# Integrated anti-electronics for positron annihilation spectroscopy

Francesco Guatieri<EMAIL_ADDRESS>Forschungs-Neutronenquelle
Heinz Maier-Leibnitz (FRM II) Technische Universität München Lichtenbergstraße
1 85748 Garching, Germany

###### Abstract

Imaging the features of a sample using Positron Annihilation Spectroscopy
(PAS) is currently achieved by rastering, i.e. by scanning the sample surface
with a sharply focused positron beam. However, a beam of arbitrary shape
(sculpted beam) would allow the application of more versatile single-pixel
imaging (SPI) techniques. I introduce the design of a microelectronic device
employing a 2D array of Zener diodes as an active positron moderator, capable
of sculpting positron beams with $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$
resolution. The re-emitted positrons are accelerated towards the sample
through a miniaturised electrostatic lens system and reaching
$100\text{\,}\mathrm{nm}$ resolution. The fast switch-on
($90\text{\,}\mathrm{p}\mathrm{s}$) and switch-off
($250\text{\,}\mathrm{p}\mathrm{s}$) time of the device would enable state-of-
the-art positron annihilation lifetime spectroscopy (PALS) and PAS imaging
with high spatial and temporal resolution.

Positron, Microelectronics, Imaging

††preprint: APS/123-QED

##  I Introduction

Slow-positron spectroscopy is at the forefront of material science, allowing,
among others, the investigation of electronic structure [1, 2], fatigue in
metals [3, 4], the presence of defects [5, 6], and the size and distribution
of pores [7]. These applications would benefit from high-resolution three-
dimensional spectroscopic reconstruction of the sample features. The depth of
the analysed feature can be determined up to a few nanometers by the well-
characterised dependency of the positron implantation depth on the positron
energy [8], a resolution that is rarely achieved with other crystallographic
techniques. However, the transversal resolution of is typically much worse,
since it is intrinsically limited by the diameter of the positron beam.

Since emittance is conserved in free-space particle transport [9], focusing a
positron beam on a smaller spot eventually requires interaction with a thermal
reservoir and entails sacrificing beam intensity. The most efficient way of
reducing the beam radius is obtained through brightness enhancement by
remoderation, i.e. by implanting the positrons in a crystal and employing the
re-emitted ones to form a new beam: this results in a reduction of about an
order of magnitude of both the beam radius and intensity [10, 11, 12, 5].
Smaller beams can be obtained through successive stages of remoderation, which
requires purposedly-designed machines and cannot be easily added to existing
apparatuses. The pinnacle of positron beam focusing has been obtained by the
SPM apparatus which achieves sub-micrometric focusing using three stages of
remoderation [13].

Conventional beam focusing techniques allow PAS two-dimensional imaging via
rastering, i.e. by scanning the sample surface with the beam. This technique
presents some non-trivial limitations. First, the imaging resolution has to be
chosen a priori, since the focusing spot size is fixed by the experimental
apparatus. Second, the correct settings to focus the beam to sub-micrometric
scale can only be determined through knife-edge methods [5], which are
extremely demanding in terms of beam time. Third, if the imaging reveals non-
round features within the sample that warrant further PAS investigation,
selectively implanting positrons into these areas requires compressing the
beam sufficiently to fit within the feature, paying the attached cost in beam
intensity. Lastly, rastering in this context is used to achieve imaging using
position-insensitive detectors, yet many other single-pixel imaging (SPI)
methods are known. Since current positron remoderators cannot sculpt beams
into arbitrary shapes, they do not allow the use of powerful SPI techniques
such as measurements in the Hadamard basis or compressed sensing (CS) methods
[14, 15].

In this work I propose to forego imaging through rastering to instead probe
the sample with positron beams that have been sculpted into specific shapes
and then produce the image through a reconstruction algorithm. An advantageous
choice of beam shapes is the Hadamard basis [16]. To achieve this goal we need
to build an active remoderator capable of re-emitting positrons selectively
from different portions of its surface, analogous to the operation of digital
micromirror devices employed in cinema projectors. We could then project the
resulting sculpted beam onto a sample’s surface with a resolution which is
only limited by the modulator’s pixel size and the precision of the optics.
Such a device is equivalent to a single remoderator paired with a slit with an
arbitrarily variable shape; as such, it does not achieve by itself the
brightness enhancement obtained by multiple remoderation stages. In a high-
resolution positron beam apparatus it would be used to replace only the last
remoderator in the stack. Finally, I contend that a reconfigurable slit with
the required flexibility and resolution can currently be realized only by
implementing it as a remoderator.

In Section II the design of a microelectronic device which can be built in a
silicon substrate using established microelectronic construction techniques is
introduced. Simulations show that this device can act as a remoderator capable
of sculpting a beam of positron at micrometeric scale. As the beam is
accelerated towards the target this resolution is then improved to sub-
micrometric scales; Section III shows how this device can be combined with a
compact set of electrostatic optics to realise a miniaturised high-resolution
PAS imager. In section IV experimental improvements in PAS imaging that would
be made possible by this device are discussed. In Section V it is shown that
the switch-on time and switch-off time of a PMZ are sufficiently fast for
positron lifetime spectroscopy. Section VI prospects some outlooks for the
future expansion of this technology.

##  II Zener diodes that modulate positron emission

In the present work I focus solely on microelectronic construction in
monocrystalline silicon as it is currently cheapest and most widely available.
Moreover, interactions of slow positrons with silicon substrates have been
well characterized, which allows for precise numerical simulations. Other
semiconductor materials which are natively good positron remoderators are
known [17] and are an interesting venue for future analyses.

Positrons implanted in silicon with a kinetic energy of a few
$\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ thermalize in the material at a
depth of a few $\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. They then
undergo diffusion for about $200\text{\,}\mathrm{n}\mathrm{m}$ before they
either annihilate with the material or reach the surface and are either
trapped or re-emitted as positrons or Positronium. Strong doping in silicon is
expected to reduce the positron diffusion range, although no effects have been
observed at concentrations below
$1\text{\cdot\\!}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ [18].

If electric fields are present inside of the silicon crystal, e.g. due to a
doping gradient, positrons will also drift in the material [19]. These
diffusion-drift phenomena could allow purpose-built microelectronic devices to
manipulate positrons within the silicon bulk. To this end, the most useful
structure I was able to identify is that of a Zener diode. A Zener diode is a
short-base diode realized with a choice of doping profile and geometry that
allows its depletion zone to reach one of the bases when the diode is reverse-
polarized causing a sharp increase in current called a Zener breakdown.

The intensity of the electric field within the depletion zone of any diode is
capable of inducing a significant positron drift. This field is always present
and, by polarizing the diode, its intensity varies. Unfortunately, this change
in intensity is not sufficient to allow by itself a significant modulation of
the positron drift. At the same time, polarizing the diode changes also the
size of the depletion zone; this phenomenon can be exploited to significantly
alter the positron drift in a portion of the substrate. I will refer to a
device designed explicitly to achieve this goal as a Positron Modulating Zener
(PMZ). Notably, metal-oxide-substrate structures also allow the production and
modulation of strong electric fields in microelectronics; however the
interfaces between materials, unavoidable in their construction, are likely to
trap positrons, which is severely detrimental to the goal of re-emitting them
in vacuum.

Figure 1: Schematic representations of the construction and principle of
operation of a PMZ device; a) axonometric representation of the PMZ structure;
b) distribution of silicon doping employed in the PMZ in transversal section;
the coating material is chosen as to impede the re-emission of positrons where
it is applied; c) when the diode is not polarized, most of the positrons
cannot reach the surface after thermalizing; d) when the diode is reverse-
polarized, the depletion zone extends further in the p-type trench, allowing
its strong electric fields to guide positrons towards the gap left in the
surface coating, which causes a strong increase in the re-emission
probability.

Figure 1a and b shows a schematic representation of how a PMZ could be
realized. A p-type trench in the shape of a half cylinder with radius
$0.8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and a doping of
$5\text{\cdot\\!}{10}^{15}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ is manufactured
into a n-type substrate having a dopant density of
$2\text{\cdot\\!}{10}^{14}\text{\,}\mathrm{c}\mathrm{m}^{-3}$. coating is
applied to the trench, made from a material (such as silicon dioxide) that
would trap any positron reaching the surface, leaving only the central
$0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ of the trench open to emit
positrons. The choice of doping density ensures that the depletion zone will
extend primarily in the substrate and that positron drift in the substrate is
not inhibited. Figure 1c and d show schematically the functioning mechanism of
the device: when the diode is not polarized (c), the depletion region is held
further away from the opening than when the diode is reverse-polarized (d). In
the latter case the electric field present in the depletion zone guides the
positrons closer to the aperture, letting a higher amount of them escape the
device than when the diode is not polarized.

Figure 2 shows a finite-element simulation of the device when non polarized
(left) and when polarized in reverse to the Zener breakdown voltage of
$2.6\text{\,}\mathrm{V}$ (right); in this regime the electric field in the
region surrounding the trench has an intensity of about
$1\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$. In these simulations a
$0.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ sigmoid-shaped doping
gradient between the p and n silicon has been employed to mimic more closely
the structure of a real device.

Figure 2: Finite element simulation of a PMZ response and Monte Carlo
simulation of positron implantation into it. Left: PMZ not polarized. Right:
PMZ reverse-polarized. The black arrows represent the direction and intensity
of the electric field, color represents the electric potential. The histograms
(in logarithmic scale) show the re-surfacing of positrons in the two cases; of
these only positrons reaching the uncoated area at the center of the diode can
be re-emitted by the PMZ.

We can predict precisely the behaviour of slow positrons implanted into the
device. Since the material is uniform and well characterized, the implantation
of positrons can be effectively approximated by the Makhov model [8] while
post-thermalization physics can be simulated using a Monte Carlo technique.
The Monte Carlo simulation has to take into account three different phenomena:
drift, diffusion and annihilation. We model all three as a series of discrete
interactions [20] after each of which the positrons are assigned a velocity
drawn from a thermal distribution independently of their velocity prior to the
interaction. Positrons travel between interactions along a parabolic
trajectory, due to the external electric field, for a time $t$ randomly drawn
from a exponential distribution with characteristic time $\tau_{I}$. Between
interactions, positrons have a chance of annihilating given by
$p_{\text{ann}}=\tau_{I}/(\tau_{I}+\tau_{\text{ann}})$, where
$\tau_{\text{ann}}$ is the lifetime of positrons in the material. We can
verify the correctness of the simulation parameters by using it to predict
positron diffusion in the material and the drift distance in presence of a
constant electric field. We used
$\tau_{I}=$0.06\text{\,}\mathrm{p}\mathrm{s}$$ and
$\tau_{\text{ann}}=$260\text{\,}\mathrm{p}\mathrm{s}$$ [18], which predict a
positron diffusion of $218\text{\,}\mathrm{n}\mathrm{m}$ and a drift of
$2.76\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ in a
$1\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ electric field, which are
in line with the values present in literature [21, 19].

The histograms placed over the device surface in Figure 2 show the amount and
location of positrons reaching the surface of the silicon die after being
implanted in the material with a kinetic energy of
$10\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. Of the positrons implanted in a
$12\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-wide region centered on the
PMZ, 0.19% emerge within the uncoated region when the diode is not polarized
and this number increases to 13.4% when the diode is reverse polarized to the
breakdown voltage. This allows the modulation of the positron re-emission with
a contrast level of 70:1.

All the PMZ construction parameters employed here were obtained through a non-
exhaustive search within the considered parameter space. It is likely that
designs with better performances could be discovered through more
comprehensive searches or by considering more complex geometries.

It is known that bare silicon is not an excellent emitter of positrons; from
the measurements of Mills [22] we can infer that about 13% of the positrons
diffusing to its surface are re-emitted into vacuum. It follows that the total
remoderation efficiency of a device comprising multiple PMZs spaced in a
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ wide grid would be only
3.5%. It might be possible to increase this efficiency by coating the surface
of the device. A natural choice could be silicon carbide (SiC), whose surface
emits 30% of the positrons reaching it [23]. This would give a theoretical
remoderation efficiency of 8.1%. However, this is not realistic since
interfaces between materials are known to act as traps for positrons with
coefficients which are difficult to predict. The actual performance of any
coating will have to be experimentally determined.

##  III A miniaturised positron beam sculpting device

As seen in Section II, a PMZ can be realized with a size smaller than
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$; thanks to modern
microelectronic construction techniques, millions of them can be printed onto
a silicon die of a few $\text{\,}\mathrm{m}\mathrm{m}^{2}$ along with the
electronic circuitry needed to address and control each one individually. I
will refer to such a device as a Positron Modulating Array (PMA). Control
electronics to address and switch each diode individually can be synthesized
as a shift-register, a grid of x-y addressed Metal Oxide Semiconductor (MOS)
transistors or with a Charge-Coupled Device (CCD)-like structure. Regardless
of the architecture of choice, the control electronics will unavoidably take
up a portion of the surface that would be otherwise available for the
realization of PMZs, in a measure that depends on the device design and the
lithographic process employed. A way to improve this metric without increasing
the resolution of the fabrication is to simply realize larger PMZs by printing
comb-shaped geometries to increase their surface area.

Figure 3: Possible application of a PMA device to produce a compact imaging
device, as described in section III. In gray, electrodes used to form the
electrostatic beam optics, in color the electric potential in the free space,
in black and white particle tracks from the simulated beam. Left: the primary
beam is accelerated to $10\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ and
implanted into a PMA device placed at the bottom. Right: a compact beam optics
designed to project and focus the positrons re-emitted from the PMA (bottom)
to a sample to be imaged (top). Both the macrospic and microscopic optics are
rotationally symmetric, along different axes. They have been designed so that
the electric field in the volume in front of the PMA is only governed by the
microscopic optics. This allows employing separate axisymmetric simulations
for the two optics, that are then merged together to describe the whole
apparatus.

Figure 3 shows a possible configuration in which a PMA could be installed to
perform high-resolution imaging of a sample surface. The setup focuses a
$10\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ positron beam onto the PMA using
three cylindrical electrodes in a standard einzel lens configuration (left).
The PMA is then used as a remoderator and to sculpt the beam; slow positrons
emitted from its surface are accelerated and focused by a custom five-
electrode lens system onto a sample placed $5\text{\,}\mathrm{c}\mathrm{m}$
away from the PMA (right). The design shown in Figure 3 was simulated using a
purpose written code to assess its expected performance in a real-world
scenario, namely the CDBS apparatus installed on the NEPOMUC positron beam [5]
operated without the use of its nickel foil remoderator. Employing a PMZ
spacing of $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a
$7\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ acceleration potential a resolution
of $100\text{\,}\mathrm{n}\mathrm{m}$ can be reached with a imaging area of
$0.1\times$0.1\text{\,}\mathrm{m}\mathrm{m}$$ ($10^{6}$ pixels). The optics in
both the macroscopic (left) and microscopic (right) portion of the machine are
completely electrostatic and realized with rotationally symmetrical electrodes
whose shapes have been optimized through procedural design, i.e. by letting an
algorithm alter them based on simulation results to progress towards more and
more efficient designs.

Given that imaging through a PMA does not entail rastering, the inclusion of
deflector coils in the microscopic optics is not mandatory and their addition
is only contingent on the necessity of correcting the beam trajectory being
expected. Moreover the active nature of the PMA allows the correction of
spherical aberrations at the source, within the resolution of the PMA; this
requires characterizing the aberrations, which can be performed by using a
CMOS detector to directly measure the secondary positron beam, as shown in a
previous work [24].

The small size of the PMA allows the construction of a very compact apparatus
to produce, extract and focus the sculpted microbeam; the advantage of this
compactness is threefold. Firstly, the small size makes it easy to keep the
sample and PMA rigidly positioned with respect to each other, even in the
presence of vibrations in the apparatus. Secondly, a shorter beam path reduces
the distortions induced on the beam by stray electromagnetic fields and makes
them easier to shield. Finally, these microbeam optics are small enough to be
installed in most existing apparatuses with minimal modifications. The main
challenge in realizing the apparatus presented here is the high voltage
gradient present between the last electrode of the microscopic optics and the
sample; this gradient is at the limit of feasibility, and as such it may be
necessary to reduce it to allow for manufacturing tolerances and imperfect
vacuum conditions.

##  IV Single-pixel imaging techniques using a postron-modulating device

The use of PMA-based imaging enables the use of several experimental
techniques which are inaccessible through rastering.

Progressively increasing resolution: It is possible to use a PMA to perform
SPI-like spectroscopy by projecting specific patterns onto the target (see
Figure 4). The Hadamard basis is particularly convenient, as it affords
discrete pixel modulations (having only $\pm 1$ entries) and, being an
orthogonal transformation, does not amplify the noise under $\ell^{2}$ norm.
Therefore, in counting experiments, as all PAS techniques ultimately are, this
technique does not entail any loss of precision in the assessment of the
brightness of each pixel in the reconstructed image [25, 16]. Since the
elements of the Hadamard basis are composed of positive and negative sign
elements, each element will require two measurements, one in which the
positive-signed portion is projected onto the sample, the other in which the
negative-signed portion is projected. The greatest advantage of imaging
through the Hadamard basis is that the imaging resolution increases as the
measurement proceeds. This would allow an experimenter to decide to stop a
measurement which has already revealed the necessary features at a lower
resolution and to proceed to higher resolutions only when needed. Figure 4
(bottom) shows a simulation depicting this progressive increase in resolution
using a test image. I have used as reference the beam of the Coincidence
Doppler-Broadening Spectroscopy (CDBS) apparatus of NEPOMUC (as in Figure 3),
without the use of the nickel remoderator, assumed that no coating was applied
to the PMA and that each element of the basis was measured for
$13\text{\,}\mathrm{s}$ as in [5]. The basis element measurements have been
modelled as Poisson draws. In the four insets it is possible to see how the
feature of the test target emerge with greater and greater resolution as more
measurement time is spent to execute the experiment.

Figure 4: Above: elements of the Hadamard basis (left) are used to sculpt the
beam before projecting it onto the sample (schematic representation, right).
Below: a simulated experiment in which sample imaging is reconstructed at
different points of the same measurement; as the measurement time increases so
does the resolution of the reconstructed image, from which smaller and smaller
details are progressively revealed.

Faster beam setup: Focusing a microbeam requires the measurement of the beam
size at the target, which requires the extremely time consuming knife-edge
technique [5]. This often results in the expenditure of comparable beam time
in the setup of the apparatus and the actual measurement. While we have
recently proposed a possible solution based on direct imaging with CMOS
sensors [24], it unfortunately does not extend to the submicrometer scale. The
knife edge technique is ultimately a counting experiment, and as such the
speed at which it can achieve a given precision is determined by the intensity
of the beam employed. As the beam is reduced in diameter a cost in intensity
needs to be paid regardless of the technique adopted to compress it. The
possibility of sculpting the beam into arbitrary shapes allows the use of
grids in place of a single knife edge, thus exploiting the entire sample
surface to determine the beam focusing, which would be orders of magnitude
faster than a traditional knife edge. In addition, a micrometric grid printed
onto a sensor similar to those in [24] would allow to test in parallel the
alignment and focusing of the beam at different points of the desired target
surface by observing the formation of Moiré patterns.

Feature isolation: Modern PAS techniques are increasingly moving towards in
operando measurements, that is, measurements in which, during the course of a
PAS measurement, a sample is manipulated mechanically, thermally, chemically
or through the exposure to light or slowly varying electromagnetic fields. In
operando measurements would benefit from the possibility of selectively
studying how a feature localized in the sample responds to the external
manipulation. The use of a PMA would allow preliminary imaging of the sample
aimed at locating the feature of interest (e.g. a crack in a solid, an
incorporation of a different material, a microelectronic structure). After the
exact shape and location of the feature had been determined, the beam could be
sculpted to implant positrons selectively into the desired feature during the
in operando measurement, thus reaching signal-to-noise ratios traditionally
inaccessible. This is of particular relevance whenever it is not known whether
the rest of the sample would also respond to the external manipulation, thus
hindering background subtraction. Particularly amenable to this technique are
man-made microstuructures, as knowledge of their expected shape makes it
possible to align the sculpted beam without the need of performing a complete
preliminary imaging of the sample. As an example, the study of the dielectric
materials employed in MOS Field-Effect Transistors through PAS [26] could be
performed, with this technique, in operando over the entirety of a HexFET
structure. Other particularly good applications would be the study of
microelectronic and micromechanic structures which can be actuated without
inducing macroscopic mechanical stresses which could misalign the sample with
respect to the PMA.

Compressed sensing: A great prominence has been gained in recent times by
compressed sensing (CS) techniques, after the development of noise resilient
compressed sensing algorithms [14]. Compressed sensing is the name of a family
of reconstruction techniques that rely on the measurement of a low-entropy
signal performed by masking it using numerous random patterns, to then
reconstruct its features, under the sole assumption of them being sparse in
some measurement basis [15], which is always the case in practical application
of PAS imaging. Unfortunately, current CS techniques are not resilient against
Poissonian noise [27]. I have tested the entire set of CS algorithms
implemented in the KL1P library [28] in a variety of scenarios using simulated
counting experiments, and none of them have shown any significant advantage
compared with the use of just the Hadamard basis. The field is nonetheless
rapidly evolving and techniques capable of working effectively with Poisson
noise are under development [29], including machine learning based approaches
[30]: the application of these techniques to PAS imaging is currently only
possible through the use of a PMA.

##  V Switching speed of the positron-modulating Zener

Among the PAS techniques, one of the most powerful is positron annihilation
lifetime spectroscopy (PALS), which works by implanting a pulsed beam of
positrons into a sample to then measure the time taken by the implanted
positrons to annihilate. In this application, the length and sharpness of the
positron pulses combined with the jitter of the detectors determines the
resolution function. Ideally, the width of the resolution function should be
kept to the order of $100\text{\,}\mathrm{p}\mathrm{s}$ [31]. As we shall see,
the expected transient time of a PMA is fast enough for it to be used either
to sharpen the output of a positron buncher/chopper, or to replace it
altogether.

Figure 5:  Top: expected resistance (orange) and capacitance (purple) per unit
length of the p-type trench. Bottom: simulated switch-on (red) and switch-off
(blue) transients of the actuation of a
$5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-long PMZ contacted on both
ends, the dashed line indicates the actuation as expected purely from an
electronic standpoint. The solid lines are the result of a full simulation
which takes into account the drift of positrons in the diode. The full
simulation exhibits both a much longer switch-off transient than the purely
electronic simulation and overshoots of the steady state emission in both
transients.

The electronic switching time of a PMZ depends on how quickly the charge
needed to polarize or de-polarize it can be transported along the p-type
channel in the center of the trench. Both the capacitance and resistance of
the channel depend on the local polarization status of the diode. These can be
computed from the number of free carriers predicted by the finite element
simulation employed in section II. The resulting curves are shown in figure 5
(top). To realize a fast-switching PMZ we want to limit its length, so for
this simulations a $5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ long PMZ
contacted on both sides was considered. In general we can expect the
electronic switching time to scale approximately with the square of the diode
length. An exact solution of the polarization/depolarization transient would
require a time-dependent three dimensional finite element simulation. However,
in this context it is practical to approximate it by using the voltage-
dependent resistance and capacitance of the trench to write a diffusion
equation which can then be solved numerically. The result is a wave of
polarization/depolarization which travels along the length of the diode with
decreasing speed; the electronic transient in the case of a
$5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-long PMZ is shown with
dashed lines Figure 5 (bottom), with the local diode polarization translated
into a positron re-emission by using the steady-state emission of the diode at
different polarization. If computed this way the switch-on time (time to reach
95% of the positron emission) is $90\text{\,}\mathrm{p}\mathrm{s}$ while the
much faster switch-off time (time to reach 5% of the emissions) is
$13\text{\,}\mathrm{p}\mathrm{s}$. The difference in time is due to the fact
that, as the diode polarizes, the channel gets pinched by the diode depletion
zone, increasing the resistance of the channel and making it more difficult to
carry the charge needed to polarize the diode further along its length. This
model does not, unfortunately, capture all the relevant physics of the PMZ
transient, since the time the positrons need to diffuse in the material is
neglected.

To simulate the diffusion process we need to run a Monte Carlo simulation in
the three-dimensional volume occupied by the PMZ using a time-dependent
electric field and a flux of implanted positrons uniformly distributed within
the simulation time window, which extends to the nanosecond prior to the
switching time to allow for the PMZ to reach a steady state before switching.
The result of this simulation is shown with solid lines in Figure 5 (bottom).
When employing this simulation technique the switch-on time is still
$90\text{\,}\mathrm{p}\mathrm{s}$, albeit with an added
$60\text{\,}\mathrm{p}\mathrm{s}$ delay, while the switch-off time increases
to $250\text{\,}\mathrm{p}\mathrm{s}$. The most striking feature is the
presence of overshoots in both transients. The overshoot in the switch-on
transient is due to the accumulation of positrons in the silicon bulk while
the PMZ is not polarized. During the steady state with the diode polarized,
positrons continuously drift towards the surface and are re-emitted with a
rate that is dictated by how quickly new positrons are implanted in the
material. This reduces the permanence time of positrons in silicon which would
otherwise be (with the exception of positrons implanted close to the surface)
given by the positron lifetime in the solid. As a consequence, the number of
positrons present in the silicon bulk at any given time is higher when the
diode is not polarized. Shortly after polarization this excess positrons are
flushed out, thus briefly overshooting the steady state emission rate. The
overshoot at switch-off is due to a different phenomenon: the electric field
within a polarized PMZ is mostly directed towards the surface (see Figure 2d)
except in extreme proximity to the surface where it changes direction and
acquires a small component pointing away from the material surface. This small
component effectively hinders the re-emission of positrons, even though most
will be able to overcome it and exit the material. When the diode polarization
is removed, this hindrance is removed too and thus positrons that were already
close to the surface will more easily escape the silicon, once again exceeding
briefly the steady state emission rate.

All considered, the expected transient time of the PMZ make it a good
candidate to act by itself as a chopper/buncher in a compact PALS apparatus or
to work in combination with an external chopper/buncher to increase its time
resolution.

##  VI Conclusions and outlook

I have presented the design of an active remoderator capable of modulating the
re-emission of positrons from any point of its surface, sculpting a positron
beam into an arbitrary shape. Numerical simulations show that a resolution of
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ can be achieved with a
contrast of 70:1. Individual emitters of the device can be switched on in
$90\text{\,}\mathrm{p}\mathrm{s}$ and switched off in
$250\text{\,}\mathrm{p}\mathrm{s}$. I have then presented a possible design of
a microbeam PAS imaging optic making use of this device. Simulations show that
when positrons are implanted into the target with an energy of
$7\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ an imaging resolution of
$100\text{\,}\mathrm{n}\mathrm{m}$ can be achieved over an imaging area of
$0.1\times$0.1\text{\,}\mathrm{m}\mathrm{m}^{2}$$. This device is extremely
compact, with a distance of $5\text{\,}\mathrm{c}\mathrm{m}$ between the PMA
and the sample, which at the same time makes it more easily shielded from
ambient noise and compatible with existing apparatuses.

The use of this device would enable several new experimental techniques.
Firstly, the possibility of selecting the desired imaging resolution as the
measurement is performed, based on the measurement results. Secondly a faster
calibration of the beam optics, in particular when sub-micrometer resolutions
are to be achieved. Thirdly, the isolation of non-round features which can
then be measured with high signal-to-noise ratios using PAS techniques.
Finally, it enables the use of compressed sensing with PAS techniques.

The modern microelectronic industry has produced a wealth of different
integrated devices, with a rich taxonomy of structures and functions. It is my
belief that many devices might exist within this assortment, with potential
applications to low-energy positron techniques performed through the
manipulation of positrons in a material bulk. These applications could span
from the construction of positron-to-positronium converters whose positronium
production can be spatially and temporally tuned, to the study of samples
embedded directly into the silicon bulk, to the formation of a positronium
Bose-Einstein condensate [32]. Two main obstacles hinder the exploration of
these opportunities: the difficulty of simulating these scenarios and the high
costs attached to the custom production of these devices. It is my hope that
the present work will spark some interest in trying to overcome both. In the
meantime, since the use of PMAs prescinds from the specific PAS techniques
employed, the identification of their feasibility is a potential benefit to
the whole of known positron spectroscopy techniques.

###### Acknowledgements.

Firstly it is imperative that I give my thanks to Dr. Davide Orsucci, for his
relentless help in reviewing the drafts of my papers, the invaluable insights
that without fail he provides me with and his unwavering patience in
tolerating my pedantry. On top of that I would like to thank Prof. Christoph
Hugenschmidt, Dr. Daniel Russell and Dr. Alessandro Moia for the advice they
provided during the preparation of this paper.

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microtypeUnable to apply patch

# Can In-context Learners Learn a Reasoning Concept
from Demonstrations?

Michal Štefánik  Marek Kadlčík
Faculty of Informatics, Masaryk University, Czech Republic
<EMAIL_ADDRESS>
###### Abstract

Language models exhibit an emergent ability to learn a new task from a small
number of input-output demonstrations. However, recent work shows that in-
context learners largely rely on their pre-trained knowledge, such as the
sentiment of the labels, instead of learning new associations from the input.
We argue that the commonly-used few-shot evaluation using a random selection
of in-context demonstrations can not disentangle models’ reliance on such
biases, as most of the randomly-selected demonstrations do not present
relations informative for prediction beyond exposing the task’s input-output
distribution.

Therefore, to evaluate models’ in-context learning ability independent of
models’ memory, we introduce a Concept-sharing few-shot learning method
choosing the demonstrations that share an underlying concept with the
predicted sample. We extract a set of such concepts from available human
explanations and measure how much models can benefit from presenting these
concepts in few-shot demonstrations.

We find that most of the recent in-context learners can not consistently
benefit from the demonstrated concepts, irrespective of the model size.
However, we note that T0 models are more sensitive to exhibited concepts,
benefiting from concept-sharing demonstrations in 7 out of 8 evaluation
scenarios.

##  1 Introduction

Figure 1: In this work, we assess in-context learners’ ability to improve when
presented with demonstrations using a reasoning concept applicable in the
prediction (§2). We extract these concepts from human explanations (§3.2) and
assess models’ ability to learn to use these concepts, as reflected in
improving their prediction quality.

In-context learning (ICL) is the alternative to the conventional training of
Large Language Models (LLMs) for specific task(s), where models are expected
to learn a new task solely from the input text. In few-shot in-context
learning that we focus on, the input text contains a set of demonstrations,
i.e. the input-output examples of the task to be learned Brown et al. (2020).

An ability to learn unseen tasks from input sequence without updating the
model has practical and theoretical implications, both of which are of great
significance; Understanding free-form user requests allow applying LLMs in
applications of restricted, or limited data availability without over-
specialization Goodfellow et al. (2014). In-context learning can provide a
handle of models’ behaviour, enabling the model to avoid specific erroneous
predictions. In theory, a training process resulting in an accurate new-task
learner defines the sufficient conditions for the emergence of a specific
level of generalization.

Recent LLMs trained on vast mixtures of tasks (Sanh et al., 2022a; Wang et
al., 2022b; Chung et al., 2022) show a certain level of new-task ICL and
gradually bring more attention and expectations in this direction. However,
counter-intuitively to the overall evaluations, in-context learners (ICLs)
also expose surprising behavioural artefacts; Liu et al. (2022) show ICLs’
sensitivity to the ordering of in-context demonstrations. Similarly, Lu et al.
(2022) find surprising sensitivity of ICLs to the specific wording of the
prompts. Min et al. (2022b) show that most of the model performance is
persisted even when the contents of the demonstrations are randomly swapped.
Contrary to the ability to learn from input, Wei et al. (2023) propose to
attribute this to the over-reliance of in-context learners on semantics of the
label tokens, especially in smaller models.

We hypothesize that the discrepancy between the expected and the perceived
abilities of ICLs might be attributed to their limited evaluation, commonly
performed with a random set of task demonstrations. However, for many open-
ended tasks, such as question answering, or translation, randomly-chosen
demonstrations rarely present a reasoning pattern which can help with the
prediction of new input (Figure 1). We argue that the evaluation with mostly
non-informative contexts also can not reflect on the ability of learning, as
observed in humans111We restrain from discussing a concept of learning in the
psychological scope, but we note that Concept learning fits well into a
definition of Associative learning (Plotnik, 2012)., as the gain of
extrapolating associations presented in non-informative demonstrations can
only bring little benefit to the practice.

We also note that in the absolute numbers, the random-demonstrations
evaluation may favour some LLMs, such as ones with a capacity to remember a
wider variety of input distributions from pre-training, that can be used for
modulating their behaviour in ICL. However, note that such behaviour differs
from learning new association(s) from the context and makes the model prone to
adversaries.

Hence, in Section 2, we propose to evaluate models’ in-context learning
ability primed with the demonstrations that exhibit a reasoning analogical to
the one required for a robust prediction of the predicted sample (Figure 1).
We measure how well can the recent few-shot learners utilize identified
concepts for more accurate predictions (§3) and find large discrepancies among
the models and concepts.

Our main contributions are following: (i) We introduce a task of Concept-
sharing few-shot learning, disentangling models’ ability to learn a new
reasoning concept from other aspects of prediction quality. We show how such
reasoning concepts can be extracted from human explanations. (ii) For a wide
variety of recent in-context learners, we measure the ability to benefit from
presented reasoning concepts. We show that while some models are better at
learning concepts on average, this ability can not be attributed to the
models’ size or training strategy.

#### Problem Definition

Given a dataset $\mathcal{D}:\\{(x_{1}\rightarrow Y_{1}),..,(x_{i}\rightarrow
Y_{i})\\}\in\mathcal{D}$ containing pairs of input $x_{j}$ with associated
label $Y_{j}$, an in-context few-shot learner $\Theta(x)\rightarrow y$ aims to
predict a correct label $y_{k+1}=Y_{k+1}$ given a sequence of $k$ input-output
demonstrations, and the predicted input $x_{k+1}$:

$\Theta([x_{1}\rightarrow Y_{1},..,x_{k}\rightarrow Y_{k}],x_{k+1})\rightarrow
y_{k+1}$ (1)

We expect in-context few-shot learner $\Theta$ to model the relation of
$x_{i}$ and $y_{i}$ by (i) identifying and (ii) applying the relations of
input and output presented in demonstrations. Each such relation is modelled
by one or more latent concepts $\mathcal{C}$:

$\forall\,(x_{i},Y_{i})\in\mathcal{D}:\exists\,\mathcal{C}:\mathcal{C}(x_{i},Y_{i})=1$
(2)

We broadly define a concept $\mathcal{C}$ as any function
$\mathcal{C}(x,y)\rightarrow\\{0,1\\}$, constraining a space of valid outputs
$y$ to the ones where $\mathcal{C}(x,y)=1$. Thus, if $\Theta$ learns a concept
$\mathcal{C}$, it will never predict for $x$ such $y$ that
$\mathcal{C}(x,y)\\!=\\!0$. In a composition
$\\{\mathcal{C}\\}\\!=\\!\\{\mathcal{C}_{1},..,\mathcal{C}_{j}\\}$, all
$\mathcal{C}_{i}\\!\in\\!\\{C\\}$ must evaluate to $1$.

Given that modelling of each $C$ valid for the task of $\mathcal{D}$ restrain
a set of possible predictions of $\Theta$ exclusively from incorrect
predictions, extending a set of concepts learned in-context with complementary
one(s) should never decrease the performance of the model $\Theta$ on
$\mathcal{D}$.

##  2 Concept-sharing Few-shot Learning

We reformulate in-context few-shot learning (1) to a Concept-sharing few-shot
learning, evaluating the ability of a few-shot learner $\Theta$ to identify
and apply a user-chosen reasoning concept $\mathcal{C}$ shown in
demonstrations. First, we classify evaluation samples such that the samples of
the same category $X_{i}$ require the concept $\mathcal{C}_{i}$ to map $x$ to
$Y$. Subsequently, in concept-sharing few-shot learning, we let the learner to
infer a prediction for input $x_{k+1}$ by presenting it with demonstrations
$(x_{j}\rightarrow Y_{j})_{1..k}\in X_{i}$, thus sharing the reasoning concept
$\mathcal{C}_{i}$ with the predicted input $x_{k+1}$:

$\displaystyle\Theta([x_{1}\rightarrow Y_{1},..,x_{k}\rightarrow
Y_{k}],x_{k+1})$ (3) where $\displaystyle\forall(x_{1..k},Y_{1..k})\in
X^{i}\text{ and }x_{k+1}\in X^{i}$

We note that $\Theta$ can rely on other features than $\mathcal{C}_{i}$, and
such reliance is not easy to disentangle. Therefore, we propose to
contextualize the results of Concept-sharing few-shot learning on a concept
$C_{i}$ as a difference to the performance obtained in a random selection of
demonstrations.

Additionally, to make the predictions based on two different sets of
demonstrations mutually comparable without systematic bias (e.g. in samples’
complexity), we perform both random and concept-sharing evaluations with the
same predicted samples $x_{k+1}$, and only change the demonstrations.222The
implementation of Concept-sharing few-shot learning is available on
https://github.com/MIR-MU/CoAT.

#### Informative Concepts Extraction

Constructing a scaled evaluation with annotated reasoning concepts
$\mathcal{C}$ is challenging since the annotations of such concepts among
datasets are very rare. However, we find the concepts inherently captured in
human explanations of some datasets, where annotators are asked to collect
answers to a question “why is [input] assigned [output]?” (Wiegreffe and
Marasović, 2021).

The form of these explanations ranges from free-text explanations, including
annotator-specific slang and stylistics, to semi-structured and structured
explanations, cast to a pre-defined format, often consisting of a set of
relations in a form “[subject1] [relation] [subject2]” that transitively maps
[input] to [output] (Jansen et al., 2018). We focus on extracting the concepts
from the subset of the semi-structured and structured explanations where the
format consistency ensures that analogical reasoning concepts will be shared
among multiple samples.

##  3 Evaluations

This section overviews few-shot in-context learners that we evaluate for
Concept-sharing few-shot learning and the datasets with explanations allowing
us to extract shared reasoning concepts.

###  3.1 Few-shot Learners

#### T0

Sanh et al. (2022b) introduce a set of in-context learning models fine-tuned
from a T5 model Raffel et al. (2020) on a variety of 35 mostly QA tasks in
zero-shot settings, aiming to perform well on a task of previously-unseen
categories. T0 is trained for sequence-to-sequence generation over a large set
of diverse tasks cast to a unified input-output format provided by task-
specific templates of Promptsource Bach et al. (2022).

#### Tk-instruct

(Wang et al., 2022a) is a set of models trained for comprehension of
annotator-like instructions, consisting of a free-text task description and a
set of input-output pairs, collected for over 1,600 tasks of
NaturalInstructions collection (Mishra et al., 2022). In comparison to T0, Tk-
instruct models can advance from a wider mix of tasks and also from fine-
tuning in the few-shot learning format matching the evaluation.

#### Flan

(Chung et al., 2022) scales the approach of fine-tuning in a few-shot learning
format to over 1,800 tasks of 146 categories including all resources of T0 and
Tk-instruct. Contrary to the former models, the training data mixture includes
several datasets with chain-of-thought labels, where the model is trained to
follow the annotated reasoning chain explicitly. We evaluate all publicly
available T5-based Flan models.

#### GPT3

(Brown et al., 2020) is a well-known causal language model that has first
shown that in-context few-shot learning ability can emerge solely from vast
amounts of unsupervised training data and parametrization, without fine-
tuning. Alternatively to other approaches, InstructGPT Ouyang et al. (2022)
fine-tunes GPT3 to follow human instructions using obtained user feedback. We
evaluate both these models through OpenAI APIs333https://beta.openai.com.

###  3.2 Datasets

Figure 2: Conceptual few-shot evaluation: Relative performance change of the
assessed in-context learners between using random demonstrations (k=3) and
concept-sharing demonstrations (§2), with concepts of the datasets described
in §3.2. Models are ordered by a number of parameters. Error bars show a 95%
confidence interval of the bootstrapped results ($100$ samples, $200$
repeats). Absolute results for both selection strategies are in Figure 4.
Figure 3: Conceptual few-shot evaluation: all concepts: Error change of the
assessed in-context learners between random demonstrations and concept-sharing
demonstrations (§2) aggregated over all assessed concepts. Experimental setup
is consistent with Figure 2.

Following is a description of datasets that we use in Conceptual few-shot
evaluation. Note that for each dataset, we highlight a single concept that we
use in Conceptual few-shot evaluation as the $\mathcal{C}$ (§2). In the case
of each model and dataset, we first evaluate all templates available in
Promptsource and report the gain of utilising the chosen concept for the best-
performing template.

#### WorldTree

(Jansen et al., 2018; Xie et al., 2020) is a collection of 5,114 science exam
questions with the explanations in the form of 9,216 shared facts supporting
the assignment of the correct answer.

We use the shared facts as the concepts $\mathcal{C}$ and evaluate with the
demonstrations of a maximal facts’ intersection with the predicted sample.
Contrary to the other datasets, in WorldTree evaluation, we prepend the facts
for all the demonstrations in the context before the demonstrations so that
the model does not have to rely on the already obtained knowledge.

#### OpenBookQA

(Mihaylov et al., 2018) is a collection of elementary-grade single-choice
questions requiring common sense knowledge about the world. A set of 4,957
explanations take the form of a triple of (object, relation, object), such as
“a stove generates heat” for a question “Which one of these can help a person
cook their food? [four options]” and a correct option “a counter cooker
appliance”.

To extract informative concepts $\mathcal{C}$, we perform syntactic analysis
of the explanation and extract the relation, identified as a root of the
sentence’s parse tree. Hence, in concept-sharing few-shot learning, we prime
the aforementioned question with other question-options-answer pairs of the
questions answerable by relating the input to output through the “generate”
function.

#### HotpotQA

(Yang et al., 2018) is a QA dataset composed of questions requiring the QA
model to jointly reason over multiple passages of multi-document contexts.
Inoue et al. (2020) enrich the dataset with explanations from three human
annotators. The explanations are structured in the form of triples
$(e_{1},r,e_{2})$, associating two entities ($e_{1}$ and $e_{2}$) through a
relation $r$, such as (“Scott Derrickson”, “is”, “an American director”).

We extract the shared concepts $\mathcal{C}$ as pairs of $(r,e_{2})$; Hence,
Conceptual few-shot will prime the prediction with questions and contexts
presenting the entities in analogical relations to the ones the model should
understand for correct prediction.

#### GLUE Diagnostic

(Wang et al., 2018) contains approximately 1,100 diagnostic samples of Natural
Language Inference intended to fool a simple statistical model. While the
concepts are heuristically extracted in other cases, GLUE diagnostic directly
annotates 30 distinct logical concepts needed in prediction, such as double
negation, conjunction, or existential quantification. We directly use these
logical concepts as the reasoning concepts $\mathcal{C}$.

###  3.3 Baseline model (Baseline-Tk-QA-1B)

To contextualize the results of existing in-context learners, we additionally
evaluate a simple newly-created few-shot in-context learner trained on a
single QA dataset. Similarly to Tk-Instruct, we construct the training
examples of the meta-learning task in the explicit few-shot learning format,
as initially proposed by Min et al. (2022a), where the model is updated to
predict correct labels with a set of randomly-selected demonstrations included
in the input (Eq. (1)). This way, we fine-tune a T5-Large model Raffel et al.
(2020) on AdversarialQA dataset Bartolo et al. (2021) until convergence on a
validation split. We assess the resulting model on Concept-sharing few-shot
learning together with other in-context learners and denote its results as
Baseline-Tk-QA-1B.

##  4 Results and Discussion

Figure 2 shows a relative change in models’ performance between a random
selection of demonstrations and Concept-sharing few-shot learning, i.e. with
demonstrations sharing a selected concept (§2), ordered by models’ size.

For each of the assessed concepts, we observe statistically significant
improvement for at least one of the models, which confirms our initial
assumption on the informativeness of the extracted concepts in prediction.

Following the ordering of the results, we see that concept learning does not
relate to the model size. Following the results for each model instance, we
see many cases where the model is able to utilise one concept but fails to
utilise, or even worsens the prediction when exposed to the other. The
variance is larger for instruction-tuned Tk-instruct models, excelling in
utilising shared reasoning logic of GLUE, but to the contrary, degrading when
being exposed to fact-sharing demonstrations of WorldTree. Contrary to these
results is the case of InstructGPT that is agnostic to concepts except for
GLUE. We speculate that this could be explained by the exposition of
evaluation samples in training.

Figure 3 shows the average of changes of Conceptual few-shot evaluation over
the inspected four concept types. The aggregation uncovers that the gain from
providing informative demonstrations largely varies among models, with T0 and
larger Flan models benefiting from the presented concepts more often. We
analyse these two cases and find that Flan’s concept learning gains need to be
mainly attributed to decreased performance on unseen tasks in random-
demonstrations settings (Figure 4), thus making this a disputable success.
However, in the case of T0, we find both model versions able to benefit from
the concept in all (8) cases with a single exception of T0-11B in OpenBookQA.
Contrary to other instructional models trained on mixtures of over 1,600
tasks, T0 was trained on a significantly smaller mix of 35 tasks of mainly QA
category with open domain of both input prompts and labels. Therefore, we
speculate that the superiority of T0 in concept learning may attribute to a
reliance of vast multi-task learners on spurious features such as the mapping
of the evaluation task to some of the previously seen tasks.

##  5 Conclusion

This work introduces a Concept-sharing few-shot learning task that reflects on
in-context learners’ ability to extract a specific reasoning concept from
demonstrations and apply it in a prediction. We assess a set of recent in-
context learners for this ability over a set of concepts extracted from human
explanations.

While we find each of the evaluated concepts to be informative for at least
one model, we show that most in-context learners can not benefit consistently
from all concepts. Despite that, we still observe some intriguing trends, such
as that the concept-learning ability does not appear to relate to the model
size, or that the T0 models are able to benefit from concepts much more
consistently than the models maintaining vastly larger mixtures of pre-
training tasks.

We believe that future work can inspire in the proposed approach of a more
detailed evaluation of the models’ behaviour. We trust that similar
evaluations, possibly scaled to a more comprehensive selection of reasoning
concepts, will allow us to better understand the capabilities of universal
language models and to refine our expectations of their behaviour accordingly.

## Limitations

#### Concepts

In this work, we extract the concepts from semi-structured explanations whose
format reassures consistency and non-ambiguity of the exploited concept(s).
The selection of datasets and corresponding concepts is primarily conditioned
by data availability, as the semi-structured explanations are available merely
for a small set of datasets.

We acknowledge that our selection of concepts is not representative for a vast
variance of concepts that users might expect models to learn from context in
interaction. Some important concepts’ features that we identify are following:
(i) a number of premises or reasoning inference steps needed to map the input
to output, (ii) the granularity of the reasoning steps, (iii) a type of the
premises; For instance, whether the familiarity with a given concept requires
a memorization of an entity property (such as “sun emits light”), or a
reasoning mechanics such as analogical reasoning (“if animals can run and cat
is an animal, then a cat can run”).

We invite future work to identify or propose a taxonomy that would better
reflect the wide variance of reasoning concepts that models are expected to
comprehend in order to serve a wide scope of unseen tasks. Such taxonomy can
motivate a more targeted collection of concepts from explanations, or
annotation of new explanations demonstrating new concepts.

#### Models

We acknowledge the limitation in a variance of evaluated models given by their
availability and our computational possibilities. We evaluate only two models
of the GPT family due to the usage limits of OpenAI API. Outside GPT models,
we do not evaluate models over 20B parameters, given the infrastructure
requirements of such settings. Nevertheless, we argue that the relevance of
the models with constrained access, or resource requirements exceeding the
limits of most organizations also remains a subject of open question.

#### Datasets

One should note that the sizes of our evaluation datasets, for which we are
able to extract concepts from explanations (Fig. 2), might be too small to
compare concept sensitivity in the absolute numbers. The sizes of our
sensitivity evaluation datasets are the following: WorldTree: 2,204 samples,
OpenBookQA: 792 samples, GLUE Diagnostics: 282 samples, HotpotQA: 182 samples.

## Ethical Considerations & Broader Impact

As outlined in Section 1, in-context learning recently presents a research
direction of broad public interest, where the outstanding results on NLP
benchmarks often do not meet the users’ expectations. It is understandable
that the focus of development in in-context learning LLMs goes to measurable
improvements on existing benchmarks, as ecologically-valid evaluations on end
use-cases are timely and difficult to compare to previous work (de Vries et
al., 2020).

Nevertheless, in this highly-exposed and fast-paced direction of research, we
identify the necessity for the emergence of fast proxy measures that can shed
light on the decision-making of the LLMs as expected by their end users.

The presented evaluation of models’ sensitivity to demonstrated reasoning
concepts introduces a technical framework for quickly assessing models’
compliance with our expected functioning; However, a selection of a
comprehensive set of concepts, that we can agree our models should be able to
learn, remains a subject of open discussion.

## Acknowledgements

We acknowledge the Centre for Biomedical Image Analysis at Masaryk University,
supported by MEYS CR (LM2023050 and CZ.02.1.01/0.0/0.0/18_046/0016045 Czech-
BioImaging) for their support in obtaining evaluations presented in this
paper.

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##  Appendix A Details of Concept-aware Evaluations

Figure 4: Conceptual few-shot evaluation: ROUGE-L of models using random
demonstrations (left) and demonstrations exploiting a concept of prediction
(§3.2; right). Boxes and confidence intervals cover 50% and 95% of the
bootstrapped results, respectively ($100$ samples, $200$ repeats). Models
marked with ∗ were exposed to the evaluation task (but not samples) in
training. Training datasets of GPT∗ models are unknown.

Unless stated otherwise, we evaluate all models over all datasets and both
demonstrations selection strategies consistently for ROUGE-L in default
settings of Lin (2004), using a number of demonstrations $k=3$ and contexts
constructed in the following format:

“Input: $x_{1}$ Prediction: $Y_{1}$ Input: $x_{2}$ Prediction: $Y_{2}$ Input:
$x_{3}$ Prediction: $Y_{3}$ Input: $x_{\text{pred}}$”

Among both random and concept-sharing evaluations, we share the same
$x_{\text{pred}}$ and only permute the demonstrations; We find cases where the
filtering of predicted samples ($x_{\text{pred}}$) to the ones sharing a
concept with sufficient amount of (3) different samples needed for
demonstrations makes the task systematically easier.

We diverge from the stated configuration only in the following cases:

  * •

Tk-Instruct-11B and HotpotQA: we limit the evaluation contexts to at most
3.500 unique words, as we can not fit longer contexts into the memory. This
might make the absolute results in this configuration overly optimistic, but
still comparable within the Conceptual few-shot evaluation.

  * •

GPT and HotpotQA: We completely exclude these evaluations given the fixed
context window size of these models will exclude the $x_{\text{pred}}$ from
prediction input in too many cases.

We choose evaluated GPT APIs based on OpenAI
documentation444https://beta.openai.com/docs/model-index-for-researchers,
picking for GPT and InstructGPT models marked as davinci and text-davinci-003.
Note that these identifiers might change in time, thus disallowing us to
guarantee the reproducibility of their evaluations.

##  Appendix B Computational Requirements

We run both training and evaluation experiments using single NVidia
A100-SXM-80GB. The time and computational requirements of evaluation depend
largely on the size of the evaluated model; We can evaluate the models up to
11B parameters on a single NVidia A100-SXM-80GB. The evaluation of Concept
Few-shot learning on all our datasets, together with the Random reference
evaluation takes approximately 2 hours for a 1B model.


††thanks: This research is based upon work supported in part by the ODNI,
IARPA, via ARO. The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the ODNI, IARPA, or
the U.S. Government.

# True Differential Superconducting On-Chip Output Amplifier

Jonathan Egan, Andrew Brownfield and Quentin Herr  Northrop Grumman Corp.,
Baltimore, MD 21240

(14 April 2021)

###### Abstract

The true-differential superconductor on-chip amplifier has complementary
outputs that float with respect to chip ground. This improves signal integrity
and compatibility with the receiving semiconductor stage. Both source-
terminated and non-source-terminated designs producing 4 mV demonstrated
rejection of a large common mode interference in the package. Measured margins
are $\pm$8.5% on the output bias, and $\pm$28% on AC clock amplitude.
Waveforms and eye diagrams are taken at 2.9-10 Gb/s. Direct measurement of
bit-error rates are better than the resolution limit of 1e-12 at 2.9 Gb/s, and
better than 1e-9 at 10 Gb/s.

On-chip output amplifiers convert the aJ signal-energy of superconductor
single-flux-quantum (SFQ) logic to levels that can be detected by standard
electronics. A design using a series-array of inductively-isolated, DC-biased
SQUIDs has proven to be uniquely effective Herr et al. (2007). The design is
amenable to distributed amplifier techniques for improved gain-bandwidth
product Herr (2010). An output voltage of 2 mV is adequate for direct
interface to a high-performance room-temperature LNA at a data rate of 10 Gb/s
Herr (2010). These findings have recently been verified independently Takeuchi
et al. (2017); Gupta et al. (2019). The only alternative is the SFQ-to-dc
converter Likharev and Semenov (1991), a smaller design that produces only 0.5
mV. This is effective at modest data rates, but 1-10 GHz rates require
multistage amplification at intermediate temperature stages Gupta et al.
(2019).

Reciprocal Quantum Logic (RQL) is an SFQ logic family that is uniquely
attractive in terms of power dissipation and latency Herr et al. (2011). RQL
is AC powered, which adds a significant constraint to the output link. The AC
power delivered at the clock frequency is typically 45-50 dB larger than the
output signals. These power levels produce strong interference in the
pressure-contact package, where isolation is imperfect. As the output
amplifiers have narrow operating margins on the applied dc bias current,
interference large enough to shift the bias and will not only obscure the
output signal but prevent it from being generated in the first place.

The solution is to make the driver differential. This is common practice in
high-speed interconnect due to its rejection of common-mode interference and
noise. Transistor-based designs typically produce the differential signals
with two independent drivers that contact chip ground. Amplitude and delay
must be well-matched, which presents a design challenge. The stacked-SQUID
design can be true-differential, meaning that the outputs are floating with
respect to chip ground. This is accomplished by wiring out both ends of the
series array. The true-differential design rejects common-mode interference
and noise on both the output signal and on the bias current.

A 2 mV output amplifier has been used routinely for lab demonstrations up to
3.5 Gb/s using high performance LNAs on the receiving end at room temperature.
The circuit converts the SFQ-based data encoding of RQL to return-to-zero (RZ)
encoding. An important feature of the output amplifiers described here is
conversion to standard non-return-to-zero (NRZ) encoding, which has twice the
bandwidth efficiency, enabling 10 Gb/s data rates. These designs produce 4 mV.
A future target is direct drive of a differential limiting amplifier (e.g.,
Hittite HMC750LP4) with improved size, weight, and power (SWAP). We report
measurements from a sampling of parts, including 10 GHz operation of a circuit
fabricated internally at Northrop Grumman in a Nb-based six-metal-layer
integrated circuit process.

##  I Design

Figure 1: True-differential output amplifiers driving load resistor $R_{L}$.
Both designs use a series-array of SQUID-based floating voltage sources
embedded in an $LC$ matching network. a) The source-terminated design has the
resistor $R_{S}=R_{L}$ in the center of the array. Half of the signal voltage
is dissipated across this resistor. Bias current flows through the optional
bypass inductor $L_{S}$. b) Without source termination, the design generates
the same voltage with only half as many stages. Both circuits have identical
JTL networks on either side of the stack to produce an NRZ signal. The second
JTL network is omitted from the figure for clarity.

The primary design trade-off is between source-terminated and non-source-
terminated. The non-source-terminated circuit is half the size and twice as
efficient. However, source termination is required to scale to the higher
power-gain product of distributed amplifiers, and to terminate the large
signal reflections at the receiving end typical of low-noise LNAs.

As shown in Fig. 1, both designs consist of a series array of SQUIDs connected
by an $LC$ network, as with the previously reported distributed amplifier Herr
(2010). The stack is DC-biased with current flowing from the +V to -V nodes.
Each SQUID is a low-impedance, floating voltage source producing a sub-mV
level. These voltages sum at the output. The lumped inductance and capacitance
at every stage produces a filter matched in frequency (and impedance) to the
output waveform, and mismatched to the internal oscillatory modes in the SQUID
and between multiple SQUIDs. This is the essential function of inductive
isolation Herr et al. (2007).

The source-terminated design is effectively two single-ended distributed
amplifiers placed back-to-back. Signal propagation time of $\sqrt{LC}$ per
stage produces a distributed design that exceeds the bandwidth-gain product of
the individual stages. By timing the trigger for each stage, the total rise
time can be as small as that of a single stage. The differential signals
generated in the array or reflected off the load are equal in amplitude and of
opposite polarity as seen by the source termination. This produces a virtual
ground, without explicit grounding. The bypass inductor eliminates power
dissipation of the bias current. To prevent distortion, the $L_{S}/R_{S}$ time
constant must be large enough to outlast DC offsets that occur in the data
pattern, e.g. in a long series of “all-1’s” or “all-0’s”.

The non-source-terminated design produces the same output voltage with half
the number of stages. This design also has a virtual ground across the central
inductor, but here differential signals will be reflected, not terminated.
Without source termination, the design is not amenable to the sequential
timing of the distributed amplifier. Instead, the fastest risetime comes by
triggering the stack broadside. While this limits the gain-bandwidth product,
the design is efficient for modest data rates and output amplitudes.

Figure 2: SQUID design. a) The four-junction SQUID has two independent input
transformers. The numbers label the switching order of the Josephson junctions
when the current goes high in the transformer on the left. b) SFQ pulses are
applied using RQL data encoding from the left (blue) and right JTLs (red) with
a relative delay of 180∘, or half a clock cycle. The integral of the voltage
transients is superconductor phase, equal to current through the primary. The
resulting output voltage (purple) is a sum of the two phase states. The
transient glitches in the output waveform at the input transitions are out-of-
band of the output.

Here we report two circuits designed to produce 4 mV with 50 ps risetime,
supporting data rates of 10 Gb/s. Both circuits convert RQL data encoding to
standard NRZ-encoded data at the output, with a fundamental frequency that is
half the data rate. As shown in Fig. 2, the conversion is done using a four-
junction SQUID that couples signals in through two independent transformers.
The RQL pattern applied to the left is delayed by half a cycle and reapplied
on the right. The sum of these two return-to-zero (RZ) voltage waveforms
produces an NRZ waveform. The design uses two separate JTL networks to feed
the output. The feeding JTL and the SQUID are isolated from each other using
an unpowered junction. This isolation reduces AC power leakage from JTLs to
the SQUID, and reduces back-action from a selected SQUID to the JTLs. A flux
bias produces the desired symmetric double-well potential so that the
unpowered junction can be triggered.

Advantages of the four-junction SQUID over a conventional two-junction design
include: 1) Smaller junctions and larger inductors in the SQUID produce more
efficient transformer coupling. These designs did not use holes in the ground
plane under the transformers in physical layout. 2) More junctions in the
SQUID improves current compliance. Total critical current of 400 $\mu$A
produced the necessary 40 $\mu$A output signal swing, corresponding to 4 mV
into the 100 $\Omega$ differential load.

The disadvantage of the four-junction SQUID is relatively modest voltage,
0.25-0.30 mV per stage using Josephson junctions with critical current density
of 100 $\mu$A/$\mu$m2. With two inputs, the SQUID is only ever partially
selected. As illustrated in Fig. 2 by the start-up switching order of the
junctions, the selected side of the compound SQUID effectively drives fanout
to the other side.

Table 1: Parameters of RQL output amplifiers Output | Source | SQUID | Area | Power | Output | Effcy
---|---|---|---|---|---|---
(mV) | Term | ($\times$) | ($\mu$m2) | (nW) | (nW) | (%)
2∗ | No | 8 | 8k | 40 | 10 | 20
4 | No | 16 | 61k | 160 | 80 | 33
4 | Yes | 32 | 190k | 320 | 80 | 16
8 | Yes | 64 | 240k | 1280 | 320 | 16
∗This amplifier produces RZ output. All others, NRZ.

Only the two junctions on the ends of the four-junction SQUID are resistively
shunted. As reported earlier Herr et al. (2007); Herr (2010) such creative
damping increases the output voltage and the reliability of the transition
back to zero. Total damping is $I_{c}R_{s}=1.2$ mV where $R_{s}$ is the local
parallel shunt resistance and $I_{c}$ is the total critical current of the
SQUID. This is well above the critical damping value of 0.7 mV. However, the
whole stack is also shunted by load resistance and source termination
resistors, with a total effective damping of $I_{c}R_{s}=0.8$ mV, This
approximates critical damping. Simulated margins on the dc bias of the SQUID
stack are about $\pm$10%.

The 4 mV design balances speed, output amplitude, and chip area. Parameters of
the 4 mV design, in comparison to 2 mV and 8 mV, are given in Table 1. RQL
output amplifiers are quite efficient with on-chip power dissipation on a nW-
scale, which is negligible compared to the heat load of the cabling. On-chip
area of 61k $\mu$m2 is dominated by that JTL feed network, and is subject to
further miniaturization in advanced fabrication processes. Even the largest
design is about the same area as that of the associated signal and ground pads
in the current package.

##  II Measurements

All test circuits had a differential input transformer and JTL chain Herr et
al. (2011) connected to the output amplifier. Circuits were powered using a
resonant network that provisioned the active area of each chip Talanov and
Strong (2017).

True-differential output amplifiers have been used on nearly all RQL digital
designs, so various configurations, clock frequencies, and fabrication
processes have been exercised. The circuits presented here were fabricated
either in the process supplied by D-Wave Berkley et al. (2010), or using the
“RQL25” process developed internally at Northrop Grumman. Both processes have
six metal layers and Josephson junctions with 100 $\mu$A/$\mu$m2 critical
current density, and support similar layout style with 0.25 $\mu$m minimum
feature size.

Figure 3: Output channel. Differential outputs are connected to micro-coax
cables via pressure contacts in a liquid He dip probe. DC bias is delivered to
the output SQUIDs via the bias-Tees. The 3 dB attenuators (50 $\Omega$)
improve the impedance match, and are used to test non-source terminated
amplifiers. Signals are amplified by Miteq JSMF4-02K180-30-10P LNAs at the
probe head. The LNAs fed a Marki BAL-0026 balun to produce signal-ended
signals for the lab instruments.  Figure 4: Measured waveform at 3.5 Gb/s of
the source-terminated circuit fabricated at D-Wave. Transition rates are
limited to 320 ps by a low-pass filter added to the output channel, not by the
part itself. In this and subsequent figures, the waveform is a signal-averaged
“4-3-2-1” chirp pattern (repetitive 11110000111000110010), and the amplitude
is that seen at the instrument after arbitrary levels of amplification.

Figure 5: Measurement at 2.9 Gb/s of the non-source-terminated circuit
fabricated in RQL25. a) Output amplitude as a function of bias current. The
rectangle corresponds to bias excursions of $\pm$8.5% that produce output
amplitude 20% less than nominal. b) The waveform is the signal-averaged
4-3-2-1 chirp pattern defined in the previous figure. c) An eye diagram
measured for a PRBS15 pattern. Data eye analysis on the Keysight J-BERT 8020A
shows an open eye directly measured down to a BER of 1e-8, and extrapolated
down to 1e-15 at the inner contour.

The output amplifiers were tested using the channel shown in Fig. 3. The room-
temperature LNAs used to amplify the on-chip signal have a typical VSWR
specification of 2:1, meaning up to half the signal amplitude is reflected.
For the non-source-terminated circuits, $-$3 dB attenuators were used to
improve the impedance match at the expense of cutting the signal-to-noise
ratio in half. These attenuators were not used for the source-terminated
circuits.

Figure 6: Measurement at 9.95 Gb/s of the non-source-terminated circuit
fabricated in RQL25. a) The waveform is again the signal-averaged 4-3-2-1
chirp pattern. b) An eye diagram measured for a PRBS15 pattern. Data eye
analysis on the Keysight J-BERT 8020A shows an open eye directly measured down
to a BER of 1e-9. The secondary rising edge in the eye corresponds to the
isolated “101” token.

Here we present measurements of waveforms, eye diagrams, and bit-error-rates
(BER). Waveform captures are useful as an initial test of signal integrity, as
clock and data feed-through, and signal reflections are visible by inspection.
Eye diagrams are useful to characterize bandwidth and signal-to-noise of the
output data link as a whole. Direct measurements of BER are required to detect
switching errors in the output amplifier. This is particularly motivated by
the history of elevated switching errors associated with superconducting
latching logic and output amplifiers Fulton and Dynes (1971); Suzuki et al.
(1988). Underlying each voltage waveform is a large, integral number of SFQ
pulses, and sometimes the circuit cannot “decide” exactly how many pulses to
produce. This is a manifestation of Buridan’s Principle Lamport (2012).

Fig. 4 shows the measured waveform of a 4 mV source-terminated output
amplifier at 3.5 Gb/s. The data show good signal integrity without clock feed-
through, confirming effectiveness of the true-differential design. This
compares quite favorably to previous results with a single-ended output, where
the clock feed-through was as large as the signal Herr et al. (2013).

Fig. 5 shows the measured output amplitude, waveform and eye diagram of the 4
mV non-source-terminated circuit at 2.9 Gb/s. Output amplitude depends on the
DC bias current applied to the SQUID stack. The peak-to-peak voltage after
amplification is 0.10 V at the nominal bias of 0.30 mA. Bias margins of
$\pm$8.5% are measured for output voltage within $\pm$20% of nominal. Margins
on the AC clock that powers the JTL feed network were measured to be 5.0 dB,
or $\pm$28% in amplitude. This agrees with simulation. The waveform shows good
signal integrity, without clock feed-through. The eye diagram at 2.9 Gb/s
shows good performance, as BER is directly measured down to 1e-8 and
extrapolated down to 1e-15. Additional direct measurements of BER using the
PRBS15 pattern found the data link to be error-free down to the resolution
limit, 1e-12.

Fig. 6 shows the measured waveform and eye diagram of the same 4 mV non-
source-terminated design, but now clocked at 9.95 Gb/s. The waveform again
shows good integrity without clock feed-through, but shows reduced amplitude
for the standalone “101” token, attributable to an observed risetime of 60 ps,
which is a little slower than the 50 ps design value. This token is visible as
a secondary rising waveform in the eye diagram. The amplified waveform is
clipped by the instrument, as the 2 V peak-to-peak input has an apparent
amplitude of about 1.6 V peak-to-peak in the eye diagram. Despite this test
artifact, BER was directly measured down to 1e-9.

##  III Conclusion

The 4 mV true-differential output amplifier achieves low BER for direct-to-
room-temperature data rates up to 10 Gb/s, in the presence of large common-
mode interference in the package. Measured margins of $\pm$8.5% on the output
bias and $\pm$28% on AC clock amplitude indicate that this component will not
limit the performance of larger digital designs. Design trades exist depending
on rate, signal amplitude, and the input impedance match at the receiving
second-stage amplifier. The simpler non-source-terminated design can be used
where explicit attenuation is added to the LNA inputs to improve the impedance
match, but this comes at the expense of reduced signal power. Source
termination handles the reflections typical of low-noise LNAs, but requires
twice as much voltage generated on chip as only half of signal goes to the
load. Source-termination also enables distributed-amplifier designs that scale
to higher data rates at higher output amplitude. Direct interface to a second-
stage differential limiting amplifier is a goal for the future.

###### Acknowledgements.

Many people contributed to the design, fabrication, and test of these
circuits. We acknowledge James Wise and Deepal Wehella Gamage for design,
Chris Kirby for fabrication, Jose Ibarrondo, Mohammed Lateef, Yamil Huertas
Morales and Jose Robinson for test support, and Anna Herr for useful
discussions.

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