GFET

    graphene field effect transistor
FET

    field effect transistor
cQD

    colloidal quantum dot
AFM

    atomic force microscope
L

    channel length
W

    channel width
EHD

    electrohydrodynamic nanoprinting
IR

    infrared
SWIR

    short wave infrared
PSD

    power spectral density
EDT

    ethane-1,2-dithiol
LbL

    layer-by-layer
NEI

    noise equivalent irradiance
NEP

    noise equivalent power
LSQ

    least squares fit
TLM

    transfer length method

## Scaling of Hybrid QDs-Graphene Photodetectors to Subwavelength Dimension

Gökhan Karaa, Patrik Rohnera,b, Erfu Wua, Dmitry N. Dirinb,d, Roman Furrera,
Dimos Poulikakosc, Maksym V. Kovalenkob,d, Michel Calamea,e, Ivan
Shorubalkoa,*

(a) Laboratory for Transport at Nanoscale Interfaces, Empa – Swiss Federal
Laboratories for Materials Science and Technology, CH-8600 Dübendorf,
Switzerland
(b) Department of Chemistry and Applied Biosciences, ETH – Swiss Federal
Institute of Technology Zurich, CH-8093 Zurich, Switzerland
(c) Laboratory of Thermodynamics in Emerging Technolohgies, Department of
Mechanical and Process Engineering, ETH – Swiss Federal Institute of
Technology Zurich, CH-8092 Zurich, Switzerland
(d) Laboratory for Thin Films and Photovoltaics, Empa – Swiss Federal
Laboratories for Materials Science and Technology, CH-8600 Dübendorf,
Switzerland
(e) Department of Physics and Swiss Nanoscience Institute, University of Basel
, CH-4056 Basel, Switzerland
* Email<EMAIL_ADDRESS>

##### Abstract

Emerging colloidal quantum dot (cQD) photodetectors currently challenge
established state-of-the-art infrared photodetectors in response speed,
spectral tunability, simplicity of solution processable fabrication, and
integration onto curved or flexible substrates. Hybrid phototransistors based
on 2D materials and cQDs, in particular, are promising due to their inherent
photogain enabling direct photosignal enhancement. The photogain is sensitive
to both, measurement conditions and photodetector geometry. This makes the
cross-comparison of devices reported in the literature rather involved. Here,
the effect of device length $L$ and width $W$ scaling to subwavelength
dimensions (sizes down to 500nm) on the photoresponse of graphene-PbS cQD
phototransistors was experimentally investigated. Photogain and responsivity
were found to scale with $1/LW$, whereas the photocurrent and specific
detectivity were independent of geometrical parameters. The measurements were
performed at scaled bias voltage conditions for comparable currents. Contact
effects were found to limit the photoresponse for devices with $L<3$ . The
relation of gate voltage, bias current, light intensity, and frequency on the
photoresponse was investigated in detail, and a photogating efficiency to
assess the cQD-graphene interface is presented. In particular, the specific
detectivity values in the range between $10^{8}$ to $10^{9}$ Jones (wavelength
of 1550nm, frequency 6Hz, room temperature) were found to be limited by the
charge transfer across the photoactive interface.

Keywords: colloidal quantum dots, graphene, phototransistors, infrared
photodetectors, scaling, subwavelength

### Introduction

Emerging photodetectors based on colloidal quantum dots (cQDs) are currently
challenging state-of-the-art infrared (IR) photodectors by low cost solution
processing1, 2, simple spectral tunability1, 2, multiband absorption3, 4, 5,
response speed6, 7, 8, and integration onto flexible substrates9, 10, 11, 12.
PbS cQDs SWIR photodiode arrays were recently fabricated with a pixel pitch of
1.62.13 This is about five times smaller than the pixel size of common InGaAs
photodetectors. Reduced dimension are, on the contrary, also limiting the
number of photons reaching a detector. The integration of photonic structures
is, thus, an attractive way to enhance light-matter interaction in cQD
photodetectors.14, 15 Another way to overcome low light levels is by coupling
a light absorber with a transistor. By introducing PbS cQDs as a photogate to
a graphene field effect transistor (GFET), photoresponsivities of $10^{6}$ to
$10^{7}$ A W were demonstrated.16, 17 The high responsivity values were also
reflected in the specific detectivity of $10^{13}$ Jones at room temperature16
that is comparable to commercial InGaAs photodetectors.

Research in hybrid phototransistors is focused on reducing dark currents by
exchanging the transistor channel material with TMDCs18, 19, 20, 21 or metal
oxides22, 12, energy barrier engineering19, 9, 23, 24, longer charge
extraction distance in the cQD films25, extending the spectral range to longer
wavelength 26, 27, 21, or technological applicability to flexible substrates9,
11, 12 and printing28, 26, 29. Although it is commonly reported that the
photogain $G_{ph}$ follows a $1/L^{2}$ dependence in those devices30, 31, 32,
limited effort has been spent to experimentally validate this prediction. In
addition, the photoresponse is highly dependent on measurement conditions
(e.g., bias, illumination), geometry, and batch-specific fabrication. Thus,
reproducibility and the cross-comparison of individual reports remain highly
involved, and derived scaling laws are inconclusive.

Here, we experimentally demonstrate the scaling of hybrid graphene - PbS cQD
phototransistors with channel length, width, and cQD film thickness. We
highlight a $1/LW$ dependence of the photogain and responsivity, and a
dimension-independent photocurrent and specific detectivity at scaled bias
voltage condition for comparable currents. We further show the photoresponse’s
gate voltage, bias current, light intensity, and frequency dependency and
derive a photogating efficiency for a cQD-graphene interface evaluation.

### Results and Discussion

#### Photocurrent Relation of Hybrid Phototransistors

Figure 1 (a) shows a cross-section of a hybrid phototransistor. Two gold
electrodes (source and drain) contact the graphene channel, and a p-Si back
gate enables charge carrier density tuning in the channel by the field effect.
The phototransistor has a channel length (L), width (W), and a cQDs film
thickness $t_{QDs}$ as the photogate on top. The incoming light creates
electron-hole (e-h) pairs in the absorbing cQD layer. The Fermi level
alignment between graphene and the cQD film causes an electric field at the
interface (depletion region) that separates the photo-generated e-h pairs and
leads to the commonly observed hole transfer to the graphene channel. The
electrons stay trapped in the film for a time $\tau_{trap}$, gating the
channel.16 The photoinitiated e-h separation current between the cQDs film and
graphene is small compared to the channel current $I_{DS}$ and is thus
neglected. The photogate modulation of $I_{DS}$ is an enhancement of the
photosignal and is described as photogain by the rate equation

$G_{ph}=\frac{\\#\;charges\;observed}{\\#\;photogenerated\;charges}=\frac{1/\tau_{transit}}{1/\tau_{trap}}.$
(1)

$\tau_{transit}$ is the transit time of a charge carrier in the channel
between source and drain contacts.30, 33, 32 Within the Drude model, the
transit time can be stated by bias condition and geometry of the channel and
leads to

$G_{ph}=\frac{1}{L^{2}}\tau_{trap}\mu V_{DS}.$ (2)

$V_{DS}$ is the bias voltage between source and drain, and $\mu$ the charge
carrier mobility in the channel. The photocurrent of the hybrid
phototransistors can thus be described by

$I_{ph}=G_{ph}\phi\eta_{Gph}e,$ (3)

where $e$ the elementary charge, and $\eta_{Gph}$ is a photogating efficiency
describing how likely an incoming photon is creating a charge photogating the
transistor channel.34, 30 The photonflux $\phi=P_{in}/E_{ph}$ is defined by
the incoming light power $P_{in}$ and photon energy $E_{ph}=hc/\lambda$. $h$
is Planck’s constant, $c$ the speed of light, and $\lambda$ the wavelength of
the incoming light. Thus, the photoresponsivity

$R=\frac{I_{ph}}{P_{in}},$ (4)

is proportional to the $G_{ph}$ and often stated as $R\sim 1/L^{2}$.

Figure 1 (a) also shows the experimental setup used to characterize the
photoresponse of the investigated detectors. A bias voltage $V_{DS}$ was
applied between the source and drain contacts, and the current $I_{DS}$ was
measured simultaneously. A gate voltage $V_{G}$ was applied to the back gate
to control the 2D charge carrier density $n=C_{SiO_{2}}V_{G}/e$ in the
graphene channel, where $C_{SiO_{2}}\approx 11.9$ nFcm^2 is the capacitance of
the gate oxide. The incoming light was chopped at a frequency $f_{chop}$. This
light modulation induced an AC photocurrent on top of $I_{DS}$ that was
extracted as a voltage drop $V_{ph}$ over a shunt resistance. The incoming
light power was characterized by a reference detector. The calculated the
photoresponsivity for a sub-wavelength dimensioned device of $0.5\times 0.5$
^2 is shown in Figure 1 (b). The inset shows the AFM image of the
characterized device before cQD sensitization. A photoresponsivity of 150AW
was reached at the first excitonic peak at 1530nm. The spin-coating of the PbS
cQDs lead to a blue shift of about 70nm compared to their excitonic peak at
1600nm (diameter $\sim$ 6nm)35 after synthesis. All measurements were
performed in vacuum and at room temperature.

Further, electrohydrodynamic nanoprinting (EHD) and layer-by-layer (LbL) spin
coating were compared to assess the influence of cQD deposition techniques on
the photoresponse. First, in Figure 1 (c), the deposition of cQDs by EHD
printing is depicted. By applying an AC voltage between the gold coated nozzle
and the substrate, cQDs accumulate at the formed meniscus, and droplets
markedly smaller than the nozzle diameter are pulled out by the electric field
from the apex of the meniscus. This enables printing features with a
resolution in the 100nm range.36, 37 For PbS cQDs in particular, a resolution
of $\sim$ 1 was previously demonstrated.28 This method enables precise spatial
control of placing cQDs in a desired geometrical pattern. After printing, a
solid-state ligand exchange treatment was performed to substitute the native
oleic acid with ethane-1,2-dithiol ligands. This single-step treatment reduces
the interparticle distance, a $t_{QDs}$ shrinkage of about 40% was observed,
leading to conductive films. Additionally, the extra control of film thickness
by EHD printing on a single sample level allowed to vary $t_{QDs}$ between 65
to 120nm. The geometry of the device channels were $L\times W=20\times 5$ ^2.

The second applied method was the layer-by-layer (LbL) spin coating. A thin
layer of cQDs was spin coated and directly treated with a solid-state ligand
exchange procedure to EDT. These two steps were repeated six times, resulting
in $t_{QDs}$ of 170nm. By this method, devices with different $L$ ranging from
20 down to 0.5 and a $W=5$ were sensitized with cQDs. In addition, the $W$
scaling was investigated by changing the width between 5 down to 0.5 with a
fixed $L=20$ . Figure 1 (d) shows the investigated devices with the LbL spin-
coating approach before deposition. In contrast to printing, the entire sample
was covered with one cQD thickness (inset). The LbL spin coating has been
proven to yield dense and crack-free conductive films38, optimally suited for
solar cells and photodetectors.39, 40, 41

Figure 1: Photodetector characterization scheme and used PbS cQD deposition
techniques. (a) the cross-section of a hybrid graphene-PbS cQD
phototransistor. Two Au electrodes contacted the graphene channel, and 285nm
SiO2 separated the p-Si back gate from the channel. A $V_{DS}$ bias was
applied, and $V_{G}$ on the back gate modulated the 2D charge carrier density
$n$ in the channel. The incoming light with energy $E_{ph}=hc/\lambda$ was
chopped at a frequency $f_{chop}$. Photocurrents were extracted with a lock-in
amplifier measuring $V_{ph}$ over a shunt resistance. (b) Photoresponsivity
$R$ of a subwavelength dimensioned photodetector of $0.5\times 0.5$ ^2. The
inset shows the AFM image of the GFET before cQD deposition. (c) EHD printing
illustration and photodetectors with a channel geometry $L\times W$ of
$20\times 5$ ^2. The PbS cQD thickness was varied. (d) LbL spin coating
illustration. The inset on the left shows an optical microscope image after
PbS cQD fabrication. The L- and W-scaling devices are shown before cQD
deposition (optical microscope and AFM).

Figure 2 (a) shows the characterized devices of EHD-printed cQDs films with
different layer thicknesses. From top to down, the plots show the source-drain
currents $I_{DS}$, transconductance $g_{m}=dI_{DS}/dV_{G}$ and the amplitude
of the photocurrent $\big{|}I_{ph}\big{|}$ while the gate voltage $V_{G}$ was
swept. $g_{m}$ was derived by a numerical derivative from $I_{DS}$. The
measurements were performed at a bias voltage $V_{DS}$ of 100mV, irradiance
$\mathds{I}_{in}$ of 120W cm^2, chopping frequency $f_{chop}$ of 6Hz, and the
curves were corrected by the individual Dirac point voltage $V_{D}$ (charge
neutrality point) of the GFETs (average $V_{D}$ of -20V). The first excitonic
peak of the cQDs shifted to a wavelength $\lambda$ of 1345nm (shift of about
250 nm). The blue shift of the absorption peak might be explained by oxidation
of the cQDs during printing. The maximum of the measured photocurrent $I_{ph}$
was found at the left, hole-doped region $p$, and right, electron-doped region
$n$, of the Dirac point. The dashed lines (red and blue) indicate where the
maximum photocurrents were measured that, on average, corresponds to 10 V
($n\approx 7\times 10^{11}$ 1/cm^2) from the Dirac point.

The solid red and blue lines (p- and n-doped region) in the $g_{m}$ and
$I_{DS}$ plots indicate where $g_{m}$ reaches about a constant maximum, as
illustrated for the device with $t_{QDs}=120$ nm. The maxima of the
photocurrent coincide with this region and thus are found where the
transconductance is the highest, as reported previously.42 The yellow shaded
region of Figure 2 (a) indicates the influenced region by random potential
fluctuations around the Dirac point in graphene. In this region, the applied
gate voltage loses its ability to tune $n$ up to a point where charges are
only redistributed locally (e-h charge puddles) and, thus, does not affect the
conductivity anymore. This causes a drop of $g_{m}$ towards the Dirac point
and depends on residual doping and temperature $n^{*}+n^{th}$.43

Figure 2 (b) depicts the four devices’ AFM images after ligand exchange
treatment. Thicker cQD films absorb more light reaching 100% absorption at
about 500nm for an absorption coefficient of $\sim 1\times 10^{5}$ cm.35
Correspondingly, an increasing trend of the photocurrent with thicker
$t_{QDs}$ films is observed in the bottom panel of Figure 2 (a). However, to
extract charges from the cQD layer, they need to be in the depletion region
close to the interface.25 This is why a saturation of the photocurrent is
expected before $t_{QDs}\approx 500$ nm is reached. In a previous study, this
saturation thickness was found to be about 160nm for EHD-printed cQD films.28

The photoresponsivity $R$ is used to assess detectors’ efficiency in
converting light into a photocurrent. However, $R$ is a derived quantity and
describes the proportionality between the photocurrent and the incoming light
power.34 Thus, in this study, the main focus was put on $I_{ph}$ as the
measured quantity at compatable conditions across the investigated devices.
Also, applying a constant bias voltage might not always be applicable over the
different length scales as the current is proportionally increasing with
$1/L$. A comparison should thus aim to keep the applied electric field
$\big{|}\vec{E}\big{|}=V_{DS}/L$ constant, which induces a constant $I_{DS}$
condition. Therefore, equation (2) can be reexpressed with a bias-current, and
further taking the channel conductivity ($\sigma=en\mu$) into account, yields

$G_{ph}=\frac{1}{LW}\frac{1}{en}\tau_{trap}I_{DS}.$ (5)

Likewise, devices observe the same irradiance $\mathds{I}_{in}$ rather than
the same light power $P_{in}$. Including (5) and $\mathds{I}_{in}$ into
equation (3) hence results in

$I_{ph}=\frac{1}{E_{ph}}\frac{1}{n}\eta_{QE}\tau_{trap}\mathds{I}_{in}I_{DS},$
(6)

and predicts a geometry-independent photocurrent. Note that $n$ states the
charge carrier density for electrons if positive and for holes if negative to
account for the ambipolar graphene FET behavior.

In Figure 2 (c), the photocurrent of two devices close to the photocurrent
saturation limited film thickness were compared. The two devices were
fabricated independently, and the cQDs deposition method was varied between
EHD printing and LbL spin coating as described in Figure 1 (b) and (c).
$I_{ph}$ was normalized by $I_{DS}$ to account for contact resistances. The
photoresponse shows a $1/n$ proportionality, and drops to zero at the Dirac
point of graphene.

Although solid-state ligand exchanged cQD films are predominantly fabricated
by a LbL approach in the literature, they do not necessarily result in a
higher conductivity than a single-step ligand exchanged approach.38
Accordingly, the devices with the EHD printed cQD films (single-step ligand
exchange) demonstrate a slightly higher photoresponse of about a factor of
1.8. The ratio of the mobilities $\mu$ is about 1.9. Using equation (6), the
photogating efficiency $\eta_{Gph}$ was estimated from the experiments as
shown in the bottom panel. The extracted $\eta_{Gph}$ also takes the different
wavelengths from the measurements (different first excitonic peaks, although
the same cQD are used for the different devices) into account. The efficiency
ratio is about 1.6 and similar to the $I_{ph}$ and $\mu$ ratios. The
photogating efficiency can further be expressed as

$\eta_{Gph}=\eta_{trans}\eta_{abs},$ (7)

where $\eta_{trans}$ describes the charge transfer efficiency, how well a
photogenerated charge can be extracted from the cQD layer to graphene, and
$\eta_{abs}$ light absorption efficiency, how many e-h pairs are generated per
incident photon.30, 33, 44 Furthermore, $\eta_{abs}$ can be estimated to be in
the range of 0.6 to 0.8. As the magnitude of $\eta_{Gph}$ is in the order of
$10^{-6}$ and one order of magnitude is lost to the absorption
($\eta_{abs}<1$), there is only about one every $10^{5}$ charge carriers that
are extracted to graphene. Thus $\eta_{trans}$ is a major source for
improvement and stresses that a single-step ligand exchange is not a limiting
factor.

Figure 2: Photoresponse transfer characteristic and comparison of devices with
EHD and LbL spin-coated films. (a) transfer characteristic of four devices
with EHD PbS cQD film of different thicknesses. From top to down, the source-
drain current $I_{DS}$, transconductance $g_{m}=dI_{DS}/dV_{G}$ and
photocurrent $I_{ph}$ from the same measurement were extracted. The gate
voltage $V_{G}$ is corrected by the Dirac point voltage $V_{D}$. $g_{m}$ was
derived from $I_{DS}$ by a numerical derivative and is highlighted as dots to
distinguish from the measured quantities. The dashed lines show the maximum
$I_{ph}$ in the p- (red) and n-doped (blue) regions of graphene. The solid red
and blue line in $g_{m}$ and $I_{DS}$ indicates the region of about constant
maxima of $g_{m}$ and is displayed for the device with $t_{QDs}=120$ nm only.
(b) AFM images of the characterized devices in (a), showing the thickness
$t_{QDs}$ of cQD films on top of graphene. (c) cross-comparison of two devices
with EHD printed and LbLs spin coated cQDs films. $I_{ph}$ (top panel) was
normalized by $I_{DS}$ to account for contact resistance and bias conditions
$V_{DS}$ of 100 (EHD) and 200 mV (LbL). Devices were measured at a wavelength
$\lambda$ matching the respective first excitonic peaks of 1345 (EHD) and 1550
nm (LbL), and $\mathds{I}_{in}=120$ μW cm^2. The bottom panel shows the two
devices’ derived photogating efficiency $\eta_{Gph}$. Yellow shading
illustrates where the highest photocurrent $I_{ph}$ has been found. The shaded
region might originate from random potential in graphene around the Dirac
point $n^{*}+n^{th}$, where a gate can not control the conductivity of
graphene.

#### Scaling of Pixel Dimensions

Next, the geometrical scaling of hybrid graphene – cQD phototransistors was
investigated. All devices were fabricated on the same sample, and the cQD film
was prepared by LbL spin coating to reduce device-to-device variations. First,
equation (6) was validated by bias, irradiance, and frequency sweeps for
devices with different $L$ and a $W$ of 5. Figure 3 (a) shows the linear
relation between the source-drain current $I_{DS}$ and $I_{ph}$ with a
proportionality factor $K_{1}$. The bias voltage was scaled according to
$V_{DS}=\big{|}\vec{E}\big{|}L$ for comparable electric fields in the
different devices. For better visualization, the measured source-drain current
$I_{DS}$ was normalized by its highest value $I_{DS,max}$. The inset shows
each device’s highest photocurrents $I_{ph,max}$ at $I_{DS,max}$. Every device
was measured at the gate voltage where the highest photocurrent was observed
in the p-doped region of graphene (red dashed line in Figure 2 (b)). The
n-doped photocurrent maxima was experimentally not accessible as the LbL spin
coating led to Dirac point voltages up to 70V.

From equation (6), there is no length dependence expected, and all curves in
Figure 3 (a) should lay on a single line. The inset shows that despite the
applied comparable electric field condition across devices, $I_{DS}$ still
scales with $L$. This can be explained by a voltage drop over the contacts
that becomes comparable to the voltage drop over the channel for smaller $L$,
and thus $\big{|}\vec{E}\big{|}$ is not the same over the different channels
despite the scaled $V_{DS}$. The contact resistance $R_{C}$ is estimated to be
in the range of about 0.6kΩ per contact by performed transfer length method
(TLM) measurements. Thus, the ratio of $2\times R_{C}$ and the channel
resistance $R_{Ch}$ become comparable ($2R_{C}/R_{Ch}\geq 0.5$) for devices
with $L<5$ . Reversibly, the contact resistance can be estimated from the
spread of the $I_{ph}$ vs. $I_{DS}$ curves assuming the same
$\big{|}\vec{E}\big{|}$ over the channel leads to the same $I_{DS}$. This also
results in $R_{C}\approx 0.6$ kΩ. With the fact that $I_{ph}$ is linearly
proportional to $I_{DS}$, this confirms the origin of the $I_{ph}$ spread from
contact limitations.

In Figure 3 (b), irradiance sweeps were performed for the same set of devices.
This $I_{ph}$–$\mathds{I}_{in}$ relation is linear with a proportionality
factor $K_{2}$. $K_{2}$ is related to the responsivity by
$R=\frac{K_{2}}{LW}$. As the curves follow a linear fit (dotted line), the
detectors were operated in the linear dynamic regime ($R=const$), reaching
responsivity values about 300AW for $\big{|}\vec{E}\big{|}=10$ kV/m and
$f_{chop}=6$ Hz. The inset highlights the noise equivalent irradiance (NEI)
for the different devices in the range of 1 to 10μWcm^2. The specific
detectivity can be calculated by32, 30

$D^{*}=\frac{\sqrt{LW}\sqrt{\Delta f}}{NEP}=R\frac{\sqrt{LW}\sqrt{\Delta
f}}{I_{noise}}.$ (8)

$NEP=NEI\times LW$ is the noise equivalent power, $\Delta f$ the frequency
bandwidth, and $I_{noise}$ the noise current. The resulting detectivities are
about $2\times 10^{8}$ Jones at 6Hz and with $\Delta f=0.026$ Hz (equivalent
noise bandwidth of lock-in measurement).

Figure 3 (c) shows the frequency response of the investigated devices. The
light chopping frequency $f_{chop}$ was varied between 5 and 200Hz. A fit to
data relates $I_{ph}=K_{3}f^{\alpha}$ dependence, where $K_{3}$ is a
proportionality constant. The curves follow $\alpha=-0.5$ ($\pm$ 0.1) up to
frequencies of about 160Hz. A differing photocurrent dynamics can be observed
for the 1 channel length device (dark brown). Around 70Hz, the trend deviates
from the $\alpha=-0.5$ as $I_{ph}$ increases with $f_{chop}$ before it drops
eventually.

The chopping frequency is defined over which period $\tau_{chop}$ the detector
sees the light. Thus, it probes $\tau_{trap}$ of the cQD film. Previous
experiments have shown that at least two trapping times can be present
underlying the photocurrent dynamics.45, 29, 46 In analogy to low frequency
$1/f$ noise47, the frequency response might originate from an envelope of
individual Lorentzian shaped trap state contributions. Thus, the slope of the
envelope $\alpha$ is a cQD film-dependent quantity and depends on the specific
film treatment as it varies across the literature (values between -0.1 to
-2).25, 11, 45 Within this frame, a possible explanation of the rise of
$I_{ph}$ around 130Hz might be the quenched escape rate from such a trap, as
new carriers are excited before the trapped ones escape. Consequently, the
traps might be constantly populated, and newly photoexcited e-h pairs can
escape without being slowed down from those traps, leading to higher
photocurrents. The energy depth of such a trap $i$ can be estimated by $\Delta
E_{trap,i}=-kTln(\tau_{0}/\tau_{trap,i})$, with $k$ the Boltzmann constant,
and $T$ the temperature.48 The free charge carrier lifetime $\tau_{0}$ is the
recombination time of a charge carrier contributing to the photocurrent,
excluding the time it spends in a trap. For $1/\tau_{trap,i}=130$ Hz an energy
depth of $\Delta E_{trap,i}\approx 250$ meV can be estimated, that corresponds
to previously found surface trap state energies for PbS cQD films between 100
to 300meV.49 For a spin coated film, this trap state is expected to be equally
distributed throughout the film. However, in Figure 3 (c), the trap appears
for the device with a channel lengt of 1, and might be explained by the
modulation of the trap in the vicinity of the source and drain electrodes.

Figure 3 (d) shows the bias current normalized power spectral density of the
measured noise current. The curves show a low frequency $1/f$-noise dependence
as typically reported47 for graphene FETs. The measurements were performed in
the dark and at the same scaled bias condition
$V_{DS}=\big{|}\vec{E}\big{|}L$. Fit to the curves reveal a mean exponent of
-1.01 $\pm$ 0.02. The inset shows the derived specific detectivity $D^{*}$
using equation (8) and taking the photocurrents from Figure 3 (c) into
account. The yellow line ($L=$ 20) has a near constant $D^{*}$ up to a
frequency of 200Hz and represents most of the investigated devices. On the
other hand, the dark brown curve ($L=$ 1) shows an increase of $D^{*}$ up to
130Hz, possibly due to quenched trapping sites leading to a rise of the
photocurrent. The $D^{*}$ values of about $2\times 10^{8}$ Jones fit well with
the estimation from NEP found in Figure 3 (b).

All the three introduced proportionality factors $K_{j}$ are related to
$\eta_{Gph}$ and can be expressed from equation (6) as

$\eta_{Gph}=nE_{ph}\frac{1}{\tau_{trap}}\frac{1}{\mathds{I}_{in}}\frac{1}{I_{DS}}I_{ph}.$
(9)

The decay time $\tau_{trap}$ describes how long it takes to recover the dark
current once the light is turned off. Thus, $\tau_{trap}$ defines the
detector’s maximum reachable photogain $G_{ph}$. To describe the
experimentally observed $I_{ph}\sim f^{\alpha}$ dependence, and the fact that
$f_{chop}$ probes $\tau_{trap,i}$, the chopping frequency can be modeled by
$\tau_{trap}=\beta f_{chop}^{\alpha}$. Here, $\beta$ is introduced to formally
correct the unit discrepancy arising from the fitting constant $\alpha$. As a
result, the estimation of $\eta_{Gph}$ can be performed with $f_{chop}$,
$\mathds{I}_{in}$, and $I_{DS}$ sweeps. Using the proportionality factors
$K_{j}$ from Figure 3 (a) to (c), the photogating efficiency was estimated in
Figure 3 (d). $\eta_{Gph}$ yield a channel length dependence and similar
values across the different measurements. Furthermore, the values of $\sim
10^{-6}$ are compatible with the observation in Figure 2 (c). A similar
decreasing trend was observed for both $g_{m}$ and $\eta_{Gph}$ towards
smaller $L$. Previously, contacting metals have been found to dope $\sim 500$
nm into the graphene channel50, thus reducing the effective gatable length of
the channel, and limiting $g_{m}$ for small devices. As both $g_{m}$ (back
gate) and $\eta_{Gph}$ (photogate) are related to the gating of the graphene
channel, the same loss of gatable channel length might be the cause for
decreasing $\eta_{Gph}$ with smaller $L$.

Figure 3: Bias, irradiance, and frequency dependence of photocurrent for a set
of devices with $W=5$ , varying $L$, and LbL spin coated cQDs PbS film of
$t_{QDs}\approx 170$ nm. (a) Linear depence of $I_{ph}=K_{1}I_{DS}$. Bias
voltage $V_{DS}$ range was scaled to reach a maximum of
$\big{|}\vec{E}_{max}\big{|}=20$ kV m. $I_{DS}$ was normalized by its maximum
value for better illustration. The highest measured $I_{ph}$ and $I_{DS}$ for
an extended set of devices are shown in the inset. (b) Linear dependence of
$I_{ph}=K_{2}\mathds{I}_{in}$. The dotted lines show linear least squares fit
(LSQ) through the data. Inset shows the noise equivalent irradiance (NEI) in
the log-log plot. (c) Frequency response of $I_{ph}$ by sweeping the light
chopping frequency $f_{chop}$. $I_{ph}=K_{3}f_{chop}^{\alpha}$, with
$\alpha=-0.5\pm 0.1$ estimated from LSQ fit to log-log plot. (d) power
spectral density (PSD) of the noise current $I_{noise}$, measured in dark. PSD
was normalized by the measured $I_{DS}$. LSQ fit the log-log plots show a mean
exponent of -1.01 $\pm$ 0.02 to the frequency $f$. (e) Photogating efficiency
$\eta_{Gph}$ estimate based on LSQ fits giving proportionality factors $K_{j}$
from (a),(b), and (c). The full formula is given in the figure. $\tau_{trap}$
was estimated as $1/\sqrt{6}s$ in agreement with fits in (c). In (b),(c), and
(d) $V_{DS}$ was scaled according to $\big{|}\vec{E}\big{|}=10$ kV m. Note
that measurements were performed on an extended set of devices including $L$
of 0.5, 1, 2.5, 5, 10, 20, but for clarity, only a distinct set is plotted in
(a) to (d). Applied gate voltage of $V_{G}=58$ V for devices with $L$ of 0.5,
2.5, 10 and $V_{G}=48$ V for $L$ of 1, 5, 20 for all the measurements.

Next, the scaling of the photorespons by $L$ and $W$ was investigated as seen
in Figure 4. For that devices with $L$(number of devices) of 0.5 ($\times$5),
1 ($\times$6), 2.5 ($\times$4), 5 ($\times$5), 10 ($\times$3), and 20
($\times$7) with a width of $W=$ 5 were measured for the $L$-scaling. For
$W$-scaling $W$(number of devices) of 0.5 ($\times$3), 1 ($\times$5), 2.5
($\times$4) and 5 ($\times$7) with a $L=$ 20 were characterized. All the
devices were on the same chip and had identical cQD film properties, allowing
for a cross-comparison. The quantities $I_{ph}$, $I_{DS}$, $n$ where extracted
from gate sweeps as described in Figure 2 (a). The values were taken at
respective maximum photocurrent points in the p-doped region of graphene. As
the photoresponse is proportional to $I_{DS}$, $V_{DS}$ was scaled for a
comparable current condition according to $I_{DS}\sim W\frac{V_{DS}}{L}$.

In Figure 4 (a), $G_{ph}$ scaling by $L$ (top) and $W$ (bottom) is presented.
A fit to the data (black dotted line) shows the $1/L$ trend and agrees with
the prediction from equation (5). The horizontal dotted line indicates the
mean value in that range. The scaling law breaks down around 1.5 (orange
shaded region), where the total graphene FET resistance $R_{T}$ is about 2kΩ.
This is comparable to the contact resistance of 1.2kΩ (600Ω per contact) and
thus points to a current injection limitation at the specified biasing
condition. The $W$-scaling follows a predicted $1/W$ trend over the evaluated
region. As the $W$-scaling devices have a $L=$ 20, the smallest resistance for
the $W=5$ is about 8kΩ such that no current injection limitation is observed.

Figure 4 (b) depicts the scaling of photoresponsivity $R$. The responsivity
was normalized by $n/I_{DS}$ to account for device-to-device variations
(different residual doping levels) and contact resistance. $R$ follows also a
$1/L$-scaling law. Although $I_{ph}$ is independent on geometry for constant
$\mathds{I}_{in}$, $P_{in}=\mathds{I}_{in}LW$ and thus

$R=\frac{I_{ph}}{\mathds{I}_{in}LW}.$ (10)

The breakdown of the scaling law, however, exceeds the current injection
limited region caused by contacts (yellow shaded) and extends up to $\sim$ 3\.
The inset shows the normalization factor additionally accounting for
$\eta_{Gph}$. As $\eta_{Gph}$ is introduced, the red shaded area appears, thus
being responsible for an additional length limitation to the scaling. The
bottom panel shows the same normalized $R$ for the $W$-scaling. In fact, as
for the gain, there is a good agreement for the $1/W$ scaling observed over
the investigated width scale. Moreover, the normalization factor, including
$\eta_{Gph}$, shows a constant trend. This highlights that $\eta_{Gph}$ might
be similarly affected by reduced gatable channel lengths due to metal doping,
as mentioned for $g_{m}$ above.

Finally, the influence of scaling on the noise and specific detectivity
$D^{*}$ was investigated in Figure 4 (c). From $I_{DS}^{2}$ normalized PSD
spectra as shown in Figure 3 (d), the values at 6Hz were extracted for every
device. The measurements were performed in dark and at the same gate voltage
where the maximum photocurrent was extracted previously. The dotted lines show
fits through the mean values of each length or width, whereas the purple
shading indicates the standard deviation. The fits suggest a $1/\sqrt{L}$ and
$1/W$ dependence for the characterized devices. For the $L$-scaling, the
distribution of points increases as $L$ gets smaller, whereas the $W$-scaling
has an overall smaller spread of the data. The inset shows derived $D^{*}$
about 3e8Jones, using the photoresponse shown in (b).

Low frequency $1/f$ noise in GFETs is primarily attributed to electrostatic
fluctuations arising from trap states at the oxide interface and leads to
conductivity fluctuations in the channel.47, 51 Conductivity $\sigma$ is
related to the product of $n\mu$. Hence, the two main models that describe
$1/f$ noise are based on $\Delta n$ (McWorther’s model) or $\Delta\mu$
(Hooge’s law) fluctuations.52, 47 Both models predict a $1/LW$ dependence of
noise in the FET channels.52 Accordingly, a volume scaling was experimentally
verified for metallic break junctions to hold in the diffusive transport
regime.53 For graphene FETs, however, the $1/LW$ scaling has been found to
experimentally not describe the data in full detail.54 Being restricted to a
surface only, graphene is very sensitive to its environment. Thus,
perturbations to the channel such as bilayers55, contacts56, 54, and
environmental exposure54 are altering the electrostatic screening potential of
the channel and influence the low-frequency $1/f$ noise. Like the
photoresponse, though, reducing $L$ is pronouncing the contact effects. Hence,
an increasing spread for smaller $L$ channels is observed. In addition,
possible regions of double layers due to non-ideal CVD graphene growth and the
additional current path through the cQD can thus be the reason for a deviation
from an overall $1/LW$ trend observed in the PSDs.

Expanding the demonstrated photocurrent characteristics of equation (6) into
the detectivity yields

$D^{*}=\frac{1}{n}\frac{1}{E_{ph}}\tau_{trap}\eta_{QE}\frac{1}{\sqrt{LW}}\frac{1}{\sqrt{PSD/I_{DS}^{2}}}.$
(11)

If now $\sqrt{PSD/I_{DS}^{2}}\sim 1/\sqrt{LW}$, there is no geometrical
dependence expected in $D^{*}$, which is consistent with the performed
measurements. It is important to mention that the experimentally determined
$D^{*}$ values between $10^{8}$ to $10^{9}$ Jones are around four orders of
magnitude lower than the highest values found in the literature16 of $10^{13}$
Jones. The difference can be explained mainly by two factors from equation
(11). First, the difference is in the found $\eta_{Gph}$ range. As
$\eta_{Gph}$ does not differ much for devices with EHD printed (single step
ligand exchange) and a LbL spin coated cQD films, the interface between
graphene and cQD film limits most likely the charge transfer. CVD graphene
processing is prone to defects and residuals from the growth and device
fabrication. On the contrary, the highest $D^{*}$ values were reached with
exfoliated single-crystal graphene. Second, $\tau_{trap}$ defines the maximum
reachable gain and is related to the photocurrent dynamics ($I_{ph}$ to
$f_{chop}$ relation). Preserving the $I_{ph}$ magnitude for increasing
$f_{chop}$ leads to increasing $D^{*}$ due to lower $1/f$ noise in the channel
as demonstrated in Figure 3 (c) and (d).

Figure 4: Length and width scaling of photoresponse. (a) photogain $G_{ph}$
scaling. LSQ fits exponents for $L$ of 1.05$\pm$0.025 and for $W$ of
0.95$\pm$0.025 (dotted black line). The orange-shaded region highlights where
the $L$-scaling breaks down as contact resistances become comparable to device
channel resistances. (b) Responisivity $R$ scaling. $R$ was normalized by
charge carrier density $n$ (different residual doping levels) and $I_{DS}$
(contact resistance). An exponent for $L$ of 0.95$\pm$0.05 and $W$ of
1.05$\pm$0.1 was fitted by LSQ. The $L$-scaling shows contact limited (orange)
and $\eta_{QE}$ limited range. The inset shows the $\eta_{QE}$ extended
normalization factor for $R$, where the red-shaded region becomes visible. (c)
power spectral density (PSD) of the noise current $I_{noise}$. The noise was
measured in dark, and PSD was normalized by the measured $I_{DS}^{2}$. LSQ
fits exponents of 0.502$\pm$0.02 for $L$-scaling and 1.05$\pm$0.03 (black
dashed line). Purple shading shows the standard deviation of the mean values.
Specific detectivity $D^{*}$ estimates between $10^{8}$ to $10^{9}$ are shown
in the insets. The bias voltage $V_{DS}$ was scaled for a constant current
condition according to $I_{DS}\sim W\frac{V_{DS}}{L}$. All values for (a) and
(b) are extracted from maximum $I_{ph}$ points aquired in gate voltage sweeps,
at a wavelength $\lambda=1550$ nm, $\mathds{I}_{in}=120$ μW cm^2, and
$f_{chop}=6$ Hz. (c) was measured at the same $V_{G}$ values as in (a) and
(b). The dotted horizontal lines indicate the mean values of the respective
data sets.

### Conclusion

Here, the scaling of hybrid graphene-PbS cQD phototransistors with channel
length $L$, width $W$, and cQD film thickness $t_{QDs}$ was experimentally
demonstrated. The photocurrent $I_{ph}$ was found to be independent of $L$ and
$W$ but increased with $t_{QDs}$. Instead, the photogain $G_{ph}$ and
responsivity $R$ scaled with $1/LW$. Although a $1/L^{2}$ dependence can be
derived for $G_{ph}$, this neglects increasing channel currents with
$I_{DS}\sim V_{DS}W/L$. Relating $G_{ph}$ to $I_{DS}$ leads to the $1/LW$
relation. For $R$, on the contrary, the geometrical dependence is evolving
from a constant irradiance $\mathds{I}_{in}=P_{in}/LW$, rather than a constant
light power $P_{in}$ seen by the detector. The $L$ scaling broke down for
$L<3$ due to contact metal contributions (contact resistance and channel
doping). Although specific detectivity is $D^{*}\sim R$, the found values in
the range between $10^{8}$ to $10^{9}$ Jones ($\lambda=1550$ nm, $f=6$ Hz,
room temperature) were independent on geometry. This is due to increasing
noise currents for smaller dimensions. The photoresponse dependency on back
gate voltage $V_{G}$, bias $I_{DS}$, irradiance $\mathds{I}_{in}$, and light
chopping frequency $f_{chop}$ were analyzed. A derived photogating efficiency
$\eta_{Gph}$ of $\sim 10^{-6}$ from those measurements pointed to a cQD-
graphene interface limited charge transfer as a source for further
photoresponse improvement.

### Methods

##### Graphene Growth

CVD graphene was grown on commercial copper foil (Foil 2017, No. 46365, Alfa
Aesar). The Cu-foil was first sonicated in acetone (15min) and rinsed with
deionized (DI) water. The foil was then immersed in nitric acid HNO3 (30min),
twice sonicated in DI-water (1min), immersed in ethanol (1min), and blow-dried
with N2. The Cu-foil was annealed in H2 (20sccm H2 and 200sccm Ar mixture) at
1000 and a pres-sure of $<$ 1mbar for 70min in a furnace. The graphene was
grown from 0.04sccm methane CH4 for 21min at 1000 and a pressure of 110mbar,
subsequentyl cooled down by opening the temperature shielding.

#####  GFET Fabrication

A p-Si substrate with 285nm chlorinated dry thermal SiO2 was prepatterned with
contacts. The contacts were fabricated by e-beam-assisted thermal evaporation
of Ti/Au (5/40 nm) on a photolithographically defined resist (AZ5214E). PMMA
50K-protected graphene was prepared for wet transfer by Cu-etching in
commercial Transene etchant (1 h), rinsed with DI water, left floating on
hydrochloric acid (5 min), and re-rinsed with DI water. Graphene was
transferred on the pre-patterned p-Si substrate from DI water, dried on a
hotplate, and put into a vacuum ($\sim$60h). Graphene channels were O2-plasma
etched using a Cu-mask, followed by Cr/Au top-contacts (2/40 nm), both defined
by e-beam lithography (PMMA 50K/PMMA 950K).

##### PbS cQD Film Fabrication

PbS QDs were synthesized following Hines et al. 57 with slight adaptations as
described in detail previously29. For electrohydrodynamic nanoprinting (EHD),
the PbS cQD were redispersed in n-tetradecane with a concentration of 40mg ml
and filtered with 0.1 PTFE syringe filter. Printing nozzles with openings
between 2 to 3 um were fabricated by pulling glass capillaries (World
Precision Instruments TW100-4) and subsequent coating Ti/Au (5/50nm). For the
printing, the nozzle was brought in $\sim 5$ proximity to the sample and an AC
voltage of 280V (250Hz) was applied between sample and nozzle. The structures
were printed with a stage speed of 6s, line-spacing about 500nm, and
subsequent overprinting was oriented in a staggered fashion. The EHD printing
system used here was previously described in more detail.58 A single-step
ligand exchange was performed by overnight soaking the cQD films in 2 vol%
ethane-1,2-dithiol (EDT) in acetonitrile. Subsequently, the samples were
rinsed in acetonitrile and dried with N2.

For the layer-by-layer (LbL) prepared film, the PbS cQDs were redispersed in
octane with a concentration of 20mg ml. A layer of PbS cQDs was spin-coated
(2500rpm, 45s), subsequently a drop of 2 vol% ethane-1,2-dithiol (EDT) in
acetonitrile was placed for 30s before spinning the sample dry, and followed
by one drop of acetonitrile and one drop of octane while sample was spinning
(2500rpm, 45s). The steps were repeated six times for a $\sim 170$ nm film
thickness.

Before film fabrication, the PbS cQDs were characterized by UV-Vis
spectroscopy (Jasco V-670). The fabricated PbS cQD films were analyzed by AFM
(Bruker Dimension ICON 3), and the device cross-section was investigated by
FIB-SEM (FEI Helios NanoLab G3 UC).

##### Device Characterization

The GFET were electrical characterization before PbS cQD film deposition in a
two-probe configuration (Keithley 4200-SCS semiconductor characterization
system). The field-effect mobility
$\mu=\frac{L}{WV_{DS}C_{G}}\frac{dI_{DS}}{dV_{G}}$ was estimated by linear
least squares fit (LSQ) to transfer curves ($I_{DS}$ vs. $V_{G}$). The contact
resistance was estimated by the transfer length method (TLM).

To characterize the photoresponse, a broadband light source (Thorlabs, SLS201)
was modulated with a chopper (Thorlabs, MC2000B-EC) and focused onto a
monochromator (Princeton Instruments, SpectraPro HRS-300 spectrometer with
grating 150 G/mm, blaze 0.8 ). Subsequent long-pass filters attenuated the
higher spectral orders (400, 600, 800, 1200, 1900 nm) at the monochromator
exit. The light was collimated with a lens and split with a 50/50 Polkadot
beamsplitter. Onto one end, a reference detector (Gentec, UM-9B-L) was placed,
and the light irradiance was determined by a lock-in amplifier (Stanford
Research System, SR865A). Onto the other light path end, the samples were
placed into an optically accessible cryostat (JANIS ST-100) with a quartz
glass window. Gate voltage and source-drain bias were applied with SMUs
(Keithley, 2614B and 2450). The photovoltage was measured over a shunt
resistance of 1kΩ with lock-in amplifiers (Stanford Research System, SR860).

Noise currents were measured in the dark with a battery-powered trans-
impedance amplifier (Stanford Research Systems, SR570). The signal was low
pass filtered (10 kHz) and subsequently acquired with a data acquisition board
(National Instruments, USB6341) at a sampling rate of 500 kHz. The DC offset
was removed for the power spectral densities estimates, and $10\times$ one-
second-long time traces were averaged.

All measurements were performed at room temperature and vacuum (4$\cdot
10^{-7}$ mbar). The line frequency of 50Hz and its harmonic 150Hz were removed
from all the frequency-dependent measurements. The measurements were repeated
throughout the study duration for a selected set of devices ($\sim 21$ days).
In between the measurements, the samples were taken out of the measurement
setup, rebonded in atmosphere, and placed back in the measurement setup. The
measurements were reproducible with a maximal deviation of $\sim$ 30%. A
device of each dimension was measured before the next one with the same
dimension was investigated to reduce the influence of possible variation from
sample alignment or degradation on the study outcome.

### Acknowledgements

The authors thank FIRST-Lab (Center for Micro- and Nanoscience) at ETH Zurich
for access to the clean-room and the Swiss National Science Foundation (SNSF,
project no. 200021 182790) for financial support.

### Author Contribution

G.K. and I.S. conceived the study and planned the experiments. G.K. fabricated
the samples, performed the measurements and analyzed the data. P.R. performed
the EHD printing with supervision by D.P. E.W. performed the FIB-SEM images.
D.D. synthesized PbS colloidal QDs. R.F. grew graphene. G.K., I.S. and M.C.
discussed the data. I.S., M.C., M.K, D.P. initated and supervised the project.
G.K. wrote the manuscript with inputs and discussions from all authors.

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  *[cQD]: colloidal quantum dot
  *[cQDs]: colloidal quantum dot
  *[IR]: infrared
  *[SWIR]: short wave infrared
  *[GFET]: graphene field effect transistor
  *[EHD]: electrohydrodynamic nanoprinting
  *[LbL]: layer-by-layer
  *[EDT]: ethane-1,2-dithiol
  *[GFETs]: graphene field effect transistor
  *[LbLs]: layer-by-layer
  *[TLM]: transfer length method
  *[LSQ]: least squares fit
  *[NEI]: noise equivalent irradiance
  *[PSD]: power spectral density
  *[FET]: field effect transistor
  *[PSDs]: power spectral density


# Bayesian Nonparametric Models for Synchronous Brain-Computer Interfaces

Jaime F Delgado Saa, Mujdat Cetin Jaime F Delgado Saa is with the Department
of Electrical and Electronics Engineering, Universidad del Norte, Colombia,
e-mail: (jadelgado@uninorte.edu.co).Mujdat Cetin is with the Faculty of
Natural Sciences and Engineering, Sabanci University, Turkey

###### Abstract

A brain-computer interface (BCI) is a system that aims for establishing a non-
muscular communication path for subjects who had suffer from a
neurodegenerative disease. Many BCI systems make use of the phenomena of
event-related synchronization and de-synchronization of brain waves as a main
feature for classification of different cognitive tasks. However, the temporal
dynamics of the electroencephalographic (EEG) signals contain additional
information that can be incorporated into the inference engine in order to
improve the performance of the BCIs. This information about the dynamics of
the signals have been exploited previously in BCIs by means of generative and
discriminative methods. In particular, hidden markov models (HMMs) have been
used in previous works. These methods have the disadvantage that the model
parameters such as the number of hidden states and the number of Gaussian
mixtures need to be fix a priori. In this work, we propose a Bayesian
nonparametric model for brain signal classification that does not require a
priori selection of the number of hidden states and the number of Gaussian
mixtures of a HMM. The results show that the proposed model outperform other
methods based on HMM as well as the winner algorithm of the BCI competition
IV.

##  I Introduction

A Brain Computer Interface (BCI) is a system that provides an alternative
communication path for people who had suffer a disease or have had an accident
that compromises their ability to perform motor tasks. Also, applications for
healthy subjects in areas of multimedia and gaming started to incorporate
these technologies in the recent years [1, 2]. BCIs make use of the brain
signals to control external devices that help the subject to communicate and
interact with the environment. A significant portion of approaches to BCIs are
based on the comparison of power values of the EEG signal during the execution
of imaginary motor tasks. However, the well-known phenomena of Event Related
Synchronization (ERS) and Event Related De-synchronization (ERD) [3] provide
more information that can be used for improving the performance of the BCIs.
This information is related not only to the difference of power of the signals
during the execution of different motor tasks but also to their change in time
in different frequency bands. In order to make use of this information more
explicity in a statistical modeling context, algorithms such as Hidden Markov
Models (HMM) [4, 5] have been used in combination with features that describe
the temporal behavior of the EEG signals [6, 7]. Although the use of EEG time-
frequency features provide good results [8, 9, 10, 11, 12] in BCI
applications, a combination of the time-frequency power features and
algorithms that take into account the temporal dynamics of those features have
been rarely reported [4, 5, 13]. One possible reason for this is that
selection of model parameters (states, Gaussian mixtures, etc. in HMM) and
frequency bands becomes problematic. In this work, we propose the use of the
signal time-frequency distribution calculated using auto-regressive models,
and a Bayesian nonparametric model for classification of two different motor
tasks. The proposed model selects the model order directly from data assuming
initially an infinite number of hidden states and an infinite number of
Gaussian mixtures for modeling of the observations. The results show that the
Bayesian approach outperforms the other HHM-based approaches used for
comparison and other methods that report results on the data set used in this
work.

##  II HMM Approach

A HMM is a finite automaton which contains a set of discrete states $H$
emitting a feature vector $x_{t}$ at each time point $t$. Given that these
kind of models are generative, it is necessary to determine the joint
probability over observations and labels, which requires all possible
observation sequences to be enumerated. In order to make the inference problem
tractable, conditional independence is assumed, meaning that future states are
independent from the past states given the current. The structure of the state
sequence is described by

$h_{t}|h_{t-1}\sim\pi_{h_{t-1}}$ (1)

where $\pi_{h_{t-1}}$ is a state dependent transition distribution for the
state $h_{t-1}$. We also have that given $h_{t}$, $x_{t}$ is independent of
observations and states at different time points:

$x_{t}|h_{t}\sim F(\theta_{h_{t}})$ (2)

that is, $x_{t}|h_{t}$ is distributed according to a function
$F(\theta_{h_{t}})$ where $\theta_{h_{t}}$ are the emission parameters related
to the state $h_{t}$.

The learning problem is defined as a search for the parameters that maximize
the log-likelihood of the observations. These parameters are the state
transition matrix $\pi$, for which each entry $\pi_{i,j}$ represents the
probability to pass from state $i$ to state $j$, the vector of initial
probabilities $\pi^{0}$ so that $h_{1}\sim\pi^{0}$, and the parameters
$\theta$ of the emission distribution $F$ for each state. In this work, the
emission distribution for each state is modeled using Gaussian mixtures.

The joint probability density function of the observations and states is given
by:

$P(h_{1:n},x_{1:n})=\pi^{0}(h_{1})p(x_{1}/h_{1})\prod_{t=2}^{n}{p(x_{t}/h_{t})p(h_{t}/h_{t-1})}$
(3)

The joint distribution of the HMM is represented by the graph in Figure 1. The
hidden states $h$ represent different states during the execution of specific
mental tasks. Such states are related to the power levels of the signal. The
assumption of these different states during the execution of the task is
justified by the time course of the EEG power in specific frequency bands.
Figure 2 shows the average spatial distribution of the scalp EEG power in the
alpha band (8-13 Hz) for one subject using bipolar montages over the
electrodes locations C3, Cz and C4 at different time points, for two different
classes. This reveals the dynamics (sequence of states) of the signal during
the execution of different motor tasks. The sequence of these states is
modeled by a HMM for each class using training data. Inference in the model is
done using the forward-backward algorithm [14] which provides an efficient
scheme for passing messages in the graph, making possible the calculation of
marginals for problems of filtering $p(h_{t}|x_{1},...,x_{t})$, prediction
$p(h_{t+\tau}|x_{1},...,x_{t})$ for $\tau>0$ and smoothing
$p(h_{t}|x_{1},...,x_{\tau})$ for $\tau>t$.

In the BCI problem studied in this work, the decision about the task executed
by the subject is made at the end of the trial. Therefore, the problem to be
solved is one of filtering. Model parameters ($\pi$, $\theta$) are learned
from training data using the Expectation-Maximization algorithm [15]. During
the expectation step the expected value of the likelihood of the model is
calculated using the forward-backward algorithm. During the maximization step,
gradient search is used to maximize the expected value of the likelihood.

The selection of the number of hidden states and the number of Gaussian
mixtures is made in two ways: 1.) Based on the description of the task, the
number of hidden states is set to three, representing start, execution, and
end of the task. The number of Gaussian mixtures was fixed to two as a
reasonable tradeoff between the expressive power of the model and its
simplicity. 2) Three-fold cross-validation is performed using the training
data, looking for a combination of the number of hidden states and the number
of Gaussian mixtures that maximizes a cost function, that in this work is
selected to be classification accuracy.

Once the number of hidden states and the number of Gaussian mixtures is fixed
one model is trained for each class using all the training data available.
During testing, the label assigned to each sequence is determined by
calculation of the likelihood of the data on each model. The model with higher
likelihood determines the assigned label.

Figure 1: Graphical model representation of a HMM

(a) Class one: Left hand

(b) Class two: right hand

Figure 2: Scalp topographical distribution of the power during the execution
of two different imaginary motor tasks.

##  III Bayesian Nonparametric HMM Approach

The motivation for using HMM-like models is to extract temporal information
from the brain signals. The underlying assumption is that during the execution
of the task the rhythms produced by the brain go through a sequence of states.
We link these states to changes in the power of the brain signals in specific
frequency bands. However, the number of states is not known a priori.
Furthermore, during each one of these states the distributions of the brain
signals are complex and therefore, are better modeled by multimodal
distributions. Again, the number of components involved in each multimodal
distribution is unknown. The Hierarchical Dirichlet Processes HMM (HDP-HMM)
with Dirichlet Processes Gaussian Mixtures [16] presents a solution to such
problems faced by conventional HMM approaches.

###  III-A The HDP-HMM and the Sticky HDP-HMM

In [17] a nonparametric Bayesian approach to HMM in which the HDP defines a
prior distribution on transition matrices over countably infinite state spaces
is presented. The HDP-HMM leads to data-driven learning algorithms which infer
posterior distributions over the number of states. One serious limitation of
the HDP-HMM presented by [17] is that the model has a tendency to produce
unrealistic rapid dynamics. In [16] a modified HDP-HMM so called sticky-HDP-
HMM is proposed, augmenting the original HDP-HMM by including a parameter for
self-transition bias and placing a separate prior on this parameter.

A DP denoted by DP($\gamma,H$), where $H$ is a base distribution and $\gamma$
is a concentration parameter, is a distribution over countably infinite random
measures:

$\displaystyle
G_{0}(\theta)=\sum_{k=1}^{\infty}{\beta_{k}\delta(\theta-\theta_{k})}$
$\displaystyle\theta\sim H$ (4)

on a parameter space $\Theta$. The weights $\beta_{k}$ are sampled via a
stick-breaking construction [18], so that $\beta\sim$ GEM($\gamma$) where
GEM($\cdot$) denotes the stick breaking construction:

$\displaystyle\beta_{k}=\beta_{k}^{\prime}\prod_{l=1}^{k-1}{(1-\beta_{l}^{\prime})}$
$\displaystyle\beta_{k}^{\prime}\sim Beta(1,\gamma)$ (5)

The HDP proposed by [17] extends the DP to cases in which groups of data are
produced by unique, generative processes [16]. The HDP places a global
Dirichlet process prior DP($\alpha$,$G_{0}$) on $\Theta$ and draws group
specific distributions:

$\displaystyle G_{j}(\theta)$
$\displaystyle=\sum_{k=1}^{\infty}{\pi_{jk}\delta(\theta-\theta_{jk})}$
$\displaystyle\pi_{j}\sim DP(\alpha,\beta)$ (6)

where $\pi_{j}$ represents the transition distribution for a specific state,
$h_{t}$ denotes the state of the Markov chain at time $t$. Figure 3 shows the
graphical model for the Sticky HDP-HMM. Given the properties in the HMM the
state $h_{t-1}$ indexes the group to which the observation $x_{t}$ is assigned
and the current state $h_{t}$ defines the parameter $\theta_{h_{t}}$ used to
generate the observation $x_{t}$. The modification proposed by [16] allows to
incorporate prior information that slow, smoothly varying dynamics are more
likely. Unlike the original HDP-HMM, this modified version does not allow
state sequences with unrealistic fast dynamics to have large posterior
probability. To that end, the work in [16] proposes to sample transition
distributions as follows:

$\displaystyle\pi_{j}\sim
DP\left(\alpha+\kappa,\frac{\alpha\beta+\kappa\delta_{j}}{\alpha+\kappa}\right)$
(7)

This modification for $\kappa$ values greater than zero has the effect of
increasing the prior probability of self transitions $E[\pi_{jj}]$. When
$\kappa$ is equal to zero the original HDP-HMM is obtained.

Figure 3: Sticky HDP-HMM graph

In many applications the distribution of the data associated with each hidden
state is complex and is better modeled with a multimodal distribution. This
motivates the use of HMM in which each hidden state is associated with a
mixture of Gaussian distributions, which poses the problem of selection of the
number of mixtures. Just like the model order problem associated with the
number of states, this problem can be solved by using DP as well. The Sticky
HDP-HMM is extended in [16] by defining a DP mixture of Gaussians, including a
new variable $s_{t}$ (see Figure 4) which indexes the Gaussian mixture
component of the $h_{t}-th$ hidden state. Therefore for the variables in
Figure 4, we have:

$\displaystyle
s_{t}\bigg{|}\\{\psi_{j}\\}_{j=1}^{\infty},h_{t}\sim\psi_{h_{t}}$ (8)
$\displaystyle x_{t}\bigg{|}\\{\theta\\}_{k,j=1}^{\infty},s_{t},h_{t}\sim
F(\theta_{h_{t},s_{t}})$ (9)

that is, $x_{t}$ is distribuited according to a function
$F(\theta_{h_{t},s_{t}})$, where $\theta_{h_{t},s_{t}}$ are the emision
parameters related to the state $h_{t}$ and mixture $s_{t}$.

Figure 4: Sticky HDP-HMM graph with DP Gaussian mixtures

We model the brain signals by fitting one HDP-HMM with a DP Gaussian mixtures
model for each class. For the Gaussian Mixtures, a Gaussian prior is placed on
the mean while an Inverse-Wishart prior is placed on the variance of the
components of each mixture following [16]. Classification is performed in the
same fashion as for the HMM approach Section II. Using test data, calculation
of the log-likelihood of the data given the model is performed. The model that
is more likely to produce the data determines the declared class.

##  IV Description of Experiments and Methods

###  IV-A Problem and Data Set Description

In typical BCI applications based on the imagination of motor activity, the
subject is requested to execute imaginary motor tasks following a visual cue.
It is known that the imagination of motor activities produces synchronization
and/or de-synchronization of the electrical signals recorded over the motor
cortex and that this process has an asymmetrical spatial distribution during
the imagination of the motor task (e.g., imagination of movement of a
particular leg produces changes in the power of electrical signals in the
contra-lateral region of the brain, see Figure 2). The models are trained
using data obtained from an initial session in which the subject is requested
to execute different motor tasks. Then, given some new (test) data, the task
is to run an inference algorithm to perform classification of the imaginary
motor task.

In this work, Data Set 2b of BCI competition IV [19] was used. The dataset
contains a total of nine subjects of which two where left out because previous
results show that no activity related to the task was detected for these
subjects (see results of the BCI competition [19]), the data contains then 7
subjects, with bipolar EEG recordings over scalp positions for electrodes C3,
Cz and C4 (see Figure 5a). The cue-based BCI paradigm involves two classes,
represented by the imagination of the movement of the left hand and the right
hand, respectively. The time scheme of the sessions is depicted in Figure 6.
At the beginning of each trial, a fixation cross and a warning tone are
presented. Three seconds later, a cue (indicating left or right movement) is
presented and the subject is requested to perform the imaginary movement of
the corresponding hand. The data set contains five sessions, three for
training and the remaining two for testing. Some of these sessions involved
feedback, indicating to the subject how well the imagination of the motor task
has been executed, and others do not. In our work we have used the sessions
with feedback because previous work has shown that temporal behavior of the
EEG signals could be modified due to the feedback influence [20].

The data were recorded in three different sessions in different days. The
training session contains 160 trails, 80 trials for each class (imaginary
right hand movement, imaginary left hand movement). The testing data consist
of two sessions, each one with 160 trials equally divided between the two
classes.

(a) EEG channels

(b) EOG channels

Figure 5: Electrode positioning for the BCI competition IV data set 2b. Figure
6: Time scheme for the experimental procedure.

###  IV-B Artifact Reduction

Linear regression was used in order to reduce the interference of
electrooculography (EOG) signals in the EEG recordings according to [21]. EOG
data recorded in three channels at electrode locations shown in Figure 5b
provide a measure of the eye movements executed by the subjects. In this
approach, the signal recorded by the EEG electrodes is modeled as the
summation of the actual underlying EEG signal and the noise, represented by a
linear combination of the EOG signals interfering into the EEG electrodes
[21].

Before the beginning of the motor task classification sessions, subjects are
requested to execute different ocular movements enabling the estimation of the
artifact-free EEG signal. As a complementary step, the obtained signals are
band-pass filtered in the frequency band of interest for real/imaginary motor
activity (8 - 35 Hz), eliminating other sources of artifact in the low and
high frequency range of the EEG recordings.

###  IV-C Spatial Filter

Spatial filters can be incorporated into the pre-processing stage by means of
Common Spatial Pattern (CSP). Given a two-class classification problem related
to motor task, CSP provides a method to maximize the power of the signal
during the execution of the class one, while minimizing it during the
execution of class two and vice versa. Given the artifact-free signal $s(n)$
obtained by the regression method described above, the spatial filtered signal
$c(n)$ is obtained by:

$c(n)=s(n)W,$ (10)

where $W$ is a matrix for which each column represents a set of subject-
specific spatial patterns. For details on the procedure for calculation of the
spatial filters with CSP the reader is referred to [22]. The signal $s(n)$
composed of three electrodes is linearly transformed into two CSP-filtered EEG
signals $c(n)$ to be used in subsequent pre-processing stages.

###  IV-D Spectral Power Estimation

The power spectrum of the signal is computed using auto-regressive (AR) models
of the signal. Burg’s method for AR model estimation was used because it
provides better stability than the Yule-Walker method, by minimizing the error
in backward and in forward direction [23]. Once the autoregressive model is
obtained we estimated the power spectrum of the signal as the frequency
response of the AR model.

For estimating the AR parameters a one-second sliding window was used over the
spatial filtered signals $c(n)$. For each signal segment of one second, the
model is estimated and the frequency response is obtained. The overlap of the
segments was fixed to 90% of the window length. This produces a time-frequency
map for each signal. From this time-frequency representation, the features to
be used in the classifiers (HMM and Sticky HDP-HMM) are selected based on
physiological information of the frequency bands related to
execution/imagination of motor tasks. The selected frequency bands used in
this work are alpha (8-13 Hz), sigma (11-15 Hz), low beta (18-23 Hz), high
beta (21-26 Hz) and low gamma (25-35 Hz). The features are calculated by
taking the average power across frequency at the indicated frequency bands.
The frequency resolution used was 1 Hz.

##  V Results

Subject | Chin | HMM-FP | HMM-CV | HDP-HMM
---|---|---|---|---
B01 | 0.40 | 0.38 | 0.43 | 0.57
B04 | 0.95 | 0.93 | 0.94 | 0.92
B05 | 0.86 | 0.81 | 0.86 | 0.83
B06 | 0.61 | 0.61 | 0.66 | 0.81
B07 | 0.56 | 0.59 | 0.63 | 0.57
B08 | 0.85 | 0.79 | 0.80 | 0.81
B09 | 0.74 | 0.70 | 0.71 | 0.79
Average | 0.71 | 0.69 | 0.72 | 0.76
TABLE I: Comparison of the proposed (Sticky) HDP-HMM approach with HMM and the
BCI’s competition winner (Chin.). HMM-FP corresponds to a HMM with parameters
fixed a priori (three hidden states, Gaussian mixtures of two components per
hidden state). HMM-CV corresponds to HMM with parameters selected by 3 Folds-
Cross-validation. HMM-FP, HMM-CV, and Sticky HDP-HMM use the same set of
features. The metric used is Cohen’s Kappa .

Figure 7 shows an example of the artifact reduction stage. Note that the EOG
interference on the EEG recordings is appreciable. However, this effect can be
reduced by linear regression. The bottom plot shows a noisy segment of EEG.
The red area corresponds to the portion of the EEG recordings that was used to
learn the regression coefficients that explain how the EOG signals propagate
through the scalp by volume conduction affecting the EEG recordings. The
segment in green shows a considerable reduction of the artifacts.

The CSPs can be projected over the scalp to visualize how the signals from
different regions in the scalp contribute to form the common spatial patterns.
These plots are presented in Figure 8. Given that the positions of the
electrodes in this data set represent left, right, and central areas over the
motor cortex, it is easy to visualize the operation of the filters. It can be
observed that for most of the subjects one of the filters gives more
importance to electrode C4 (right) and the other filter gives more importance
to the electrode C3 (left). This is consistent with the fact that the
representation of the hand is contra-lateral, i.e., imagination of the
movement of the left hand produces ERD in the right hemisphere while
imagination of movement in the right hand produces ERD in the left hemisphere.
According to the spatial filters in Figure 8, the electrode Cz does not
provide significant information for classification in most of the subjects.

Classification results are summarized in Table I. We use the kappa values [24]
as the metric for comparing different methods. The kappa values are defined
according to:

$\kappa=\frac{C\times P_{cc}-1}{C-1}$ (11)

where $C$ is the number of classes and $P_{cc}$ is the probability of correct
classification111Equation (11) takes this simple form given that the same
number of samples for each class is available for each subject in each
session.. Relatively larger kappa values indicate better performance.

The average performance of the Sticky HDP-HMM is higher than the results
obtained with all other methods. Also, compared to the results of the BCI
competition, where the maximum kappa value reported was 0.60 the Sticky HDP-
HMM provides better results. It is worth noting that the value reported by the
BCI competition corresponds to the maximum value obtained along the execution
of the task. In the case of the Sticky HDP-HMM and the HMM-like models
presented in Table I, the time of maximum performance is expected to be at the
end of the trial. The HMM with fixed parameters (HMM-FP column in Table I)
makes use of three hidden states representing beginning, execution, and end of
the motor task, and Gaussian mixtures of two components were allowed per each
hidden state. The HMM-CV refers to the HMM approach with the number of hidden
states and Gaussian mixtures learned using cross-validation. HMM-CV method
uses three-fold cross-validation for selection of the number of states and the
number of components of the Gaussian mixtures. The space of the parameters for
the HMM-CV is 1 to 3 hidden states and 1 to 3 Gaussian mixtures.

Figure 7: EOG artifact removal Figure 8: Topographical projection of the
spatial filters.

##  VI Conclusion

In this work a Bayesian nonparametric model that makes use of hierarchical
Dirichlet processes has been presented for classification of sensorimotor
rhythms in synchronous BCIs. First, preprocessing methods including artifact
reduction by linear regression, spatial filtering by common spatial patterns
and spectral power estimation have been performed on the data. The proposed
method aims to model the dynamics of the EEG signal during the execution of
the motor task showing that this information can increase the performance of
the system. Although methods based on HMM that aim to model the dynamics of
the EEG has been introduced before, the main disadvantage of these classic HMM
methods is that the number of hidden states and the parameters of the
distribution that models the the data have to be set a priori. The proposed
method based on HDP-HMM provides a solution for the selection of the number of
hidden states and the number of components of the mixture of Gaussians used to
model the distribution of the data. This is achieved by inferring a posterior
distribution over the number of hidden states by means of Hierarchical
Dirichlet Process, which allows an infinite number of states. Also, Dirichlet
processes are used to model the number of components of the mixture of
Gaussians. The results show that these methods provide good classification
results. In particular, the proposed HDP-HMM approach with self bias
transition (Sticky HDP-HMM) provides better results outperforming the
classical HMM approaches and the method proposed by the winner of the BCI
competition IV.

## Acknowledgment

This work was partially supported by Universidad del Norte (Colombia), the
Scientific and Technological Research Council of Turkey under Grant 111E056,
and by Sabanci University under Grant IACF-11-00889.

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