DERAEDT, H

MICHIELSEN, K

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ELECTRON FOCUSING BY MULTIPLE-QUANTUM-POINT CONTACTS
MICHIELSEN, K; de Raedt, Hans
Published in:
Journal of Physics-Condensed Matter
DOI:
10.1088/0953-8984/4/34/011
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1992
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MICHIELSEN, K., & DERAEDT, H. (1992). ELECTRON FOCUSING BY MULTIPLE-QUANTUM-POINT
CONTACTS. Journal of Physics-Condensed Matter, 4(34), 7121-7126. DOI: 10.1088/0953-8984/4/34/011
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J.  Phys.: Condens. Matter 4 (1992) 7121-7126 Printed in the UK 
Electron focusing by multiple-quantum-point contacts 
K Michielsen and H De Raedt 
Institute for Theoretical Physics, University of Groningen, Nijenborgh 4, NL-9747 AG 
Groningen, The Netherlands 
Received 8 May 1992, in Bnal tom 1 July 1992 
Abstract. A device for focusing ballistic eleitrons is pmposed. It consists of an array 
of quantum point contacLs and a potential barrier drawn up by a gate. Computer 
simulalion analysis of this device shows that it can focus electrons in a cone of ZOO,  
with a throughput of z 0.6%. 
Recent progress in manufacturing nanometre structures in two-dimensional electron 
gas (ZDEG) systems have made it possible to perform ‘electron-optics’ experiments in 
solid-state devices [l]. In an ideal device the motion of the electrons is not affected 
by interactions with impurities, phonons etc, Le. the electrons travel ballistically, just 
as they would do in ultra-high vacuum. In real devices, typical distances for ballistic 
motion can be as large as 250A,, A, being the Fermi wavelength of the electrons 
[2]. If the dimension of the point contacts becomes comparable to A,, a classical 
‘billiard-ball’, treatment of the motion of the electrons is no longer valid. TO explore 
the properties of electron transport through quantum point contacts a proper, wave- 
mechanical, description is required. 
The ability to focus a beam of light by means of lenses is very important for a 
wide variety of applications and this holds for the focusing of electrons as well [3]. 
In solid-state devices, the electrons emerging from the quantum point contact can be 
directed to a small area by applying a magnetic field [2], by putting a gate behind the 
point contact [4,5] or by refraction of the electrons by a convex lens (i.e. a region in 
which the electrons have a longer wavelength) [6]. 
In this paper we propose a device which focuses the electrons in a cone of 20’ 
or less, is controlled by a gate and has a high throughput, thereby alleviating most of 
the shortcomings [3] of the above-mentioncd focusing techniques. The layout of the 
device is depicted in figure 1. Several quantum point contacts are put in parallel to 
form a grating. The grating is followed by a tunnelling barrier (drawn up by a gate). 
The height of the barrier is to be chosen such that it is slightly below the Fermi 
energy E,. ’RI be concrete we consider a ‘typical’ grating of thickness A,, period 
a + b = 2AF and linewidth b = A,. 
To investigate the focusing capability of the device we envisage the following 
thought experiment. Imagine a wave, with a well-defined direction and energy EF, 
impinging on the gratingtbarrier system. A detector is placed far away from the 
latter. The (normalized) transmitted intensity P(0)  recorded by this detector is 
plotted as a function of the angle (e)  with respect to the normal to the exit plane 
of the device. ’RI characterize the collimation we calculate A0 3 IO1 - @,I where 
0953-8984/9U341121+06$04.50 @ 1992 IOP Publishing Ltd 7121 
7122 K Michielsen and H De Raedt 
Figore 1. Electron foeusing device consisting of quantum paint contacts, put in parallel. 
followed by a tunnelling barrier drawn up by a gale. Also shown is the potential energy 
diagram along the (2, y = 0) line. 
0l,2 are determined such that the normalized angular distribution P(O1,,) = l / e .  
We define the angular spread 0, E A0/2, because due to the reflection symmetry 
of the device, P ( 0 )  for a positive angle of incidence I) equals P(-0)  for -I). The 
angular distribution P(f3) thus obtained provides a quantitative characterization of 
the focusing capability of the device. 
To implement this thought experiment, we have simulated the motion of electron 
waves in the proposed device. The size of the complete system consisting of the 
emittor, grating+barrier and collector was taken to be lOOX, x 51X,, i.e. comparable 
to the mean free path of the electrons in ZDEG systems. Incident waves were chosen to 
be Gaussians of width 6X, x 6X,, the largest wave packets that can be accommodated 
by the device. The equation of motion of the electrons, i.e. the time-dependent 
Schrodinger equation (TDSE), was solved by an accurate and unconditionally stable 
numerical technique [7]. The TDSE approach is flexible in the sense that it can handle 
arbitraly geometries and potentials. The angular distribution of the transmitted wave 
was calculated by the method described in [8]. A typical simulation takes about three 
hours of CPU time on a CRAY 2-YMP. 
A single +wide quantum point contact is a strong scatterer. This is illustrated 
by figure 2 which shows the angular distribution for the case of normal incidence 
(I) = On). The angular spread 8. FZ 35O. In the ZDEG system the electrons impinge 
on the entrance plane of the point contact from all possible directions. Hence the 
total angular distribution is an incoherent superposition of distributions for each 
angle of incidence. Calculations [4,5] for I) f Oo show that for the single point 
contact Os does not significantly depend on I), although the shape and the position 
of the maximum of the angular profile does. Therefore the angular spread of the 
total angular distribution is larger (0, FZ 60') than the angular spread for normal 
incidence [4,5]. By putting quantum point contacts in parallel the waves leaving each 
opening interfere so as to change the angular distribution. This is demonstrated in 
Electron f0cUs;lg by multiple-quonrum-point contacfs 7123 
i o , ,  I , ,  1 ,  ~, I 
06 - 
m 
L1. 
-
! n I /  n 
0 8 1  m L1. ,,/\ .: 
, , .~ .........,..... .................... ...................  04 
Figure 2. Angular distribution P(0)  Venus angle 0 for a wave impinging on a single 
XF-wide constriclion. The direction of the incident wave is along the z-axis ($h = 0"). 
The angular spread 0. sz 35O. The dashed line mrresponds to l / e .  
O . * t  , Ah A\, jl , , , 1~ 
0 
-100 -80 -Ea -40 -a 0 20 4a 60 60 100' 
0 
Feure 3. Angular dislribulion P(!) ve,nus angle 8 for a wave transmilted by the grating 
(without barrier, V = 0). The angle of inmming wave with respecl to the z-axis is 
$h = 20'. The dashed line corresponds 10 i/e. 
figure 3, where we have chosen an angle of incidence $ = 20°. Instead of one broad 
distribution with some additional structure, the angular distribution now consists of 
several, rather narrow, peaks. As expected the current through the grating is, in 
general, larger than through one of its openings. The grating serves to compress 
the electron wave but does not, by itself, focus the electron beam in a particular 
(in our case, normal to the grating) direction. The expression obtained by applying 
conservation of transvese momentum in the presence of a periodic potential, i.e. 
sin(Bpat) = sin(Binmmi,,,.) + . / ( a  + b) where n = 0, &l, . . . , correctly predicB the 
positions of all peaks if we take into account the reflections caused by the simulation 
box boundaries. For instance according to this formula, there should be a peak at 
B = 57O but not at B = -57' as indicated in figure 3. This is entirely due to the 
fact that our simulation box is finite. Those components of the outgoing wave that 
contribute to the signal at B = 57" have already been reflected by the simulation 
box boundary before the analysis of the wave packet was carried out. The signals 
at B = -20' and B = 41°, the latter being too weak to show up in figure 3, are 
ako due to reflections by the simulation box boundary. The standard expression for 
the intensity profile of a grating [9] does not predict the occurrence of the peak at 
0 = -41". This is to be expected because the width of each opening is of the order 
of the wavelength 191. 
To eliminate all non-normal components of the outgoing wave as much as possible, 
7124 
a potential barrier is put behind the constriction(s). The height of the barrier should 
be chosen such that it is slightly below the Fermi energy Ep and the width of the 
barrier should be as large as possible (in theory) [4]. Then the potential barrier will 
reduce the intensity of all transmitted waves with non-zero transverse momentum. In 
practice the width of the barrier is limited by the size of the device, essentially the 
distance for ballistic motion. For our calculations presented in this paper we have 
taken W = 3X, and V = 0.98Ep. The precise values of these parameters are 
not important as long as they satisfy the qualitative criteria mentioned above. For a 
device consisting of one constriction only the angular spread, for a wave incident in 
the normal direction, is reduced considerably (from 8, c 35O to 8, c 10") as shown 
in figure 4. The transmission coefficient T c 0.006 is much larger than the one of a 
device proposed earlier [4]. 
K Michielsen and H De Raedt 
Figurt 4. Angular distribution P(6) versus angle 0 for a wave impinging on a single 
XF-wide constriction tollowed by the ptential described in the text. The direction of 
the incident wave is along the z-axis ($ = On). The angular spread 0, s loo.  The 
dashed line corresponds to 1 f e. 
0 8  - 
0.6 - - 
m 
a 
-
-30 -20 -10 0 IO 20 30 e 
... ... 
Figure 5. Full angular distribution for the grating+bamer system, obtained by summing 
over 19 different inmming waves: $ = O', 39.5 ' .  &So, M O ,  57.5'. *go, +16', 
AZOo, f30'. f50'. Essentially all intensity is mncentrated in a mne of 20'. The 
dashed line corresponds to l /e.  
The efficiency of the focusing device can be further enhanced by putting point 
contacts in parallel, as shown in figure 1. The transmission coefficient for normal 
incidence T FJ 0.047. The full angular distribution of this device is obtained by 
integrating over all incident waves. In practice we approximate this integral by a 
Electron focusing by muhiple-quantum-point contacts 7125 
sum over a (small) number of incoming waves. The angular distribution, obtained by 
summing over 19 different angles of incidence, is depicted in figure 5. The wiggles in 
P(6)  are a direct consequence of replacing the integral by a finite sum. Increasing 
the number of incident waves yields an intensity profile which is smoother but has 
the same shape. The incoming waves effectively 'hit' about six constrictions. The 
throughput is approximately 0.6%. It is clear that this device is strongly focusing. 
Unlike for the single-constriction set-up, essentially all transmitted intensity is con- 
centrated in the interval [-loo, IOo]. As for all devices of this kind, the focusing 
effect can be further enhanced by applying an electric field, for instance by changing 
the Fermi wavelength in the collector of the device [51. 
Some remarks about the 'ideality' of the device are in order. In our simulations 
we assumed that the geometry of the device is perfect, made of ideal point contacts. 
One advantage of having a number of quantum point contacts in parallel may be that 
conductance fluctuations and localization effem caused by impurities are avcraged out 
[IO]. Variation of the width and position of the quantum point contacts reduces the 
transmission and does not alter the collimation. We now argue that in all respects 
(excluding imperfections, phonons etc), what we have simulated is a 'worst case' 
situation. In practice the corners of the point contact will be rounded. This rounding 
leads to an extra collimation of the wave [ll-131. To investigate the influence of 
smoothening the entrance and exit region of the point contacts, we have attached 
to the grating horn-like structures but found no noticable effects. This is in concert 
with our earlier findings [SI that for constrictions of minimal width ES A,, rather 
long horns (> 5AF) are required to improve the focusing. Unlike in the device we 
have simulated, the potential inside a constriction or grating in a real device will 
not be zero. It has been suggested [ll] that a potential inside a constriction can 
also provide a mechanism for focusing the wave. 'Able 1 demonstrates that this is 
not the case. The diffraction pattern is very similar to that of a constriction without 
internal potential. Physically, in the absence of a potential behind the constriction, 
the collimation is determined by the scattering from the exit plane of the constriction, 
i.e. by a slit. It follows that the two effecB mentioned do not significantly improve the 
focusing. However, they may have inauence to the throughput. A potential inside the 
constriction will reduce the transmission through it, whereas rounding the entrance 
and exit regions of the grating will increase the transmission coefficient 
Table 1. Influence on the angular spread 8, of the presence of a constant potential V 
inside a single Xp-wide conslriaion. The angle of incidence $ = 0". 
0.00 3 4 Y  4' 
0.75 34Of 4 O  
0.98 33'* 4' 
1.00 3 4 9  4' 
Acknowledgments 
We are grateful to N Garcia for a critical reading of the manuscript and for many 
useful suggestions. We have profited from stimulating discussions with T Klapwijk. 
7126 
This work is partially supported by the Stichting voor Fundamenteel Onderzoek 
der Materie (FOM), which is financially supported by the Nederlandse Organisatie 
voor Wetenschappelijk Onderzoek (NWO), a supercomputer grant of the NCF (The 
Netherlands), and the EEC Science Plan wntract SC1-0395-C (MB). 
K MichieLEen and H De Raedt 
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ELECTRON FOCUSING BY MULTIPLE-QUANTUM-POINT CONTACTS

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ELECTRON FOCUSING BY MULTIPLE-QUANTUM-POINT CONTACTS

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