awschalom, D.D., Loss, D., & Samarth, N. Semiconductor Spintronics and Quantum 
Computation. Springer (2002). 
20. Saraga, D.S., Altshuler, B.L., Westervelt, R.M., & Loss, D. Coulomb Scattering in a 2D 
Interacting Electron Gas and Production of EPR Pairs. Phys. Rev. Lett. 92, 246803 (2004). 
21. Schliemann, J., Loss, D., & Westervelt, R.M.  Zitterbewegung of Electronic Wave 
Packets in III-V Zinc-Blende Semiconductor Quantum Wells.  Phys. Rev. Lett. 94, 206801 
(2005). 
22. Sohn, L.L., Kouwenhoven, L.P., & Schon, G. Mesoscopic Electron Transport. Kluwer 
Academic Publishers (1997), and references therein. 
23. van Wees, B.J., et al. Quantized conductance of point contacts in a two-dimensional 
electron gas. Phys. Rev. Lett. 60, 848 (1988). 
24. Shepard, K.L., Roukes, M.L., & Van der Gaag, B.P. Direct measurement of the 
transmission matrix of a mesoscopic conductor. Phys. Rev. Lett. 68, 2660 (1992). 
25. Crommie, M.F., Lutz, & C.P., Eigler, D.M. Confinement of Electrons to Quantum 
Corrals on a Metal Surface. Science 262, 218 (1993). 
26. Kikkawa, J.M., & Awschalom, D.D. Lateral drag of spin coherence in gallium arsenide.  
Nature 397, 139 (1999). 
27. Katine, J.A., et al., Point Contact Conductance of an Open Resonator. Phys. Rev. Lett. 79, 
4806 (1997). 
28. Aidala, K.E., Parrott, R.E., Heller, E.J., & Westervelt, R.M. Imaging electrons in a 
magnetic field. Physica E 34, 409 (2006). 
29. Heller, E.J. et al. Thermal averages in a quantum point contact with a single coherent 
wave packet. Nano Letters 5, 1285 (2005). 
Author Contributions: Katherine E. Aidala conducted the experiments with R.M. Westervelt; Robert E. Parrot 
and Tobias Kramer did classical and quantum simulations of electron flow with E.J. Heller, and M.P. Hanson 
grew the semiconductor heterostructure with A.C. Gossard. 
Acknowledgements: This work has been performed with support at Harvard University from the ARO, the 
NSF-funded Nanoscale Science and Engineering Center (NSEC), and the DFG (Emmy-Noether program).  
Work at Santa Barbara has been supported in part by the Institute for Quantum Engineering, Science and 
Technology (iQUEST).  We would also like to thank the National Nanotechnology Infrastructure Network 
(NNIN) and the Harvard CrimsonGrid for computing resources. 
Correspondence and requests for materials should be addressed to R.M.W. (e-mail: 
westervelt@deas.harvard.edu). 
Figure 1 (a) Schematic showing a conducting scanning probe microscope (SPM) tip 
scanned above the surface of a device. (b) Scanning electron micrograph of the 
magnetic focusing device; the inset shows an SPM image of the electron flow at zero 
magnetic field, displaying branches in electron flow. (c) Quantum simulation showing 
the scattering of an incoming plane wave by an SPM tip, with 2DEG Fermi energy EF 
= 13 meV, and a Gaussian tip potential with height 0.2 EF and width 50 nm. (d-f) 
Classical simulations of magnetic focusing in a magnetic field: (d) in a flat potential, 
showing circular cyclotron orbits, (e) with small-angle scattering, showing added 
branches, and (f) in a flat potential with an SPM tip. 
Figure 2  Experimental SPM images of magnetic focusing in a 2DEG at 4.2 K 
recorded near the first three magnetic focusing peaks: (a-c) Weakly focusing 
(η ≈ 0.5) tip 90 nm above the 2DEG recorded at B = 100 mT, B = 174 mT, and B = 
262 mT with cyclotron radii rc = 970 nm, rc = 560 nm and rc = 370 nm, respectively.  
The left QPC is on the first conductance plateau, the right is on the third. (d-f) 
Strongly focusing (η ≈1.0) tip 60 nm above the 2DEG recorded at B = 74 mT, B = 
169 mT, and B = 254 mT with cyclotron radii rc = 1310 nm, rc = 580 nm and rc = 380 
nm, respectively.  Both QPCs are on the second conductance plateau.  The color 
scale shows the change in transmission between the QPCs induced by the tip.  (g,h) 
Closeup surface plots in the yellow rectangles of (d,f) that show the regularity and 
consistency of the quantum fringe structure. 
Figure 3  Quantum simulations of SPM images of the first magnetic focusing peak (B 
= 77 mT) for λF = 40 nm at 1.7 K, showing the change in transmission ΔT between 
QPCs as the tip is scanned above, including small-angle scattering.  (a) For a weak 
tip (η = 0.2) that scatters into small angles, the dark area of reduced transmission 
(ΔT < 0) shows a classical cyclotron orbit.  For (b) moderate (η = 0.6) and (c) strong 
(η = 1.2) tips, quantum interference fringes become visible.  (d) Correspondence 
between the simulated SPM image (beige surface) from (a) and ray tracing 
calculations of the originally transmitted electron trajectories (red lines) before the tip 
was present; regions with ΔT < 0 are omitted for ease of comparison. The blue 
surface is the smoothly disordered background potential.  The dark areas with 
reduced transmission (ΔT < 0) line up very well with the original classical trajectories.  
For stronger tips, imaging of the original classical trajectories becomes more difficult, 
but fringing reveals regions of high coherent flux of electron waves.  
Figure 4  Direct comparison of interference fringes between experiment and theory 
for different tip strengths. (a) How interference occurs between a path deflected by 
the SPM tip with a direct path between the two QPCs.  (b-d) Quantum simulations of 
interference fringes at 1.7 K and B = 77 mT for weak (η = 0.2), moderate (η = 0.6) 
and strong (η = 1.0) tip strengths; the Fermi wavelength is λF = 40 nm.  The panels 
are located in the black box in Fig. 3c.  (e-g) SPM images at B = 173 T showing 
fringes that appear and move closer as the tip strength η  increases, in good 
agreement with the simulations.  The images have dimensions 600 x 450 nm2 and 
are located 750 nm above and 500 nm to the right of the midpoint between the two 
QPCs.

  The magnetic focusing of electrons has proven its utility in fundamental
studies of electron transport. Here we report the direct imaging of magnetic
focusing of electron waves, specifically in a two-dimensional electron gas
(2DEG). We see the semicircular trajectories of electrons as they bounce along
a boundary in the 2DEG, as well as fringes showing the coherent nature of the
electron waves. Imaging flow in open systems is made possible by a cooled
scanning probe microscope. Remarkable agreement between experiment and theory
demonstrates our ability to see these trajectories and to use this system as an
interferometer. We image branched electron flow as well as the interference of
electron waves. This technique can visualize the motion of electron waves
between two points in an open system, providing a straightforward way to study
systems that may be useful for quantum information processing and spintronics.

Microsoft Word - Aidala_focusing_arXiv.doc
Imaging Magnetic Focusing of Coherent  
Electron Waves 
Katherine E. Aidala,1* Robert E. Parrott,2 Tobias Kramer,2 E.J. Heller,2,3  
R.M. Westervelt,1,2 M.P. Hanson4 and A.C. Gossard4 
1Div. of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 
2Dept. of Physics, Harvard University, Cambridge, MA 02138 
3Dept. of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 
4Dept. of Materials Science, University of California, Santa Barbara, CA 93106 
* Present Address: Dept. of Physics, Mount Holyoke College, South Hadley, MA 01075 
The magnetic focusing of electrons has proven its utility in fundamental studies of 
electron transport.1-4  Here we report the direct imaging of magnetic focusing of 
electron waves, specifically in a two-dimensional electron gas (2DEG).  We see the 
semicircular trajectories of electrons as they bounce along a boundary in the 2DEG, as 
well as fringes showing the coherent nature of the electron waves.  Imaging flow in open 
systems is made possible by a cooled scanning probe microscope.5-17  Remarkable 
agreement between experiment and theory demonstrates our ability to see these 
trajectories and to use this system as an interferometer.  We image branched electron 
flow11 as well as the interference of electron waves.10,11,18  This technique can visualize 
the motion of electron waves between two points in an open system, providing a 
straightforward way to study systems that may be useful for quantum information 
processing and spintronics.19-21 
For nanoscale devices at low temperatures, both the particle and wave aspects of 
electron motion are important.  Electrons can travel ballistically through an unconfined two-
dimensional electron gas (2DEG) while their quantum phase remains coherent.  The 
development of unconfined or open devices in this regime1,22-26 clears the way for new 
applications in electron interferometry, spintronics, and quantum information processing.18-
21,27  The most basic question for open systems is: "What is the pattern of flow for electron 
waves?"  Imaging flow is difficult, because the electrons are often buried inside a 
heterostructure, and because low temperatures and strong magnetic fields are needed to 
observe quantum phenomena.   
A cooled scanning probe microscope (SPM) provides a way to image flow: a 
conducting SPM tip above the sample capacitively couples to the 2DEG below.5-17  Pathways 
for electron waves emerging from a quantum point contact (QPC) were revealed10 and found 
to have a dramatic branched form.11  The branching was shown to be the real-space 
consequence of small-angle scattering by charged donor atoms - the well known mechanism 
that limits the electron mobility.  A charged tip was able to image this flow, because it 
backscattered electrons; some of them follow a time-reversed path back through the QPC, 
measurably reducing its conductance.  Images of flow were obtained by displaying the QPC 
conductance as the tip was scanned in a plane above the sample.10,11  However, this 
backscattering imaging technique cannot be used in a perpendicular magnetic field, because 
the electrons follow curved paths and no longer reverse.  Magnetic fields are important for 
spintronics and quantum information processing,19-21 so it is important to understand electron 
flow with a field present.   
In this letter, we demonstrate a new way to image the flow of electron waves between 
two points in an open system using a cooled SPM, and we use this approach to image 
magnetic focusing of electron waves in a 2DEG.  Magnetic focusing plays an important role 
in the study of ballistic transport.1-4  Images were obtained by using an electron lens formed 
in the 2DEG beneath a charged SPM tip to redirect flow, by throwing a shadow downstream.  
We obtain clear pictures of semicircular bouncing-ball patterns of particle flow that cause 
magnetic focusing.  Branching of the flow is visible.  In addition, we see fringes created by 
the interference of electron waves bouncing off the tip with those travelling directly between 
the two QPCs.  In this way, the cooled SPM acts as a new type of electron interferometer for 
solids.  The observed bouncing-ball patterns of flow and interference fringes are in excellent 
agreement with full quantum simulations that include the tip-induced electron lens and small-
angle scattering, demonstrating that this new imaging technique accurately views the flow of 
electron waves through the device. 
Figure 1 illustrates the device geometry and the imaging technique.  Two QPCs are 
formed in a 2DEG by surface gates (Fig. 1a); a scanning electron micrograph of the device is 
shown in Fig. 1b.  The separation between the QPC centers is L = 2.7 μm.  A 2DEG with 
density 3.8x1011 cm−2 and mobility 500,000 cm2/Vs is located 47 nm beneath the surface of a 
GaAs/AlGaAs heterostructure with the following layers: 5 nm GaAs cap, 20 nm Al0.3Ga0.7As 
barrier, Si delta-doping, then a 22 nm Al0.3Ga0.7As barrier next to the 2DEG in GaAs.  Metal 
gates that define the QPCs were fabricated using e-beam lithography.  The device is mounted 
in the SPM, inside a superconducting solenoid, and cooled to 4.2 K.  A computer controls the 
SPM and records the images.  
Magnetic focusing (Fig. 1d) occurs because electrons leaving one QPC over a range of 
angles circle around and rejoin at the second QPC, when the spacing L is close to the 
diameter of a cyclotron orbit.  For GaAs the electron cyclotron orbit is circular with radius: 
     rc = kF eB ,       (1) 
where e is the electron charge, B is the perpendicular magnetic field, and kF  is the Fermi 
wavevector.  As B is increased, the first focusing peak occurs when L = 2rc .  Additional 
peaks occur at higher fields when L = 2nrc  is an integer multiple of the cyclotron diameter at 
fields: 
     Bn = 2n kF eL .      (2) 
The shape, clarity and spacing of magnetic focusing peaks provide information about ballistic 
flow in the sample material.  The effects of small-angle scattering are shown in the simulation 
of Fig. 1e (and in supplementary material):  magnetic focusing still occurs, but the flow now 
contains branches11 similar to flow in the SPM image in Fig. 1b for B = 0.   
To image electron flow from one point to another, a small movable electron lens is 
created by creating a dip in electron density immediately below the charged SPM tip.  The 
lens deflects electrons (Fig. 1c), throwing a V-shaped shadow downstream.  An image of 
electron flow is obtained by displaying the transmission T between the QPCs as the tip is 
scanned across a plane 10 nm above the surface.  T is measured by recording the voltage 
across the second QPC as a known current is passed through the first.  When electrons are 
ballistically injected into the second QPC from the first, a voltage develops that drives a 
current in the opposite direction to cancel the influx of electrons; this voltage is proportional 
to T. 
Simulations of electron flow that show how the imaging technique works are presented 
in Figs. 1c-f.  A Gaussian ϕ tip = Vo exp − r − rtip( )
2a2( ) is used to model the tip potential in 
the 2DEG, where rtip is the tip position, a is the width, and Vo is the height.  For this paper Vo 
> 0, and the tip creates a dip in electron density.  The relative strength of the tip potential is 
   η =
,        (3) 
where EF is the Fermi energy.  For η <1, the electron gas is partially depleted, and an 
imperfect diverging lens is created with a focal length determined by η.  When η ≥1, the 
2DEG is fully depleted, and electrons can backscatter.  Simulations (Fig. 1c) at B = 0 show 
how a weak lens (η = 0.2) creates a V-shaped shadow downstream by forcing electrons to the 
sides, where they form two caustics in flow along the legs of the V.  Figure 1f shows how we 
image magnetic focusing: when the shadow cast by the lens beneath the tip hits the second 
QPC, T is reduced.  The amount of reduction ΔT is proportional to the original flux, before 
the tip was introduced.  By displaying ΔT vs. tip position rtip as the tip is raster scanned in a 
plane above, an image of the original flow is created.28 
Experimental images of magnetic focusing near the first, second, and third peaks are 
presented in Figs. 2a-c for a weak tip (η ≈ 0.5), and in Figs. 2d-e for a strong tip (η ≈1.0).  
The bouncing semicircular cyclotron orbits imaged here are an experimental visualization of 
the origins of magnetic focusing.  Figure 2a shows a single, semicircular crescent 
characteristic of the first peak.  Branches in flow from small-angle scattering are also visible.  
For Fig. 2b the bounce characteristic of the second peak is clearly seen.  For Fig. 2c, recorded 
near the third focusing peak, it becomes difficult to see distinct bounces, although circular 
features are evident, with radii comparable to rc.  The bouncing-ball orbits begin to form the 
semi-classical equivalent of an edge state.  
With a strongly scattering tip (η ~ 1.0) the SPM images (Figs. 2d-f) have new and 
distinctly finer features, resulting from the interference of electron waves, some traveling 
along new pathways created by scattering at the tip. In the absence of the tip, electrons move 
from one QPC to the other by simultaneously traveling along multiple paths, which add up 
with a particular overall phase. As quantified below, a strongly scattering tip introduces new 
trajectories (and removes some of the old).  These new trajectories interfere with the original 
ones, and create interference fringes as the tip is scanned that can be seen in the experimental 
images. The images in Figs. 2d-f also show bouncing ball orbits similar to their counterparts 
in Figs. 2a-c.  The striking difference is the appearance of narrow fringe-like features.  In 
some locations, a noticeably periodic structure of fringes exists, shown in the blowups Figs. 
2g and 2h.  
We can understand both the classical and quantum behavior by using full, thermally 
averaged quantum simulations of an SPM image including tip scattering.  Figures 3a-c show 
simulations of an image on the first magnetic focusing peak for weak (η = 0.2), moderate 
(η = 0.6) and strong (η =1.2) tips.  These were obtained at 1.7 K using a thermal wavepacket 
calculation (26) with λF = 40 nm, and they include a random background potential from 
donor ions.  For a weak tip in Fig. 3a, the dark area of reduced transmission (ΔT < 0) 
corresponds to a classical cyclotron orbit.  For moderate and strong tips in Figs. 3b and 3c, 
quantum interference fringes created by tip scattering become visible.  The increase in fringe-
like structure as the tip strength is increased is in excellent qualitative agreement with the 
behavior of the experimental SPM images in Figs. 2a-c and Figs. 2d-f.   
To understand the source of contrast in the experimental images, it is very useful to 
compare quantum simulations with classical trajectories.10-11  Figure 3d joins quantum 
simulations from Fig. 3a (beige surface) with classical trajectories (red lines) computed 
without the tip present using ray tracing.  The background potential is shown in blue.  Areas 
with ΔT < 0 are eliminated for ease of comparison.  We see that the original trajectories line 
up with paths of decreased transmission, because the flow is blocked by the tip.  These 
simulations reveal the unusual, and yet very informative way that electron flow is encoded in 
the experimental images.   
The origin of fringing can be understood by a simple semiclassical argument pictured in 
Fig. 4a.  When a new trajectory with phase φ deflects from the tip and reaches the target 
QPC, it contributes an amplitude φiae  which interferes coherently with the background 
amplitude Aoe
iφo  from the nascent trajectories, such that the change in transmission depends 
on the phase difference ΔT ∝ cos(φ − φo) .  This is illustrated in Fig. 4a by the interference of 
a direct path between the QPCs along a single cyclotron orbit, with a path deflected by the 
tip, composed of two cyclotron orbit segments.  The phase of the deflected trajectory   φ = S  
is proportional to the classical action S accumulated along the trajectory.  When the tip is 
moved, S changes, and so does φ .  This leads to a simple equation for the fringe spacing d:  
   d =
 ,      (4) 
where θ is the angle by which the trajectory was deflected by the tip, shown in Fig. 4a. When 
the tip backscatters by θ = π , the fringe spacing is λF /2 as has been seen in previous 
experiments.10,11  When using a weak tip withη <1, the maximum possible scattering angle is 
θ < π , because it cannot backscatter.  This implies that the minimum fringe spacing is 
dmin > λF /2 .  For sufficiently weak tips η ~ 0.1, the fringe spacings are so large that they are 
indistinguishable from classical structures such as branches. 
A direct comparison of fringing between theory (Figs. 4b-d) and experiment (Figs. 4e-
g) shows remarkable agreement.  A series of quantum simulations in a small region (outlined 
in black in Fig. 3c) is compared with SPM images of a comparable area.  The simulations in 
Figs. 4b-d are for η = 0.2, 0.6 and 1.0 respectively, with the corresponding experimental 
images for these tip strengths displayed below in Figs. 4e-g.  The agreement is excellent.  The 
fringe spacing for a strong tip is comparable to λF , to be expected when the electrons are 
scattered by θ ~ π 3.  In this geometry the SPM acts as an interferometer that could be used 
to extract information from the fringes about the momentum and energy of the electrons.  
We have successfully visualized coherent electron transport patterns in a 2DEG 
magnetic focusing experiment and provided theoretical explanations of phenomena not 
previously predicted nor measured.  The complex interaction of focusing, branching, and tip 
scattering has been unraveled, revealing the true nature of electron pathways in a real device. 
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Author Contributions: Katherine E. Aidala conducted the experiments with R.M. Westervelt; Robert E. Parrot 
and Tobias Kramer did classical and quantum simulations of electron flow with E.J. Heller, and M.P. Hanson 
grew the semiconductor heterostructure with A.C. Gossard. 
Acknowledgements: This work has been performed with support at Harvard University from the ARO, the 
NSF-funded Nanoscale Science and Engineering Center (NSEC), and the DFG (Emmy-Noether program).  
Work at Santa Barbara has been supported in part by the Institute for Quantum Engineering, Science and 
Technology (iQUEST).  We would also like to thank the National Nanotechnology Infrastructure Network 
(NNIN) and the Harvard CrimsonGrid for computing resources. 
Correspondence and requests for materials should be addressed to R.M.W. (e-mail: 
westervelt@deas.harvard.edu). 
Figure 1 (a) Schematic showing a conducting scanning probe microscope (SPM) tip 
scanned above the surface of a device. (b) Scanning electron micrograph of the 
magnetic focusing device; the inset shows an SPM image of the electron flow at zero 
magnetic field, displaying branches in electron flow. (c) Quantum simulation showing 
the scattering of an incoming plane wave by an SPM tip, with 2DEG Fermi energy EF 
= 13 meV, and a Gaussian tip potential with height 0.2 EF and width 50 nm. (d-f) 
Classical simulations of magnetic focusing in a magnetic field: (d) in a flat potential, 
showing circular cyclotron orbits, (e) with small-angle scattering, showing added 
branches, and (f) in a flat potential with an SPM tip. 
Figure 2  Experimental SPM images of magnetic focusing in a 2DEG at 4.2 K 
recorded near the first three magnetic focusing peaks: (a-c) Weakly focusing 
(η ≈ 0.5) tip 90 nm above the 2DEG recorded at B = 100 mT, B = 174 mT, and B = 
262 mT with cyclotron radii rc = 970 nm, rc = 560 nm and rc = 370 nm, respectively.  
The left QPC is on the first conductance plateau, the right is on the third. (d-f) 
Strongly focusing (η ≈1.0) tip 60 nm above the 2DEG recorded at B = 74 mT, B = 
169 mT, and B = 254 mT with cyclotron radii rc = 1310 nm, rc = 580 nm and rc = 380 
nm, respectively.  Both QPCs are on the second conductance plateau.  The color 
scale shows the change in transmission between the QPCs induced by the tip.  (g,h) 
Closeup surface plots in the yellow rectangles of (d,f) that show the regularity and 
consistency of the quantum fringe structure. 
Figure 3  Quantum simulations of SPM images of the first magnetic focusing peak (B 
= 77 mT) for λF = 40 nm at 1.7 K, showing the change in transmission ΔT between 
QPCs as the tip is scanned above, including small-angle scattering.  (a) For a weak 
tip (η = 0.2) that scatters into small angles, the dark area of reduced transmission 
(ΔT < 0) shows a classical cyclotron orbit.  For (b) moderate (η = 0.6) and (c) strong 
(η = 1.2) tips, quantum interference fringes become visible.  (d) Correspondence 
between the simulated SPM image (beige surface) from (a) and ray tracing 
calculations of the originally transmitted electron trajectories (red lines) before the tip 
was present; regions with ΔT < 0 are omitted for ease of comparison. The blue 
surface is the smoothly disordered background potential.  The dark areas with 
reduced transmission (ΔT < 0) line up very well with the original classical trajectories.  
For stronger tips, imaging of the original classical trajectories becomes more difficult, 
but fringing reveals regions of high coherent flux of electron waves.  
Figure 4  Direct comparison of interference fringes between experiment and theory 
for different tip strengths. (a) How interference occurs between a path deflected by 
the SPM tip with a direct path between the two QPCs.  (b-d) Quantum simulations of 
interference fringes at 1.7 K and B = 77 mT for weak (η = 0.2), moderate (η = 0.6) 
and strong (η = 1.0) tip strengths; the Fermi wavelength is λF = 40 nm.  The panels 
are located in the black box in Fig. 3c.  (e-g) SPM images at B = 173 T showing 
fringes that appear and move closer as the tip strength η  increases, in good 
agreement with the simulations.  The images have dimensions 600 x 450 nm2 and 
are located 750 nm above and 500 nm to the right of the midpoint between the two 
QPCs.
