Fast and Flexible Selection with a Single Switch

Tamara Broderick1,∗, David J. C. MacKay2

1 Department of Statistics, University of California, Berkeley, California,
USA
2 Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
$\ast$ E-mail<EMAIL_ADDRESS>
## Abstract

Selection methods that require only a single-switch input, such as a button
click or blink, are potentially useful for individuals with motor impairments,
mobile technology users, and individuals wishing to transmit information
securely. We present a single-switch selection method, “Nomon,” that is
general and efficient. Existing single-switch selection methods require
selectable options to be arranged in ways that limit potential applications.
By contrast, traditional operating systems, web browsers, and free-form
applications (such as drawing) place options at arbitrary points on the
screen. Nomon, however, has the flexibility to select any point on a screen.
Nomon adapts automatically to an individual’s clicking ability; it allows a
person who clicks precisely to make a selection quickly and allows a person
who clicks imprecisely more time to make a selection without error. Nomon
reaps gains in information rate by allowing the specification of beliefs
(priors) about option selection probabilities and by avoiding tree-based
selection schemes in favor of direct (posterior) inference. We have developed
both a Nomon-based writing application and a drawing application. To evaluate
Nomon’s performance, we compared the writing application with a popular
existing method for single-switch writing (row-column scanning). Novice users
wrote 35% faster with the Nomon interface than with the scanning interface. An
experienced user (author TB, with $>$ 10 hours practice) wrote at speeds of
9.3 words per minute with Nomon, using 1.2 clicks per character and making no
errors in the final text.

## Introduction

In single-switch communication, user input consists of repeated clicks,
distinguished only by timing information; these clicks might be generated by
pressing a button or blinking. For instance, the range of movement of
individuals with severe motor impairments may be limited to a single muscle.
Alternatively, a crowded or jostled mobile technology user may be able to
click precisely while other actions are difficult or sloppy. A single switch
may also be useful when information conveyed, such as a PIN, is sensitive and
hand location on a normal keyboard might betray this content. Our method,
Nomon (Figures 1, 2), expands the application scope of existing methods and
facilitates faster writing than the most common single-switch writing
interface.

Existing single-switch communication methods include scanning [1, 2, 3, 4, 5,
6, 7, 8, 9] and One-Button Dasher [10, 11, 12, 13, 14, 15]. (Morse Code does
not fall under the strict definition of a single switch interface since it
requires either click duration information or multiple switches.) Scanning is
the most popular single-switch selection method. In a scanning interface,
options such as letters are arranged in a grid (Figure 3). For standard row-
column scanning, each row of the grid is highlighted in turn, with the
highlight moving to the next row at fixed time intervals, a.k.a. scanning
delays. When a click is made, the columns of the selected row are then
highlighted in turn, typically iterating at the same fixed time intervals. To
select a column, and thereby make a final selection, the user clicks when the
highlight is on that column. A variety of customizable commercial scanning
software exists for writing and computer navigation [16, 17, 18, 19] although
customization is often not single-switch accessible. The Gnome Onscreen
Keyboard [19], by contrast, can generate a grid for new applications “on the
fly.”

While the scanning method can be used to select anything that can be arranged
in a grid, One-Button Dasher is limited to writing with alphabetic character
sets. Dasher works by arranging all possible character strings in alphabetic
order and having the user zoom in on the desired string. More likely strings,
according to the language model, are given relatively more space and are thus
easier to select.

Scanning and One-Button Dasher require options to be arranged in a particular
configuration. By contrast, traditional operating systems, web browsers, and
free-form applications such as drawing place options at arbitrary points on
the screen. Scanning, the most popular single-switch communication method, is
limited in further ways by its grid structure. For instance, the grid options
may theoretically be reordered after any selection to allow the most likely
options to be selected the most quickly. However, in practice this reordering
requires that users either learn many grid arrangements or search the grid for
their desired option upon each reordering.

Even scanning a grid that maintains a fixed layout at all times has drawbacks.
Previous studies suggest that, at least among children, scanning a fixed grid
demands a higher cognitive load than direct selection [20, 21, 22, 23]—though
an earlier study found no difference [24]. One implicated factor is the need
for a user to divide her attention between the scanning highlight and the
desired option [20, 23]. Another issue in scanning is the possibility of
distraction, and loss of the target from working memory, while highlighting
progresses [25, 23].

Therefore, we seek a single-switch selection method that is not limited to
certain forms of option placement. We want our method to work for any number
of options; to be able to effectively reorder the set of selections without
imposing additional cognitive load; and to allow the user to attend only to
the desired target.

Below, we begin by describing such a method, which we call “Nomon.” We also
describe how our method can adapt to individuals’ clicking abilities and how
it can incorporate prior beliefs about option selection frequency. In order to
evaluate our method’s performance, we note that much single-switch research
has focused on optimizing writing speed [1, 2, 3, 4, 5, 6] and the number of
clicks per output symbol [7, 8, 9] in scanning interfaces. In light of these
studies, we developed a writing application, the Nomon Keyboard (Figure 2),
using our method and compared its performance with a popular commercial
scanning interface, The Grid 2 [16] (Figure 3). We examined the study
participants’ writing speeds, error rates, and number of clicks made per
character as well as the subjective ratings of their experiences.

The full technical report describing Nomon is available online at
http://www.inference.phy.cam.ac.uk/nomon/files/nomon_tech_report.pdf. The
Nomon Keyboard, as well as a drawing application (Nomon Draw) and instructions
for the use of both applications, is available for download at
http://www.inference.phy.cam.ac.uk/nomon/ under the GNU General Public License
3.0.

### A New Method

Figure 1: An example Nomon application for selecting between 16 points on
screen (screenshot). The horizontal and vertical positions of the option
points were chosen uniformly at random in the box shown to illustrate the
flexibility of the method.

Nomon, a new single-switch communication method, does not limit the user to
selecting options that can be arranged in a grid or alphabetically. Rather, it
can be used to select among any points of interest on a screen. The trademark
of a Nomon application is a set of small clocks, one clock associated with
each selectable option. Each clock appears alongside its corresponding option
on the screen. For instance, Figure 1 illustrates clocks corresponding to 16
arbitrary option locations. Another example might be a drawing application
where a clock appears at every “pixel” on the canvas and also next to each
menu option. In a writing application (the Nomon Keyboard), a clock appears
next to each character, word completion, or text editing function (Figure 2).

Figure 2: The Nomon Keyboard, a writing application (screenshot). Words that
are prefixed by the concatenation of the current context and the letter X
appear next to the letter X. Underscore represents a space. Options for
period, a character-deletion function, and an undo function are also
available.

Just as menu options and drawing tools in a point-and-click interface are
accessed in the same way by the mouse, all Nomon clocks are selected in the
same way by a single switch. Each Nomon clock features a moving hand and a
fixed line at noon. All moving hands rotate at the same, fixed speed but, at
any time, are located at a variety of angles relative to noon. The user tries
to click precisely when the moving hand on her desired clock is at noon. She
repeats this action until the clock is selected. Selection is signalled by the
desired clock being highlighted with a darker color and the entire application
flashing a lighter color; there may also be audio feedback. Between clicks (if
more than one click is required to select a clock), the clock angular offsets
are adjusted by a heuristic to maximize the expected information content of
the user’s next click.

Row-column scanning can be viewed as a special case of the Nomon selection
method where clocks are arranged in a grid, moving-hand angular offsets are
aligned alternately across rows and columns of clocks, and each selection is
based only on the times of the last two clicks. But this synchrony does not
take full advantage of the continuous, periodic representation of the clock
and imposes an order on the set of options relative to their positions
onscreen. Rather, by allowing more general clock hand positions, we can,
effectively, completely reorder the set of selections after each click without
demanding any extra cognitive load from the user.

Similarly, the independent movement of the clock hands frees the user to
attend only to the desired target, in contrast to the need, in scanning, for
the user to attend both to the desired target and the moving highlight.
Further, the scanning user may forget her target as highlighting progresses.
But in Nomon, once the target is located visually, the user is free (without
suffering a performance penalty) to focus on selecting a single, fixed clock.
Since the clock periods are usually much shorter than a full scanning
rotation, there is also no significant penalty for missing a potential click
time.

In Nomon, by contrast with scanning, we assume that the user will not always
click perfectly at the desired time. The details of Nomon operation are
described more fully in the Nomon Operation section below and outlined here.
Nomon can learn a user’s probability of clicking at different (typically
small) offsets relative to noon. This learning is accomplished via an
approximate Parzen window estimator, with contributions from more recent
clicks weighted more strongly to allow adaptation to a user whose skill
changes over time. We can also specify a prior probability distribution over
clocks according to a predictive model of user choices. For instance, in the
writing application tested below, our language model assigned prior
probabilities to letters and word completions based on the British National
Corpus word-frequency list [26]. These prior probabilities could also be
adaptive and context dependent.

During a particular selection process, the posterior probability of any clock
given the clicks thus far can be calculated from Bayes’ theorem. When the
probability of a single clock is sufficiently high, we declare it the winner.
The probability threshold for winning is an adjustable parameter of the model;
it can vary according to context or from clock to clock. A higher threshold
can ensure greater safety for critical actions.

## Results

Figure 3: The scanning grid from The Grid 2 used in this comparison study
(screenshot). The six long rectangles on the left hold word completions. The
remaining options are fixed and include letters, an underscore for space, a
period, a character-deletion function, and a word-deletion function.

We developed a writing program using the Nomon method, the Nomon Keyboard
(Figure 2), and conducted a study to compare writing with Nomon to writing
with a popular commercial scanning interface, The Grid 2 [16] (Figure 3). To
that end, sixteen study participants with no previous experience of either
interface wrote with Nomon and The Grid 2. In each of two sessions, a
participant used one of the interfaces to write short phrases appearing on
screen. A session was divided into four blocks, each lasting approximately
$14$ minutes. During the first three (of four) writing blocks, each
participant was allowed to adjust the rotation-period or scanning-delay
parameter, as appropriate to the current interface, at the end of each written
phrase. No changes were allowed during the final block. For each interface,
cash prizes were won by the faster half of participants in the final block.

In total, we collected 34 hours of data from 16 novice participants and one
experienced single-switch user (TB, with $>10$ hours experience in each
interface). We compared three objective measures of the novice participants’
performance between the two interfaces: text-entry rate, error rate, and click
load (clicks per character). We also examined subjective ratings of the two
interfaces given by the novice participants.

### Text-entry Rate

Figure 4: Mean entry rate (left) and click load (right) across interface
blocks. Mean entry rate is measured in words per minute, and click load is
measured in clicks per output character. In both panels, error bars represent
95% confidence intervals for the novice user means, and the average
experienced user (TB) performance is illustrated by horizontal lines for
comparison.

We calculated text-entry rate in words per minute, where a word is defined as
five consecutive characters in the output text. At the beginning of each
fourteen-minute block, the participants were asked to write two periods “..”
using the interface for that session. This action signalled that they were
ready to begin and initiated the display of the first target phrase. Timing
started once the two periods were written. After every phrase, participants
wrote two periods to signal that they were ready for a new phrase. Timing
stopped after the final two periods following the last phrase were written.
All periods except the first two in a block were counted as characters in what
follows, and the time spent writing them was counted as well.

The left panel of Figure 4 shows the novice participants’ mean entry rates
across the four blocks for each interface. Also shown, for comparison, is the
performance of the experienced user. Participants wrote faster with Nomon than
with The Grid 2 during the first block ($F_{1,15}=129$, $p=9.3\cdot 10^{-9}$).
The total session time was short for both interfaces, but participants’
writing speed with each interface improved with practice. Participants became
faster at writing using the Nomon Keyboard during the Nomon session
($F_{3,45}=59$, $p=1.4\cdot 10^{-15}$) and became faster at writing using The
Grid 2 during the scanning session ($F_{3,45}=122$, $p<10^{-15}$). In the
final block we see that participants remained faster at writing with Nomon
than with The Grid 2 ($F_{1,15}=135$, $p=6.8\cdot 10^{-9}$). In this fourth
block, participants wrote $35\%$ faster with Nomon than with the scanning
interface; participants were writing at $4.3$ words per minute on average with
The Grid 2 and $5.8$ words per minute with the Nomon Keyboard. The experienced
user wrote, on average, at $9.3$ words per minute with the Nomon Keyboard and
$5.9$ words per minute with The Grid 2.

While the alphabetic layout was easy for novices to use, a computer simulating
writing from a conversational corpus with no errors has been shown to achieve
a $19\%$ faster writing speed with a frequency-ordered layout than with an
alphabetic layout [6]. Even if we artificially inflate the novice writing
speeds using The Grid 2 by $19\%$, novices remain faster at writing with Nomon
($F_{1,15}=19.14$, $p=5.4\cdot 10^{-4}$).

### Error Rate

To find the error rate during a block, we begin by computing the character-
level Levenshtein distance [27] $d_{i}$ between the $i^{\rm th}$ target phrase
in the block and the text written by the participant; $d_{i}$ is also known as
the edit distance. We define the error rate for the block to be
$\sum_{i}d_{i}/\sum_{i}n_{i}$, where $n_{i}$ is the number of characters in
the $i^{\rm th}$ target phrase.

The average novice character-level error rate (over all blocks) for the Nomon
Keyboard was $0.43\%$, and the average novice error rate for The Grid 2 was
$0.34\%$. There was no significant difference in novice error rate between the
two interfaces ($F_{1,15}=0.71$, $p=0.41$). The experienced user made no
errors while using Nomon ($\sum_{i}d_{i}=0$) and made one error while using
The Grid 2, for a mean scanning block error rate of $0.06\%$.

We believe that the participants’ output errors were mostly caused by poor
recall of the target sentence. For instance, one participant pluralized “head”
in “head_shoulders_knees_and_toes” and wrote “reading_week_is_almost_here”
instead of “reading_week_is_just_about_here”.

### Click Load

The click load is the number of clicks per output-text character. Other names
for this measure include “keystrokes per character” [28] and “gestures per
character” [14]. The click load is calculated as the number of button presses
in a block divided by the number of characters in the output. Clicking often
can be tiring for any user and especially so for some users with specific
motor impairments.

While the inclusion of word-completion options in a scanning grid has been
shown to have no positive effect on writing speed with a scanning interface
[7], other studies confirm that word completion options yield substantial
click-rate savings over the baseline (mistake-free) row-column click load of
two clicks per character [8, 9]. Therefore, we included six word-completion
options in the leftmost row of our scanning grid (consistent with the default
layouts in The Grid 2 [16]). These were ordered from top to bottom and filled
in automatically by the software.

Click loads are illustrated in the right panel of Figure 4. The average novice
rate (over all blocks) for the Nomon Keyboard was $1.58$ clicks per character,
and the average novice rate for The Grid 2 was $1.55$ clicks per character.
There was no significant difference in novice click load between the two
interfaces ($F_{1,15}=0.49$, $p=0.49$).

While the experienced user required, on average, $1.51$ clicks per character
in The Grid 2, she required only $1.18$ clicks per character using the Nomon
Keyboard. For comparison, writing with the same character set on a normal
keyboard requires at least one key press for each character and thus at least
1 click per character (possibly more due to error correction). To compare to
Morse code, we find letter, space, and period frequencies directly from our
phrase set. We assume the Morse encoding of [17, 29]. In this case, an error-
free Morse code click load estimate is $3.0$ clicks per character. This load
is over twice as high as the click load of the experienced user on the Nomon
Keyboard.

### Subjective Ratings

We assessed novice participants’ opinions with a questionnaire immediately
after writing with an interface was completed. The questionnaires for each
interface were identical (except for the name of the interface). Participants
were asked to rate how much they agreed with a series of statements on a scale
from 1 (strongly disagree) to 7 (strongly agree). These statements were
largely the same as those in [30]. Participants were encouraged to write any
thoughts about the interfaces in an “Open Comments” box.

Participants’ responses to selected statements are summarized in Table 1. Not
only did participants like using the Nomon Keyboard in aggregate, but every
participant individually liked using Nomon at least as much as The Grid 2.
Contributing factors for why the Nomon Keyboard was preferred became apparent
in the remaining responses. Participants found it easier to select word
completions and easier to correct errors with the Nomon Keyboard. These
responses corroborate our objective findings above.

While many written comments agreed with participants’ numerical ratings,
unique to the open comments section was the sentiment that Nomon looks unusual
at first but is worth getting to know. One participant remarked,
“Surprisingly, I found this more user-friendly.” Another noted, “The writing
system looks intimidating when it first comes up on screen but is actually
very easy to use.”

Table 1: Subjective ratings of the two interfaces by novice participants.

Statement | Nomon | The Grid 2
---|---|---
| mean (sd) | mean (sd)
I liked writing using X. |  5.6 (1.4) | 3.9 (1.5)
It was easy to select word completions (the, and, cat, …). |  6.1 (0.7) | 4.8 (1.3)
It was easy to correct errors. |  4.5 (1.8) | 3.9 (1.7)

Each response to the lefthand statements was on a scale from 1 (strongly
disagree) to 7 (strongly agree). In the questionnaires, the interface name was
substituted for X. Mean responses are shown with standard deviations in
parentheses. Boldface is used to highlight the means corresponding to a more
positive user experience.

## Discussion

Nomon benefits in this comparison from its nice scaling properties and clock-
position flexibility. Our posterior-based selection method implies that the
time taken to make a selection in Nomon scales logarithmically with the number
of clocks if the prior over clocks is uniform. The entropy of the discrete
uniform distribution, which happens to be the highest-entropy (finite)
discrete distribution, scales logarithmically with respect to the number of
points in the support. Figure 5 shows that, generally, $2$ clicks are required
by an experienced Nomon user (TB, with $>10$ hours experience) to make a
selection in a 30-clock application. In a Nomon application with uniform prior
and $401$ clocks, $3$ clicks are generally required for this user to make a
selection. The difference in entropy between the prior for the $401$-clock
application and the highest-entropy prior for the $30$-clock application is
about $3.5$ bits, in agreement with $\log_{2}(401/30)=3.7$.

Not only does the number of clicks to selection in Nomon scale well, but
including additional options with small prior probabilities has little effect
on clicks-to-selection for more-likely clocks. Therefore, we could place many
more word completions on screen than would be feasible for a scanning
interface. We limited ourselves to three per character so as to allow fast
reading of the three relevant options. Placing word completions next to
letters in Nomon was feasible since clock position onscreen does not affect
Nomon operation. Interspersing word completions with letters in row-column
scanning would increase the number of scanning steps required to reach many
options.

While a Nomon writing application allows a straightforward comparison of Nomon
with existing single-switch communication methods, the Nomon selection method
is not limited to writing. For example, Nomon can be used for internet
browsing by placing a Nomon clock next to each link. Or Nomon can be used for
drawing by placing a dense grid of, say, hundreds of clocks on a canvas. (The
Nomon Draw application works in this way.) A user can draw a line by selecting
points directly from the canvas. Options for colors, shape drawing, saving,
and printing can likewise be accessed with clocks. A general graphical user
interface can be navigated with Nomon by placing clocks at the points where a
user might traditionally point and click.

It is worth pointing out that the flexibility of Nomon is not specific to our
clock display choice. Other local periodic representations of the global set
of options would also allow the arbitrary placement of options onscreen. For
instance, the clocks could be replaced by bouncing balls at different points
in their trajectories; instead of clicking at noon, the user would click when
the desired ball hits the ground. It remains to be studied whether such
alternative display choices might facilitate even faster or easier use of this
system.

Figure 5: Entropy of the estimated probability distribution over clocks for
two Nomon applications. Entropy is shown as a function of clicks remaining to
selection. Each solid line represents a single selection process. Dotted lines
decreasing to zero at respective rates of $1$ (lower) and $3$ (upper) bits per
click are illustrated for reference. Left: 25 selections on the Nomon
Keyboard: 30 clocks, non-trivial prior $p(c)$, clock period $2.0$ seconds,
switch input from joystick button. Right: 25 selections on another Nomon
application: 401 clocks, uniform prior $p(c)$, clock period $2.0$ seconds,
switch input from space bar. Data was generated by the experienced user (TB).

## Materials and Methods

We begin by detailing the experimental method used in the study above and
follow with a description of how Nomon functions.

### Experiment

#### Participants

We recruited sixteen participants from the university community across a wide
range of academic disciplines. All participants gave written informed consent.
In accordance with the University of Cambridge ethical review procedure as
defined in the Cambridge Psychology Research Ethics Committee Handbook
(http://www.bio.cam.ac.uk/sbs/psyres/), the experimental design received an
internal peer review within the department, where it was decided that ethics
approval from the committee was not necessary.

The participants’ ages ranged from 22 to 39 (mean = 26, sd = 4). Eight were
women, and eight were men. Participants were screened for motor or cognitive
difficulties; in particular, no participant had dyslexia or RSI. None of the
participants had used a scanning or Nomon interface before. No participant had
regularly used any single-switch interface before. Twelve of the participants
had used word completion (e.g. on cell phones).

In addition to the sixteen novice participants, an experienced user of Nomon
and The Grid 2 ($>10$ hours writing with each interface) was run through the
same experimental procedure for comparison.

#### Apparatus and Software

All sessions were run on a Dell Latitude XT Tablet PC with a partitioned hard
drive. The 12.1 inch color screen had a physical screen size of 261 $\times$
163 mm. The single-switch hardware device in all cases was the trigger button
of a Logic3 Tornado USB joystick. Participants operated the trigger button
with the first finger of their left or right hand. None of the other joystick
inputs was used. For both writing interfaces, automated spoken feedback was
provided as the user wrote.

##### Nomon Keyboard

We ran the Nomon Keyboard (Figure 2) on an Ubuntu 8.10 operating system
running the Linux kernel. The screen resolution was 1280 $\times$ 800 pixels,
and the physical size of the keyboard display was 224 $\times$ 85 mm (1125
$\times$ 416 pixels). The interface was docked in the upper part of the
screen. A text box and phrase box were located below the keyboard in the same
window. The keys of the keyboard were arranged in six rows and five columns.
Each key contained a principal character, with letters in alphabetical order
(across and then down) first, followed by four special characters: an
underscore (representing space), a period, a character-deletion function, and
an undo function. Each letter key also contained up to three word completions.
The undo function undid the previous selection if it was a character
selection, word-completion selection, or deletion.

The clock rotation period $T$ could be set to $2.0\cdot 0.9^{j}$ seconds for
$j\in\\{-4,-3,\ldots,18\\}$. Higher $j$ corresponded to faster rotation. The
initial setting of the period for novices was $T=2.0$ ($j=0$). The experienced
user initially chose $T=1.06$ ($j=6$).

##### The Grid 2

We ran The Grid 2 (Figure 3) on a Windows Vista Service Pack 1 operating
system. The screen resolution was again 1280 $\times$ 800 pixels. The physical
size of The Grid 2 display, using the scanning grid we designed for this
experiment, was 261 $\times$ 102 mm (1280 $\times$ 500 pixels). The interface
was docked in the upper part of the screen, and the text box and phrase box
were docked immediately below. Six word-completion boxes appeared on the left
side of the main interface. The remaining space was divided into six rows and
five columns of keys. Each key contained a single character. First were
letters arranged in alphabetical order (across and then down), followed by an
underscore, a period, a character-deletion function, and a word-deletion
function.

The Grid 2 allowed scanning delay values $d$ at $0.1(10-j)$ for
$j\in\\{\ldots,-1,0,1,\ldots,9\\}$. Higher $j$ corresponded to faster
scanning. The initial setting of the delay for novices was $d=1.0$ ($j=0$).
The experienced user initially chose $d=0.5$ ($j=5$).

#### Procedure

The experiment consisted of two identical sessions, one for each interface.
The starting interface was balanced across participants, and sessions were
spaced at least four hours apart.

Each session proceeded according to the same schedule. The first ten minutes
were introductory. First, the supervisor either explained or reviewed the
experimental procedure according to the session number. Then the participant
was shown how to use one of the interfaces. The demonstration included basic
writing, word completion, and error correction.

The next hour was divided into four 14-minute blocks, separated by short
breaks. During the blocks, participants were asked to write phrases drawn from
a modified version of the phrase set provided by [31], with British spellings
and words substituted for their American counterparts. For each participant, a
different random ordering of the initial phrase set was generated. Phrases
appeared one at a time in the phrase box at the bottom of the screen. Once a
participant finished a phrase, writing the period character twice would cause
a new target phrase to appear and the text box to empty. Participants were
instructed that no changes relevant to a particular phrase could be made after
the two periods were written.

During the first three (of four) writing blocks, each participant was allowed
to adjust the rotation-period or scanning-delay parameter at the end of each
written phrase. In particular, immediately after writing two periods and
receiving the new target phrase, the participant could increment or decrement
$j$ (defined above) by one. The experienced user incremented to $j=7$
($T=0.96$) after two blocks using the Nomon Keyboard and incremented to $j=6$
($d=0.4$) after two blocks using The Grid 2. No other changes were made by
this user.

Novice participants were paid £10 for each of the two sessions; the
experienced participant was not paid. Novice participants were informed at the
beginning of the study that they could receive a £5 bonus for achieving a
writing speed among the top half of novice participants for each interface.
They were further informed that, for the purposes of the bonus, writing speed
would be measured only during the final writing block. They were told that
they would not be allowed to change the rotation-period or scanning-delay
parameter during this block and thus would have to calibrate it as they saw
fit during the previous blocks. Information about their own writing speeds
across full blocks and also phrase-by-phrase was made available to
participants during the break after each block.

We performed seven significance tests with a family-level significance of
$0.05$. Observing the Bonferroni correction, we performed each individual test
at a significance level of $\alpha=0.007$. Wherever $F$ values are quoted, an
analysis of variance (ANOVA) test for repeated measures was performed.

### Nomon Operation

We here describe the prior over clocks, click likelihood (given a clock), and
the resulting posterior over clocks in turn. While we focus on a prior for a
specific application (the Nomon Keyboard), the likelihood and posterior
discussions are germane to a general Nomon application.

#### Prior

In the absence of information about clock probabilities, we use a uniform
prior $p(c)$ over clocks $c:1\leq c\leq C$. We can choose a more informative
prior for our Nomon writing application, the Nomon Keyboard (Figure 2). This
interface features four special characters (underscore representing space;
period; Delete; and Undo), 26 letters, and up to three word completions per
letter. We assign fixed prior probabilities to the special characters and
assign the remaining priors according to Laplace smoothing out of the leftover
probability mass $p_{\rm alpha}$. Let $l_{1}\cdots l_{N}$ ($N\geq 0$) be the
context (all letters from the end of the current output text) before the user
begins to make another selection. Let $\mathcal{W}_{\rm on}$ be the set of
word completions appearing on screen, and set $C_{\rm on}=|\mathcal{W}_{\rm
on}|+26$. To form our corpus, we begin with the British National Corpus word
list [26], then we remove single-letter words besides “I” and “a” and keep
only words appearing with some small minimum frequency ($>5$ appearances in
the corpus).

When an appropriate word completion is offered, the user may nevertheless
choose the next single letter; the following model assumes that the user is
equally likely to choose either of these options. If $f_{w}$ is the number of
occurrences of word $w$ in our corpus, we define a context frequency
$f(l_{1}\cdots l_{N})=\sum_{w}f_{w}\mathbbm{1}\\{l_{1}\cdots l_{n}\textrm{
prefixes }w\\}$ and a screen word-completion summed frequency
$f(\mathcal{W}_{\rm on})=\sum_{w^{\prime}\in\mathcal{W}_{\rm
on}}f(w^{\prime})$. If $c(l^{\prime})$ is the clock corresponding to letter
$l^{\prime}$ and $c(w)$ the clock corresponding to word $w$,

$\displaystyle p(c(l^{\prime}))$ $\displaystyle=$ $\displaystyle p_{\rm
alpha}\times\frac{f(l_{1}\cdots l_{N}l^{\prime})+1}{f(l_{1}\cdots
l_{N})+f(\mathcal{W}_{\rm on})+C_{\rm on}}$ (1) $\displaystyle p(c(w))$
$\displaystyle=$ $\displaystyle p_{\rm alpha}\times\frac{f(w)+1}{f(l_{1}\cdots
l_{N})+f(\mathcal{W}_{\rm on})+C_{\rm on}}$ (2)

To model an ideal user, we would subtract the count of words onscreen prefixed
by $l_{1}\cdots l_{N}l^{\prime}$ from the numerator of $p(c(l^{\prime}))$, and
both denominators would equal $f(l_{1}\cdots l_{N})+C_{\rm on}$. Finally,
while the number of letters is fixed at 26, $C_{\rm on}$ is variable since,
for any letter, we include only those word completions among the three most
probable above a certain threshold. It was judged that requiring
$f_{w}/f(l_{1}\cdots l_{N})>0.001$ yielded a reasonable balance between
displaying common words and not cluttering the screen.

#### Click Distribution

Any particular clock $c$ defines a desired click time at noon. We wish to
estimate a user’s click time distribution relative to noon $p(t|c)$, where we
distinguish $t$ only up to the clock period $T$ and set $t|c$ to zero at noon.
To that end, we begin with a broad, and slightly offset, initial setting of
our estimate for $p(t|c)$: $\hat{g}_{0}(t)=\mathcal{N}(t;0.05T,(0.14T)^{2})$.
The $T$-dependence ensures the estimate will be nontrivial at any user-chosen
period. We update the $\hat{g}_{0}$ distribution with a (modified) Parzen
window estimator—with width given below—and a damping factor $\lambda$ that
allows learning to continue over time. After any selection is made, we modify
the distribution estimate with the data from the $n_{\rm delay}^{\rm th}$
selection before the latest one (here $n_{\rm delay}=2$). This delay allows
the user to choose Undo after a selection, in which case we do not use the
clicks toward that selection for learning. Once a selection occurred $n_{\rm
delay}$ rounds in the past, we assume that it was correctly chosen. With the
clock choice $c$ known for the $s^{\rm th}$ selection, we are able to
calculate click times around noon $t_{s,r}$ for each click that was made
toward this selection. We treat these as data from the distribution $g$ we are
estimating. To calculate our estimate $\hat{g}_{s}$ for $g$ after the $s^{\rm
th}$ selection, we make use of the unnormalized distributions $\tilde{g}_{s}$.

$\tilde{g}_{s}(t)=\lambda\tilde{g}_{s-1}(t)+\sum_{r=1}^{R_{s}}\mathcal{N}\left(t;t_{s,r},\hat{\sigma}^{2}_{\rm{NS},s}\right)\textrm{
with }\tilde{g}_{0}(t)=n_{\lambda}\hat{g}_{0}(t)$ (3)

The update equation specifies that, after each selection, $\tilde{g}$ is
damped by the factor $\lambda$. The next term is a sum over clicks $r$ leading
to the $s^{\rm th}$ selection. Within the summation is a normal density
centered at the click time $t_{s,r}$, as in Parzen window estimation. The
width for this Parzen-window term is given by $\hat{\sigma}_{\textrm{NS},s}$,
which is derived from the normal scale rule estimate [32, 33] for the Parzen
window. That is,

$\hat{\sigma}_{\textrm{NS},s}=1.06n_{\lambda}^{-0.2}\hat{\sigma}_{s},$ (4)

where $\hat{\sigma}^{2}_{s}$ is the standard (Gaussian maximum likelihood)
variance estimator obtained from the last $n_{\lambda}$ clicks before the
$s^{\rm th}$ round. The factor $n_{\lambda}=(1-\lambda)^{-1}$ in the initial
$\tilde{g}_{0}$ definition is an effective number of samples derived from the
damping factor. Using this factor and the unnormalized update, we ensure that
the initial estimate $\hat{g}_{0}$ dominates $\hat{g}_{s}$ even after the
first few selections. Without the $n_{\lambda}$ factor, the Parzen window term
for the first click,
$\mathcal{N}\left(t;t_{1,1},\hat{\sigma}^{2}_{\rm{NS},1}\right)$, would have
nearly equal weight with the initial estimate.

This estimate for $p(t|c)$ allows us to save the estimated distribution and
update it quickly and easily during operation of the application. As a result,
users can start the Nomon application immediately, without a waiting or
calibration period, but they can also enjoy an experience tailored to their
abilities. For instance, a user need not click at noon (or any offset)
exactly. Their personal offset, reflecting reaction time, is learned by this
method rather than hard-coded and, as long as it is not too close to 6
o’clock, will make no difference to program operation. The precision around
this personal offset determines the number of clicks necessary to make a
selection.

#### Posterior

With a prior and likelihood, we may calculate the posterior probability of
each clock $c$ given the $R$ clicks thus far using Bayes’ theorem:
$p_{c,R}=p(c|t_{1:R})\propto p(c)\prod_{r=1}^{R}p(t_{r}|c)$. In practice, we
store the unnormalized log probabilities for each $p_{c,R}$. Checking that the
highest clock probability $p_{(C),R}$ exceeds some threshold would require
exponentiating every stored value and summing over the results. Noting that
$p_{(C),R}>1-p_{\rm error}$ is equivalent to
$p_{(C),R}>\alpha\sum_{c\neq(C)}p_{c,R}$ for some $\alpha$, we instead declare
a winner when $p_{(C),R}>\alpha p_{(C-1),R}$. The choice of $\alpha=99$
represents a desired upper bound on error fraction, per selection, of $0.01$.
In a sample of 1,714 consecutive selections made by an experienced Nomon user
(TB) on the Nomon Keyboard under this setting, the average value of
$p_{(C),R}/p_{(C-1),R}$ over all selections after the deciding click was
$0.001$, and the average value of $p_{(C),R}/\sum_{c\neq(C)}p_{c,R}$ was
$0.002$, suggesting our heuristic stopping criterion is a reasonable
approximation to the desired one. In the 1,714 selections, 3 (non-consecutive)
selections were Undo, indicating mistakes and giving an empirical error rate
of about $0.002$, in line with the calculated rate.

## Acknowledgments

This research was supported by donations from the Nine Tuna Foundation and
Nokia. We thank Sensory Software for generously providing the 60-day free
trial of The Grid 2 used in our performance comparison. We are grateful to Per
Ola Kristensson, Geoffrey Hinton, Keith Vertanen, Philipp Hennig, Carl
Scheffler, and Philip Sterne for helpful discussions. TB’s research is
supported by a Marshall Scholarship.

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# Oblique and asymmetric Klein tunneling across smooth NP junctions or NPN
junctions in 8-$Pmmn$ borophene

Zhan Kong  School of Science, Chongqing University of Posts and
Telecommunications, Chongqing, 400065, China  Jian Li  School of Science,
Chongqing University of Posts and Telecommunications, Chongqing, 400065, China
Laboratory of Quantum Information Technology, Chongqing University of Posts
and Telecommunications, Chongqing, 400065, China  Yi Zhang  School of Science,
Chongqing University of Posts and Telecommunications, Chongqing, 400065, China
Shu-Hui Zhang<EMAIL_ADDRESS>College of Mathematics and
Physics, Beijing University of Chemical Technology, Beijing 100029, China
Jia-Ji Zhu<EMAIL_ADDRESS>School of Science, Chongqing University of
Posts and Telecommunications, Chongqing, 400065, China  Laboratory of Quantum
Information Technology, Chongqing University of Posts and Telecommunications,
Chongqing, 400065, China

###### Abstract

The tunneling of electrons and holes in quantum structures plays a crucial
role in studying the transport properties of materials and the related
devices. 8-$Pmmn$ borophene is a new two-dimensional Dirac material that hosts
tilted Dirac cone and chiral, anisotropic massless Dirac fermions. We adopt
the transfer matrix method to investigate the Klein tunneling of massless
fermions across the smooth NP junctions and NPN junctions of 8-$Pmmn$
borophene. Like the sharp NP junctions of 8-$Pmmn$ borophene, the tilted Dirac
cones induce the oblique Klein tunneling. The angle of perfect transmission to
the normal incidence is $20.4^{\circ}$, a constant determined by the
Hamiltonian of 8-$Pmmn$ borophene. For the NPN junction, there are branches of
the Klein tunneling in the phase diagram. We find that the asymmetric Klein
tunneling is induced by the chirality and anisotropy of the carriers.
Furthermore, we show the oscillation of electrical resistance related to the
Klein tunneling in the NPN junctions. One may analyze the pattern of
electrical resistance and verify the existence of asymmetric Klein tunneling
experimentally.

##  I Introduction

Two-dimensional (2D) materials have been the superstars for their novel
properties in condensed matter physics since its first isolation of graphene
in 2004ref-1 . Right now, the booming 2D materials family includes not just
graphene and the derivatives of graphene but also transition metal
dichalcogenides (TMDs)ref-2 ; ref-3 ; ref-35 , black phosphorusref-4 ; ref-33
; ref-34 ; ref-38 , Indium selenideref-5 ; ref-6 ; ref-37 , staneneref-7 ;
ref-8  and many other layered materialsref-9 ; ref-10 . Among these 2D
materials, the so-called Dirac materials host massless Dirac fermions always
in the spotlight. Carriers in 2D Dirac materials usually have chirality or
pseudospin from two atomic sublattices. Together with chirality, the linear
Dirac dispersion gives rise to remarkable transport properties, including the
absence of backscatteringref-1 ; ref-11 ; ref-36 . Due to the suppression of
backscattering, massless Dirac fermions could tunnel a single square barrier
with 100% transmission probability. This surprising result has been known as
Klein tunnelingref-11 ; ref-31 ; ref-40 ; Hyungju Oh (2016); ref-99 . Klein
tunneling is the basic electrical conduction mechanism through the interface
between $p$-doped and $n$-doped regions. Klein tunneling’s elucidation plays a
key role in designing and inventing electronic devices based on 2D Dirac
materials.
Recently, several 2D boron structures have been predicted and experimentally
fabricated ref-24 ; ref-13 ; ref-14 ; ref-25 . The 8-$Pmmn$ borophene belongs
to the space group $Pmmn$, which means an orthorhombic lattice has an $mmm$
symmetric point group (three-mirror symmetry planes perpendicular to each
other) combine with a glide plane at one of the mirror symmetry planes ref-24
; ref-book5 . This kind of structure is the most stable symmetric phase of
borophene and may be kinetically stable at ambient conditions. It revealed the
tilted Dirac cone and anisotropic massless Dirac fermions by first-principles
calculations ref-15 ; ref-16 .These unique Dirac fermions attracts people to
explore the various physical properties such as strain-induced pseudomagnetic
field ref-21 , anisotropic density–density response ref-29 ; ref-71 ; ref-72 ;
ref-73 , optical conductivity ref-74 ; ref-75 , modified Weiss oscillation
ref-28 ; ref-41 , borophane and its tight-binding model ref-41 , nonlinear
optical polarization rotation ref-76 , oblique Klein tunneling ref-19 ; ref-77
; ref-78 , few-layer borophene ref-79 ; ref-80 , intense light response ref-81
; ref-82 , RKKY interaction ref-83 ; ref-84 , anomalous caustics ref-85 ,
electron–phonon coupling ref-86 , valley–contrast behaviors ref-87 ; ref-88 ,
Andreev reflection ref-89 , and so on. The oblique Klein tunneling, the
deviation of the perfect transmission direction to the normal direction of the
interface, is induced by the anisotropic massless Dirac fermions or the tilted
Dirac cone ref-18 ; ref-19 . However, the on-site disorder or smoothing of the
NP junction interface or the square potential may destroy the ideal Klein
tunneling, which means the sharp interface strongly depends on high-quality
fabrication state-of-the-art technology ref-20 . Therefore, the detailed
discussion of the smooth NP junction and the tunable trapezoid potential would
be helpful for the promising electronic devices based on 2D Dirac materials.
In this paper, we study the transmission properties of anisotropic and tilted
massless Dirac fermions across smooth NP junctions and NPN junctions in
8-$Pmmn$ borophene. Similar to the sharp NP junction, the oblique Klein
tunneling retains due to the tilted Dirac cone. This conclusion does not
depend on the NP junctions’ doping levels as the normal Klein tunneling but
depends on the junction direction. We show the angle of oblique Klein
tunneling is $20.4^{\circ}$, a constant determined by the Hamiltonian
parameters of 8-$Pmmn$ borophene. For the NPN junction, there are branches of
the Klein tunneling in the phase diagram. We find that the asymmetric Klein
tunneling is induced by the chirality and anisotropy of the carriersref-42 .
The indirect consequence of the asymmetric Klein tunneling lies in the
oscillation of the electrical resistance. The analysis of the pattern of the
oscillation of electrical resistance would help verify the existence of
asymmetric Klein tunneling experimentally.
The rest of the paper is organized as follows. In Secion II, we introduce the
Hamiltonian and the energy spectrum for the 8-$Pmmn$ borophene, the NP and NPN
junction’s potential, and present the transfer matrix method for the detailed
derivation of transmissions across the junctions. In Section III, we
demonstrate perfect transmission numerically, showing that the oblique Klein
tunneling in NP junctions and the asymmetric Klein tunneling in NPN junctions.
Then, we calculate the electrical resistance from the Landauer formula for the
NPN junction. Finally, we give a brief conclusion in Section IV.

##  II Theoretical Formalism

###  II.1 Model

The crystal structure of 8-$Pmmn$ borophene has two sublattices, as
illustrated in Fig. 1(a) by different colors. It is made of buckled triangular
layers where each unit cell has eight atoms under the symmetry of space group
$Pmmn$(No. 59 in ref-book1 ) , the so called 8-$Pmmn$ structure. The tilted
Dirac cone emerges from the hexagonal lattice formed by the inner atoms
(yellow in Fig. 1(a)).ref-16  This hexagonal structure is topologically
equivalent to uniaxially strained graphene, and the Hamiltonian of 8-$Pmmn$
borophene around one Dirac point is given by ref-21 ; ref-28 ; ref-29 ; ref-41

Figure 1: (a) Crystal structure of 8-$Pmmn$ borophene. The unit cell of
8-$Pmmn$ borophene contains two types of nonequivalent boron atoms, the ridge
atoms (orange) and the inner atoms (yellow). (b) The schematic diagram of the
smooth NP junction in 8-$Pmmn$ borophene. Note that the true tilted Dirac cone
is along $y$ direction but $x$ direction. (c)The schematic diagram of the
smooth NPN junction in 8-$Pmmn$ borophene. Here we choose $n=6.25$ and
$m=12.5$ for the numerical calculations.

$\displaystyle\hat{H}_{0}=\upsilon_{x}\sigma_{x}\hat{p}_{x}+\upsilon_{y}\sigma_{y}\hat{p}_{y}+\upsilon_{t}\mathbf{I}_{2\times
2}\hat{p}_{y}$ (1)

where $\hat{p}_{x,y}$ are the momentum operators, $\sigma_{x,y}$ are $2\times
2$ Pauli matrices, and $\mathbf{I}_{2\times 2}$ is a $2\times 2$ unit matrix.
The anisotropic velocities are
$\upsilon_{x}=0.86\upsilon_{F},\upsilon_{y}=0.69\upsilon_{F},\upsilon_{t}=0.32\upsilon_{F},\upsilon_{F}=10^{6}$
m/sref-21 .The energy dispersion and the corresponding wave functions of
$\hat{H}_{0}$ are

$\displaystyle
E_{\lambda,\mathbf{k}}=\upsilon_{t}p_{y}+\lambda\upsilon_{x}\sqrt{p_{x}^{2}+\gamma_{1}^{2}p_{y}^{2}},\gamma_{1}=\frac{\upsilon_{y}}{\upsilon_{x}}$
(2)
$\displaystyle\psi_{\lambda,\mathbf{k}}\left(\mathbf{r}\right)=\frac{1}{\sqrt{2}}\left[\begin{array}[c]{c}1\\\
\lambda\frac{k_{x}+i\gamma_{1}k_{y}}{\sqrt{k_{x}^{2}+\gamma_{1}^{2}k_{y}^{2}}}\end{array}\right]e^{i\mathbf{k}\cdot\mathbf{r}}$
(5)

Here, $\lambda=\pm 1$, denoting the conduction $\left(+1\right)$ and valence
$\left(-1\right)$ band, respectively. For 8-$Pmmn$ borophene, the shape of
Fermi surface for the fixing energy is elliptical with eccentricity $e$
determined by $\upsilon_{x}$, $\upsilon_{y}$ and $\upsilon_{t}$, which differs
from the circular shape with radius $E_{F}/\hbar v_{F}$ of graphene. We can
rewrite Eq. 2 in following wayref-19 ; ref-26 :

$\displaystyle\frac{p_{x}^{2}}{a_{\lambda,E}}+\frac{\left(p_{y}+c_{\lambda,E}\right)^{2}}{b_{\lambda,E}}=1$
(6) $\displaystyle
a_{\lambda,E}=\frac{\upsilon_{y}^{2}E_{\lambda,\mathbf{k}}^{2}}{\upsilon_{x}^{2}\left(\upsilon_{y}^{2}-\upsilon_{t}^{2}\right)},\,\,\,b_{\lambda,E}=\frac{\upsilon_{y}^{2}E_{\lambda,\mathbf{k}}^{2}}{\left(\upsilon_{y}^{2}-\upsilon_{t}^{2}\right)^{2}},\,\,\,c_{\lambda,E}=\frac{\upsilon_{t}E_{\lambda,\mathbf{k}}}{\left(\upsilon_{y}^{2}-\upsilon_{t}^{2}\right)}$
(7)

The eccentricity of the Fermi surface can be determined by
$e=\sqrt{\upsilon_{x}^{2}-\upsilon_{y}^{2}+\upsilon_{t}^{2}}/\upsilon_{x}$. As
a direct consequence, the eccentricity is not depend on the energy and the
center of ellipse is at

$\displaystyle\hbar k_{x}$ $\displaystyle=$ $\displaystyle 0,\,\,\,\hbar
k_{y}=-\frac{\upsilon_{t}E_{\lambda,\mathbf{k}}}{\left(\upsilon_{y}^{2}-\upsilon_{t}^{2}\right)}$
(8)

Notice that the center of ellipse is not at the origin and it moves with
increasing the Fermi levels. In a NP junction setup, the translation symmetry
preserves along the $y$ axis, so the $k_{y}$ is always a good quantum number.
When the momentum $p_{y}$ is given, the $p_{x}$ in different regions of
8-$Pmmn$ borophene NP junction is

$\displaystyle
p_{x}=\pm\frac{1}{\upsilon_{x}}\sqrt{\left(E_{\lambda,\mathbf{k}}-\upsilon_{t}p_{y}\right)^{2}-\left(\upsilon_{y}p_{y}\right)^{2}}$
(9)

Like the graphene NP junction, one can implement a bipolar NP junction or
tunable NPN-type potential barriers in 8-$Pmmn$ borophene by top/back gate
voltages, and the potential function of the NP junction (as depicted in Fig.
1(b)) has the form:

$\displaystyle U_{NP}(x)=\left\\{\begin{matrix}V_{0}&,&x>na/2\\\
2V_{0}x/na&,&\\-na/2\leq x\leq na/2\\\ -V_{0}&,&x<-na/2\end{matrix}\right.$
(10)

where $a=\hbar\upsilon_{F}/0.04$ eV is a unit length and $n>0\wedge
n\in\mathbb{R}$. The NPN junction depicted in Fig. 1(c) has the form

$\displaystyle U_{NPN}(x)=\left\\{\begin{matrix}-V_{0}&,&3na/2+ma<x&\\\
\frac{-2V_{0}(x-ma-na)}{na}&,&na/2+ma\leq x\leq 3na/2+ma&\\\
V_{0}&,&na/2<x<na/2+ma&\\\ \frac{2V_{0}x}{na}&,&-na/2\leq x\leq na/2&\\\
-V_{0}&,&x<-na/2&\end{matrix}\right.$ (11)

where $m>0\wedge m\in\mathbb{R}$. Next, we will utilize the transfer matrix
method to solve the ballistic transport problem in smooth NP/NPN junctions of
8-$Pmmn$ borophene.

###  II.2 Transfer matrix method

The transfer matrix method is a powerful tool in the analysis of quantum
transport of the massless fermions in 2D Dirac materialsref-30 ; ref-31 ;
ref-39 . The central idea lies in that the wave function in one position can
be related to those in other positions through a transfer matrixref-27 .

We adopt a transfer matrix method to study quantum transport in the smooth NP
or NPN junction in 8-$Pmmn$ borophene. There are two different matrices in
transfer matrix method: one is the transmission matrix and the other is
propagating matrix. Transmission matrix connects the electrons across an
interface and the propagating matrix connects the electrons propagating over a
distance in the homogeneous regions. As we can see below the propagating
matrix can be derived by the transmission matrix. We define the transmission
matrix $T$ as follows:

$\displaystyle T\left(\begin{array}[]{c}A_{R_{m+1}}\\\
A_{L_{m+1}}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}A_{R_{m}}\\\
A_{L_{m}}\end{array}\right)$ (16)

where $A_{R_{m}}$ ($A_{L_{m}}$) represents the right (left) traveling wave
amplitude in $m$ region. The transmission matrix connects the wave function’s
amplitude of two different regions. The condition of connecting amplitude
coefficients between adjacent regions is the continuity of the wave functions
at the interface. We can treat the smooth potential as the sum of infinite
slices of junctions and figure out the wave function from the Schrödinger
equation. Since the energy dispersion of 8-$Pmmn$ borophene is linear, we only
need the continuity condition of the wave functions at the interface. Then the
transmission matrices $T$ can be constructed from matrices $M$ of each slice,

$\displaystyle
M\left(k_{m+1},x_{m}\right)\left(\begin{array}[]{c}A_{R_{m+1}}\\\
A_{L_{m+1}}\end{array}\right)=M\left(k_{m},x_{m}\right)\left(\begin{array}[]{c}A_{R_{m}}\\\
A_{L_{m}}\end{array}\right)$ $\displaystyle
M\left(k_{m},x_{m}\right)^{-1}M\left(k_{m+1},x_{m}\right)\left(\begin{array}[]{c}A_{R_{m+1}}\\\
A_{L_{m+1}}\end{array}\right)=\left(\begin{array}[]{c}A_{R_{m}}\\\
A_{L_{m}}\end{array}\right)$ $\displaystyle
M\left(k_{m},x_{m}\right)^{-1}M\left(k_{m+1},x_{m}\right)=T$

Figure 2: Potential profile of (a) NP junction, (b)PN junction, (c)NN
junction, and (d)PP junction in each slice of the junctions.

Suppose that an n-doped region $m$ is next to a p-doped region $m+1$ and
carriers go through from n-doped region to p-doped region like in Fig. 2(a),
the wave functions at interface can be connected in the way of

$\displaystyle\frac{A_{R_{m+1}}}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
e^{i\theta_{m+1}}\end{array}\right)e^{ik_{z,m+1}x_{m}}+\frac{A_{L_{m+1}}}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-e^{-i\theta_{m+1}}\end{array}\right)e^{-ik_{z,m+1}x_{m}}$ (23)
$\displaystyle=\frac{A_{R_{m}}}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
e^{-i\theta_{m+1}}\end{array}\right)e^{-ik_{z,m}x_{m}}+\frac{A_{L_{m}}}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-e^{i\theta_{m+1}}\end{array}\right)e^{ik_{z,m}x_{m}}.$ (28)

Here, we define $k_{x,m}\left(x\right)$ and $\theta_{m}$ as

$\displaystyle k_{x,m}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar\upsilon_{x}}\sqrt{\left(-U_{m}\left(x\right)+\hbar\upsilon_{t}k_{y}\right)^{2}-\left(\hbar\upsilon_{y}k_{y}\right)^{2}}$
(29) $\displaystyle e^{i\theta_{m}}$ $\displaystyle=$
$\displaystyle\frac{k_{x,m}+i\gamma_{1}k_{y}}{\sqrt{k_{x,m}^{2}+\gamma_{1}^{2}k_{y}^{2}}}$
(30)

where $U_{m}\left(x\right)$ is the doping level in $m$ region and $k_{x}$ may
take positive or negative imaginary values when
$\left(-U_{m}\left(x\right)+\hbar\upsilon_{t}k_{y}\right)^{2}-\left(\hbar\upsilon_{y}k_{y}\right)^{2}<0$.
The phase $e^{i\theta_{m}}$ in Eq. (28) is defined as the wave function phase
difference between the two sublattices. The sign of the $k_{x}$ defines the
propagating direction of the carriers. Without loss of generality, we can take
only positive imaginary value for the transmission matrix, which means the
positive propagating direction of electrons is defined on right-going state.
Here, the potential profile $U_{m}\left(x\right)$ in adjacent regions within
NP junction is linear but not rectangular; we treat the potential as a series
of step potential to solve the tunneling problems by the transmission
matrices. For convenience, we choose $a=\hbar v_{F}/0.04$ eV to be the length
unit and $0.01$ eV to be the energy unit, where $0.04$ eV is the maximum of
the doping level.

Then we rewrite the Eq. (28) to construct the transmission matrices

$\displaystyle\left(\begin{array}[c]{ll}e^{-ik_{x,m+1}x_{m}}&\,\,e^{ik_{x,m+1}x_{m}}\\\
e^{-i\theta_{m+1}}e^{-ik_{x,m+1}x_{m}}&\,\,-e^{i\theta_{m+1}}e^{ik_{x,m+1}x_{m}}\end{array}\right)\left(\begin{array}[c]{c}A_{R_{m+1}}\\\
A_{L_{m+1}}\end{array}\right)$
$\displaystyle=\left(\begin{array}[c]{ll}e^{ik_{x,m}x_{m}}&e^{-ik_{x,m}x_{m}}\\\
e^{i\theta_{m}}e^{ik_{x,m}x_{m}}&-e^{-i\theta_{m}}e^{-ik_{x,m}x_{m}}\end{array}\right)\left(\begin{array}[c]{c}A_{R_{m}}\\\
A_{L_{m}}\end{array}\right).$

Therefore the transmission matrix between $m$ and $m+1$ region is

$\begin{array}[]{ccc}\begin{array}[]{c}T_{m,m+1}^{n\rightarrow p}\\\ \\\ \\\
\end{array}&\begin{array}[]{c}=\\\ \\\ \\\
\end{array}&\begin{array}[]{l}\left(\begin{array}[]{ll}e^{ik_{x,m}x_{m}}&e^{-ik_{x,m}x_{m}}\\\
e^{i\theta_{m}}e^{ik_{x,m}x_{m}}&-e^{-i\theta_{m}}e^{-ik_{x,m}x_{m}}\end{array}\right)^{-1}\\\
\left(\begin{array}[]{ll}e^{-ik_{x,m+1}x_{m}}&e^{ik_{x,m+1}x_{m}}\\\
e^{-i\theta_{m+1}}e^{-ik_{x,m+1}x_{m}}&-e^{i\theta_{m+1}}e^{ik_{x,m+1}x_{m}}\end{array}\right)\end{array}\end{array}$
(33)

while the transmission matrices of the carriers going through from p-doped
region $m$ to n-doped region $m+1$ and between two n-doped or p-doped region
(shown in Fig. 2) are

$\begin{array}[]{ccc}\begin{array}[]{c}T_{m,m+1}^{p\rightarrow n}\\\ \\\ \\\
\end{array}&\begin{array}[]{c}=\\\ \\\ \\\
\end{array}&\begin{array}[]{l}\left(\begin{array}[]{ll}e^{-ik_{x,m}x_{m}}&e^{ik_{x,m}x_{m}}\\\
e^{-i\theta_{m}}e^{-ik_{x,m}x_{m}}&-e^{i\theta_{m}}e^{ik_{x,m}x_{m}}\end{array}\right)^{-1}\\\
\left(\begin{array}[]{ll}e^{ik_{x,m+1}x_{m}}&e^{-ik_{x,m+1}x_{m}}\\\
e^{i\theta_{m+1}}e^{ik_{x,m+1}x_{m}}&-e^{-i\theta_{m+1}}e^{-ik_{x,m+1}x_{m}}\end{array}\right)\end{array}\end{array}$
(34)

$\begin{array}[]{ccc}\begin{array}[]{c}T_{m,m+1}^{n\rightarrow n}\\\ \\\ \\\
\end{array}&\begin{array}[]{c}=\\\ \\\ \\\
\end{array}&\begin{array}[]{l}\left(\begin{array}[]{ll}e^{ik_{x,m}x_{m}}&e^{-ik_{x,m}x_{m}}\\\
e^{i\theta_{m}}e^{ik_{x,m}x_{m}}&-e^{-i\theta_{m}}e^{-ik_{x,m}x_{m}}\end{array}\right)^{-1}\\\
\left(\begin{array}[]{ll}e^{ik_{x,m+1}x_{m}}&e^{-ik_{x,m+1}x_{m}}\\\
e^{i\theta_{m+1}}e^{ik_{x,m+1}x_{m}}&-e^{-i\theta_{m+1}}e^{-ik_{x,m+1}x_{m}}\end{array}\right)\end{array}\end{array}$
(35)

$\begin{array}[]{ccc}\begin{array}[]{c}T_{m,m+1}^{p\rightarrow p}\\\ \\\ \\\
\end{array}&\begin{array}[]{c}=\\\ \\\ \\\
\end{array}&\begin{array}[]{l}\left(\begin{array}[]{ll}e^{-ik_{x,m}x_{m}}&e^{ik_{x,m}x_{m}}\\\
e^{-i\theta_{m}}e^{-ik_{x,m}x_{m}}&-e^{i\theta_{m}}e^{ik_{x,m}x_{m}}\end{array}\right)^{-1}\\\
\left(\begin{array}[]{ll}e^{-ik_{x,m+1}x_{m}}&e^{ik_{x,m+1}x_{m}}\\\
e^{-i\theta_{m+1}}e^{-ik_{x,m+1}x_{m}}&-e^{i\theta_{m+1}}e^{ik_{x,m+1}x_{m}}\end{array}\right)\end{array}\end{array}$
(36)

For the case of NPN junction, a trapezoidal potential profile as in Fig. 1(c),
we can also treat the trapezoidal potential into infinite slices of connected
step potentials. The transmission matrices define at the interface between
each step potentials. Multiplying all the transmission matrices would give the
propagation matrices,

$\displaystyle T_{all}=T_{0,1}^{n\rightarrow n}T_{1,2}^{n\rightarrow
n}...T_{k-1,k}^{n\rightarrow n}T_{k,k+1}^{n\rightarrow
p}T_{k+1,k+2}^{p\rightarrow p}...\times$ (37) $\displaystyle
T_{k^{\prime}-1,k^{\prime}}^{p\rightarrow
p}T_{k^{\prime},k^{\prime}+1}^{p\rightarrow
p}...T_{k^{\prime\prime}-1,k}^{p\rightarrow
p}T_{k^{\prime\prime},k^{\prime\prime}+1}^{p\rightarrow
n}T_{k^{\prime\prime}+1,k^{\prime\prime}+2}^{n\rightarrow
n}...T_{m-2,m-1}^{n\rightarrow n}T_{m-1,m}^{n\rightarrow n}$

then we reach the formula

$\displaystyle T_{all}\left(\begin{array}[c]{c}A_{R_{m}}\\\
A_{L_{m}}\end{array}\right)=\left(\begin{array}[c]{c}A_{R_{0}}\\\
A_{L_{0}}\end{array}\right)$ (42)

When incident electrons go from the leftmost side of the NPN junction to the
rightmost side, there are no reflection states in the rightmost side, i.e.
$A_{L_{m}}=0$. We can connect the amplitude of incident states to the
amplitude of reflection states

$\displaystyle\left(\begin{array}[]{ll}T_{11}&T_{12}\\\
T_{21}&T_{22}\end{array}\right)\left(\begin{array}[]{c}A_{R_{m}}\\\
0\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}A_{R_{0}}\\\
A_{L_{0}}\end{array}\right)$ $\displaystyle\frac{A_{R_{m}}}{A_{R_{0}}}$
$\displaystyle=$ $\displaystyle\frac{1}{T_{11}}$

Finally, the transmission probability is
$T=\left|t\right|^{2}=\left|A_{R_{m}}/A_{R_{0}}\right|^{2}=\left|1/T_{11}\right|^{2}$.

Figure 3: Fermi surface at different doped regions $\pm\varepsilon_{doping}$.
The blue (red) ellipse represents the electron (hole) Fermi surface in n-doped
(p-doped) region. The solid vectors $\mathbf{k}_{in}$ (blue) and
$\mathbf{k}_{out}$ (red) are the wave vector of incident carriers and
transmitted carriers, respectively. The dashed vectors $\mathbf{v}_{KTin}$
(blue) and $\mathbf{v}_{KTout}$ (red) are the group velocity of incident
carriers and transmitted carriers, respectively. The green dotted lines
indicate the values of the good quantum number $k_{y}$ posed restrictions for
the NP junction and the NPN junction.

There is a trick in constructing the propagation matrices from the
transmission matrices. As shown in Fig. 3, the incident states at the left-
hand side of the junction have a different Fermi surface from the transmitted
states at the right-hand side in the NP junction. Suppose the NP junction is
sharp. The good quantum number $k_{y}$ should be restricted between the top
dotted green line and the middle dotted green line, since the incident states
and the transmitted states are propagating only in this scenario. While
supposing the NP junction is smooth, the Fermi surface in the region of
varying potential would shrink to the Dirac point, and the
$E_{\lambda,\mathbf{k}}$, $a_{\lambda,E}$, $b_{\lambda,E}$, and
$c_{\lambda,E}$ from Eq. (7) reduce to zero as well. Therefore, $k_{x}$
vanishes to diverge the transmission matrices when the carriers approaching
the NP junction center. However, we could play a trick by properly segmenting
the region of varying potential and jumping the diverging point. The trick
lies in the fact that the carriers would not experience any singularity when
going through an infinitesimal interval around the diverging point. For
instance, the transmission matrix at the Dirac point cannot be well defined
with incident states $k_{y}=0$, whereas the carriers are well-defined decay
states at the Dirac point. We can ignore the decay states of the carriers
going through infinitesimal intervals around the Dirac point, and it would
eliminate any possible ambiguity.

##  III Results and Discussions

In this section, we present the numerical results for the transmission
probability and electrical conduction of the massless Dirac fermions across
the borophene NP junction and NPN junction.

Figure 4: (a) Potential profile of smooth NP junctions and (b) the angular
behavior of the transmission probability for different NP junctions
corresponding to different colors at (a).

###  III.1 The oblique Klein tunneling in smooth NP junctions

Various smooth NP junctions with fixing n/p doping level but different slopes
are depicted in Fig. 4(a). We set the length of the varying region in
different NP junctions as $6.25$ $a$, $12.5$ $a$, $25$ $a$, $50$ $a$,
respectively, where $a=\hbar\upsilon_{F}/0.04$ eV, and plot the angular
transmission probability for different NP junctions. As shown in Fig. 4(b),
the shaper the NP junction is, the wider the angular transmission probability
spans. This phenomenon is caused by the decay states in the varying region and
is similar to the graphene smooth NP junction. In the varying region,
$(-U_{m}(x)+\hbar\upsilon_{t}k_{y})^{2}-(\hbar\upsilon_{y}k_{y})^{2}<0$, so
that the propagating states degenerate to the decaying states when the
carriers gradually approach the junction’s center. Therefore the transmission
probability increases with increasing the slope of potential in the varying
region. If we take $k_{y}=0$, i.e., the normal incident case, we can see the
perfect transmission, the Klein tunneling.

Figure 5: The angular transmission amplitude for $k$ vector (red) and for
group velocity (blue). The doping level is $0.04$ eV and the length of varying
region is $6.25$ $a$.

Fig. 5 shows that the angular transmission amplitude of the $k$ vector is
different from the one of group velocity. The actual incident angle across the
junction is based on the group velocity of carriers. The actual angular
transmission probability for group velocity shown in Fig. 5 indicates a
rotation of the Klein tunneling, the oblique Klein tunneling. It means that
the perfect transmission does not occur in the normal incident but with a non-
zero angle $\theta_{K}$.

The value of $\theta_{K}$ can be determined from the elliptical Fermi surface
of 8-$Pmmn$ borophene. The angle for the group velocity is
$\theta_{v}=\arctan\left[v_{y}\left(\varepsilon,k_{y}\right)/v_{x}\left(\varepsilon,k_{y}\right)\right]$,
where $v_{y}\left(\varepsilon,k_{y}\right)$ and
$v_{x}\left(\varepsilon,k_{y}\right)$ can be obtained by

$\displaystyle v_{x}\left(\varepsilon,k_{y}\right)$ $\displaystyle=$
$\displaystyle\frac{\partial E_{\lambda,\mathbf{k}}}{\hbar\partial
k_{x}}=\frac{\lambda
k_{x}\upsilon_{x}}{\sqrt{k_{x}^{2}+\gamma_{1}^{2}k_{y}^{2}}}$ (44)
$\displaystyle v_{y}\left(\varepsilon,k_{y}\right)$ $\displaystyle=$
$\displaystyle\frac{\partial E_{\lambda,\mathbf{k}}}{\hbar\partial
k_{y}}=\upsilon_{t}+\frac{\lambda\gamma_{1}^{2}k_{y}\upsilon_{x}}{\sqrt{k_{x}^{2}+\gamma_{1}^{2}k_{y}^{2}}}$
(45)

Combined with above equations and let $k_{y}=0$, we can find the angle of
Klein tunneling for group velocity,

$\displaystyle\theta_{K}=\arctan\left(\frac{\upsilon_{t}}{\upsilon_{x}}\right)\approx
20.4^{\circ}$ (46)

This oblique Klein tunneling can also be found in sharp NP junctions of
8-$Pmmn$ borophene.ref-18 ; ref-19

###  III.2 The asymmetric Klein tunneling in the smooth NPN junctions

The NPN junction, as shown in the Fig. 1(c) can be seen as a trapezoid
potential barrier. We set the length of the varying regions as $6.25$ $a$ and
the length of the flat potential barrier as $12.5$ $a$.

In Fig. 6, we plot the transmission probability depending on different doping
levels and $k_{y}$. Note that $n-$ and $p-$ regions have the same absolute
value of doping level. We can see the Klein tunneling in several branches. The
number of branches increases by lifting the doping level, which could also be
observed in the graphene NPN junctionsref-11 ; ref-31 . Also, we can see the
Klein tunneling is asymmetric. The asymmetric Klein tunneling results from the
carriers’ chirality and anisotropyref-17 . It is not surprising to see it here
because the carriers of 8-$Pmmn$ have both chirality and anisotropy.
The blue lines in Fig. 6(a) denote the forbidden zones, where the transmission
probability vanishes. The equation of the boundary of the forbidden zone is
$k_{y}=\pm\varepsilon_{doping}/\hbar(\upsilon_{t}+\upsilon_{y})$. There are
two types of the forbidden zone: (I) the no-incident zone and (II) the
vanishing transmitted zone. In the no-incident zone
$k_{y}\geq\varepsilon_{doping}/\hbar(\upsilon_{t}+\upsilon_{y})$, there is no
incident states since the parameters $k_{y}$ and doping level is beyond the
Dirac cone; In the vanishing transmitted zone
$k_{y}\leq-\varepsilon_{doping}/\hbar(\upsilon_{t}+\upsilon_{y})$, the
transmitted carriers severely decay in the region of barrier.

Figure 6: (a) Transmission probability versus the doping level and the $k_{y}$
in NPN junction. Blue lines denote the forbidden zone, where transmission
probability vanishes, and there are only the decaying states in the p-doped
region. (b) The tansmission probability depending on $k_{y}$ when the doping
level is $4\times 0.01$ eV. Figure 7: Transmission probability depends on top
edge of the trapezoid potentials. The top edge varies from 0 to $24$ $a$ and
the bottom edge is fixed as $25$ $a$. The height of the trapezoid potentials
or the absolute value of n/p doping level is fixed as $0.04$ eV.

Next, we fix the bottom edge and the height of the trapezoid potential (NPN
junction) and plot the transmission probability versus the potential’s top
edge.
When the top edge’s length varies from $0$ to the bottom edge’s length, the
NPN junction experiences a change from triangle potential to trapezoid
potential and finally to a square potential. We can see from Fig. 7 the number
of branches increases with increasing the top edge’s length. It is somehow
counterintuitive that the square potential favors Klein tunneling more than
the triangle potential. The reason is that the carriers would have more
chances to degenerate to decaying states when incident into a slope of
potential, in fact, a smooth NP junction.

###  III.3 The electrical resistance of the smooth NPN junctions

One can create the NPN junction by implementing a design with two
electrostatic gates, a global back gate and a local top gate. A back voltage
applied to the back gate could tune the carrier density in the borophene
sheet, whereas a top voltage applied to the top gate could tune the density
only in the narrow strip below the gate. These two gates can be controlled
independentlyref-32 .

To clarify the effect of the Klein tunneling on the transport property, here
we discuss the electrical conduction of the NPN junction in 8-$Pmmn$
borophene. In the ballistic regime, we apply the Landauer-Buttiker formula
$G=2e^{2}MT/h$ to calculate the electrical conductanceSupriyo Datta (1999). In
our setup, the Landauer formula can be written asref-23

$\displaystyle
G_{fet}=\frac{4e^{2}}{h}{\textstyle\sum\limits_{ch.}}T_{ch}\approx\frac{4e^{2}}{h}\int_{k_{y\min}}^{k_{y\max}}\frac{dk_{y}}{2\pi/W}T\left(k_{y}\right)$
(47)

where $k_{y\max}=\varepsilon_{doping}/\hbar(\upsilon_{t}+\upsilon_{y})$ and
$k_{y\min}=\varepsilon_{doping}/\hbar(\upsilon_{t}-\upsilon_{y})$. We choose
the width of the junction $W=10\mu m$ and calculate the electrical resistance
by the Landauer formula. To reveal the link of the resistance with the Klein
tunneling, we plot the transmission probability versus the doping level in
Fig. 8(a) and the resistance depending on doping levels in Fig. 8(b). We can
see the resistance oscillation when increasing the doping level from $0$ to
$0.08$ eV. The oscillation pattern indicates the effect of the Klein
tunneling. When the doping level varies from $0$ to $-0.08$ eV, the NPN
junction becomes a NNN junction so that the curves of resistance are flat in
the negative doping regime.

Figure 8: (a) Transmission probability depends on $k_{y}$ and the height of
the trapezoid potentials (doping levels of the NPN junctions). The top edge’s
length is $12.5$ $a$ and the bottom edge’s length is $25$ $a$. The doping
level of n-doped region (outside the NPN junction) is set $-0.04$ eV. (b) The
electrical resistance of the NPN junction depending on the doping level.

##  IV Conclusions

This work investigates the transport properties of massless fermions in the
smooth 8-$Pmmn$ borophene NP and NPN junctions by the transfer matrix method.
Compare with the sharp junction, the smooth NP junction also shows that the
oblique Klein tunneling induced by the tilted Dirac cones. We can calculate
from the parameters of the Hamiltonian that the angle of oblique Klein
tunneling is $20.4^{\circ}$. We also show the branches of the NPN tunneling in
the phase diagram, which indicates the asymmetric Klein tunneling. The
physical origin of the asymmetric Klein tunneling lies in the chirality and
anisotropy of the carriers, and we can verify the asymmetric Klein tunneling
experimentally by analyzing the pattern of the electrical resistance
oscillation. For the oblique Klein tunneling, we have discussed the
experimental feasibility in detail in our previous study ref-19 . The present
numerical demonstration in smooth junctions proves the effectiveness of our
previous discussion and favors the observation in future experiments.

###### Acknowledgements.

This work was supported by the Scientific Research Program from Science and
Technology Bureau of Chongqing City (Grant No. cstc2020jcyj-msxm0925,
cstc2020jcyj-msxmX0810), the Science and Technology Research Program of
Chongqing Municipal Education Commission (Grant No. KJQN202000639), and the
key technology innovations project to industries of Chongqing (cstc2016zdcy-
ztzx0067)..

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