Розробка математичної моделі фоточутливого сенсора Холла на основі CdS
The paper proposes a mathematical model of a photosensitive Hall sensor based on a CdS single crystal which uses the internal photoelectric effect. The model describes the change in the concentration and mobility of charge carriers under the influence of irradiation and allows us to estimate the inc...
Збережено в:
| Дата: | 2025 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
PE "Politekhperiodika", Book and Journal Publishers
2025
|
| Теми: | |
| Онлайн доступ: | https://www.tkea.com.ua/index.php/journal/article/view/TKEA2025.3-4.15 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Technology and design in electronic equipment |
| Завантажити файл: |  |
Репозитарії
Technology and design in electronic equipment| _version_ | 1867569667487301632 |
|---|---|
| author | Sergiichuk, Viktor Oliinyk, Ostap |
| author_facet | Sergiichuk, Viktor Oliinyk, Ostap |
| author_institution_txt_mv | [
{
"author": "Viktor Sergiichuk",
"institution": "Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine"
},
{
"author": "Ostap Oliinyk",
"institution": "Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine"
}
] |
| author_sort | Sergiichuk, Viktor |
| baseUrl_str | https://www.tkea.com.ua/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-06-09T12:16:19Z |
| description | The paper proposes a mathematical model of a photosensitive Hall sensor based on a CdS single crystal which uses the internal photoelectric effect. The model describes the change in the concentration and mobility of charge carriers under the influence of irradiation and allows us to estimate the increase in the sensitivity of the sensor to a magnetic field. The mathematical model accurately describes the increase in Hall voltage when the sensor current exceeds 40 mA (error margin of less than 5%). Experiments showed a 2.8-fold increase in sensor sensitivity in the low-current sensor mode, which is partially explained by noise and additional effects in the crystal. The authors propose the materials and sensor topology that will maximize Hall voltage to obtain maximum sensor sensitivity. The comparison of theoretical results and experimental data confirmed a twofold increase in the Hall voltage under illumination. The developed approach can be used to optimize planar sensors in systems that require high accuracy and energy efficiency. |
| doi_str_mv | 10.15222/TKEA2025.3-4.15 |
| first_indexed | 2026-02-08T08:11:05Z |
| format | Article |
| fulltext |
Teсhnology and design in electronic equipment, 2025, N 3 – 4 15ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
1
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
UDC 681.586.728
DEVELOPMENT OF A MATHEMATICAL MODEL
FOR A CdS-BASED PHOTOSENSITIVE HALL SENSOR
In modern sensor technology, there is a sustained
increase in requirements for accuracy, energy efficiency,
and functional integration of measurement devices.
In particular, in the fields of automotive electronics,
biomedical systems, and wireless sensor networks, the
demand for multifunctional sensors capable of operating
simultaneously based on several physical principles is
steadily growing [1], [2]. One of the promising directions
is the integration of the Hall effect with the photoelectric
effect within a single semiconductor element, which opens
up opportunities to enhance magnetic field sensitivity
through additional photogeneration of charge carriers.
Conventional Hall sensors are characterized by
high reliability and accuracy and are widely used in
magnetometry, positioning systems, and current sensing
applications [3]. At the same time, their performance
remains limited by a constant charge carrier concentration
and geometric dimensions, without accounting for
temperature effects, magnetic field configuration, carrier
concentration fluctuations, and related factors. Recent
studies have demonstrated that optical irradiation
can significantly alter the electrical conductivity of
semiconductors, affecting both the concentration and
mobility of charge carriers [4].
In [5], it was demonstrated that directed illumination
during Hall measurements opens new possibilities for
material diagnostics, including defect spectroscopy,
evaluation of photogeneration processes, and assessment
of the contributions of electrons and holes along with their
mobilities. Hall photoeffect spectroscopy with a controlled
radiation flux has shown high sensitivity to recombination
centers and even the occurrence of negative differential
photoconductivity (NDPC) in semiconductors, which is
consistent with the concept of an “active layer” and with
The paper proposes a mathematical model of a photosensitive Hall sensor based on a CdS single crystal which uses the
internal photoelectric effect. The model describes the change in the concentration and mobility of charge carriers under
irradiation and allows estimating the increase in the sensor’s sensitivity to a magnetic field. The model accurately describes
the increase in Hall voltage when the sensor current exceeds 40 mA (error margin of less than 5%). Experiments showed a 2.8-
fold increase in sensor sensitivity in the low-current sensor mode, which is partially due to noise and additional effects in the
crystal. The proposed materials and sensor topology can help maximize Hall voltage to obtain maximum sensor sensitivity. The
comparison of theoretical results and experimental data confirmed a twofold increase in the Hall voltage under irradiation.
Keywords: Hall effect, photosensitive Hall sensor, CdS, mathematical modeling, planar Hall sensor.
changes in the concentrations of electrons and holes, as
well as their mobilities, under external irradiation.
A useful extension in the context of the present
work is the light-modulated Hall effect (LMHE) [6].
By introducing optical modulation and employing
phase-sensitive detection of the Hall voltage, it becomes
possible to separate the photogenerated contribution and
to enhance the signal-to-noise ratio in systems with low
charge carrier mobility.
Previous studies on the modeling of photoinduced
effects in semiconductors [7], [8], [9] have demonstrated
the influence of photon flux, quantum efficiency, and
charge carrier lifetime on the transport properties of
materials. However, most existing models consider
static conditions or isolated optical phenomena without
comprehensively accounting for their interaction with
the classical Hall effect. This creates a scientific niche
for the development of a more generalized mathematical
model that integrates both effects and enables prediction
of changes in sensor sensitivity under illumination.
Despite the existence of studies addressing individual
aspects of the photoinduced Hall effect, the majority of
research focuses either on spectroscopic diagnostics of
materials or on static analysis of the Hall effect without
considering the influence of irradiation. At present, there
are no models that consistently describe the combined
impact of photogeneration, recombination processes,
and variations in charge carrier mobility on the Hall
voltage in practical structures based on cadmium sulfide
(CdS) single crystals. These circumstances motivated the
development of the mathematical model proposed in this
work, which accounts for the effect of irradiation on the
parameters of a Hall sensor.
DOI: 10.15222/TKEA2025.3-4.??
Viktor SERGIICHUK, Ostap OLIINYK
Ukraine, Igor Sikorsky Kyiv Polytechnic Institute
E-mail: ostap.oliinyk@gmail.com
DOI: 10.15222/TKEA2025.3-4.15
Teсhnology and design in electronic equipment, 2025, N 3 – 416 ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
2
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
In this work, CdS single crystals are considered
a promising sensing material, as they combine high
photoconductivity in the visible spectral range —
compared with other materials — with relatively high
electron mobility (100 – 600 cm²/(V·s)), low hole
mobility (15 – 70 cm²/(V·s)), and n-type conductivity,
which ensures an increase in the Hall voltage under
illumination. The bandgap width is Eg ≈ 2.4 eV, making
CdS particularly effective for photoelectron generation
in the visible wavelength range (480 – 550 nm), where
maximum photoconductivity and minimal losses due
to thermodynamic processes are achieved. In addition,
CdS is characterized by a relatively low intrinsic charge
carrier concentration in the dark state, which enables clear
separation of the photogenerated EMF contribution within
the Hall voltage. CdS-based structures are well compatible
with planar microfabrication technologies, exhibit stable
electrophysical parameters at constant temperature,
and provide reproducible results, making this material
suitable for the investigation of photosensitive Hall
sensors and their subsequent practical implementation.
Theoretical Analysis of the Effect of Optical
Irradiation on CdS Parameters and the Hall
Voltage
The classical physical model accounting for the
topology of a Hall sensor is based on the phenomenon
of transverse voltage occurring when current passes
through a conductor (or semiconductor) placed in an
orthogonal (crossed) magnetic field. This phenomenon,
first discovered by Edwin Hall in the late nineteenth
century, is widely employed for measuring magnetic flux
density and for the indirect determination of a range of
other physical quantities [1], [2].
The classical expression for the Hall voltage can be
written as
H H
IB IBU R
ned d
= = , (1)
where I is current through the sensor;
B is magnetic flux density (magnetic field intensity);
n is charge carrier concentration in the material;
e is elementary charge (electron or hole);
d is thickness of the sensing element along the direction
in which the Hall voltage is generated;
RН is Hall coefficient, RН = 1/(ne).
The classical Hall sensor model assumes that the
semiconductor or metallic plate is homogeneous in
thickness and composition, that charge carriers move
within it in accordance with Ohm’s law under the
action of the Lorentz force in a magnetic field, and that
edge effects do not have a significant influence and are
therefore neglected.
In practice, the shape and dimensions of a sensor
may deviate from the idealized model, and the thickness
and material properties of the crystal may be non-
uniform. This is commonly accounted for by introducing
a dimensionless geometrical correction factor G into
equation (1):
H H
IBU R G
d
= . (2)
In more complex systems, it is essential to account
for the charge carrier mobility μ and recombination
processes. In the case of doped semiconductors, where
different types of charge carriers (electrons and holes)
coexist, the Hall coefficient is expressed as the combined
contribution of each carrier type. Under these conditions,
equation (2) takes the form:
( )
2 2
2
μ μ
μ μ
h e
H
h e
p n IBU G
dq p n
-
=
+
, (3)
where μh, μe is the mobilities of holes and electrons,
respectively.
The classical model provides a convenient and
efficient means for estimating the key parameters of
a Hall sensor, such as sensitivity and output signal,
which is particularly useful in the early stages of
device design [2]. However, the classical model has
certain limitations, as it does not take into consideration
temperature effects, magnetic field intensity, or external
optical irradiation.
Several models [10], [11], [12] proposed to describe
Hall sensors incorporate additional effects or parameters,
such as anisotropic conductivity in the semiconductor
crystal, surface scattering, and charge carrier transport
phenomena. Nevertheless, parameters such as carrier
concentration, sensor thickness, and mobility are
typically assumed constant, being determined during the
design and fabrication stages. Given that these parameters
can vary under optical irradiation, it is necessary for the
model to account for the internal photoelectric effect and
its influence on the Hall voltage.
Over the past two decades, numerous experimental
and theoretical studies have investigated the effects
of charge carrier photogeneration in semiconductor
structures. Photogeneration significantly increases the
charge carrier population in a semiconductor, thereby
modifying its electrical conductivity [13]. In particular,
studies [14], [15] have demonstrated that the contributions
of electrons and holes can be accurately separated, and
their mobilities can be determined in high-mobility
organic semiconductors under optical excitation.
Photogeneration has been shown to substantially affect
both carrier concentration and effective mobility, which
is consistent with the assumptions underlying the present
work.
In the context of contactless conductivity measurement
techniques, it is also important to note the phenomenon
known as the optical hall effect, which enables evaluation
of free carrier parameters (concentration, mobility, and
effective mass) without direct electrical contact, by
employing a magnetic field and an optically excited
medium. For example, the authors of [16] presented a
Teсhnology and design in electronic equipment, 2025, N 3 – 4 17ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
3
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
detailed model of the optical hall effect dielectric tensor
function for multilayer semiconductor structures and
explained how data analysis based on Mueller matrices
and 4×4 matrix formalisms provide access to the
electrical characteristics of the material.
Optical irradiation affects not only carrier concentration
but also carrier mobility, both of which are critical for
sensor operation. The expression for the change in the
excess carrier concentration Δn in a semiconductor under
external irradiation is given by:
d Δ
d τ
n nD
t
= = , (4)
where D is charge carrier generation rate;
τ is lifetime of nonequilibrium charge carriers.
The mobility of electrons and holes is determined by
the expression:
m = et / m*, (5)
where m* is the effective mass of a charge carrier.
From expression (4), the time-dependent change in
the concentration of nonequilibrium charge carriers ∆n(t)
relative to the initial (intrinsic) carrier concentration n(0)
can be derived [17]:
∆n(t) = n(0)·exp(– t/τ). (6)
The photogeneration parameters, in particular the
quantum efficiency and the absorption coefficient,
determine the magnitude and linearity of the photosensitive
signal and can be incorporated into the sensor model.
Since the recombination rate is inversely proportional
to the carrier lifetime, the quantum efficiency can be
expressed as follows:
τη
τ τ
nr
q
r nr
=
+ , (7)
where τnr, τr is carrier lifetimes in the absence of
irradiation and under irradiation, respectively.
The amplitude of the output signal is determined by
the change in the resistance R of the active region of the
sensor and depends on the quantum efficiency η and the
absorption coefficient α:
η (1 ехр( α ))eR d
hv
= - - , (8)
where hv is the energy of the incident photon.
Photogeneration exhibits pronounced dynamic
properties that affect the sensor response time and its
frequency characteristics. This implies that models
accounting only for the static influence of light
may be insufficiently accurate for describing sensor
operation under time-varying irradiation conditions
[18]. Accordingly, the frequency dependence of the
photoinduced current can be expressed as follows:
( )
( )
0
2
μω
1 ωτ
ph
D EI =
+
, (9)
where D0 is amplitude of the charge carrier generation rate;
Е is the electric field;
ω is the angular modulation frequency.
Despite significant progress in understanding photo-
generation in semiconductors, several key aspects remain
insufficiently studied or generalized in contemporary
models, such as the nonlinear interaction between
photogeneration and the classical Hall effect [19], [20]
dynamic variations of photogeneration and their impact
on the temporal characteristics of sensors [21], [22],
as well as the interplay between optical and magnetic
effects.
The next step is the development of a mathematical
model capable of combining the internal photoelectric
effect and the Hall effect in order to achieve a higher
amplitude of the sensor output voltage UH.
The planar topology proposed in this work (Fig. 1)
enables uniform distribution of the near-surface current
within the sensing element and ensures measurement
stability. A key feature is that the photosensitive region
of the sensor overlaps the carrier transport paths; so that
even small variations in illumination lead to significant
changes in carrier concentration. This creates favorable
conditions for enhancing magnetic field sensitivity
without the need to modify the sensor geometry.
The low temperature dependence of carrier mobility
and concentration for the selected CdS sample (less
Fig. 1. Topology and external view of the proposed planar Hall
sensor with a photosensitive active region:
1 — semiconductor plate; 2, 2′ — metallic contacts on the surface
of the plate; 3 — irradiation source (λ = 532 nm, P = 100 mW);
d, d′— thickness of the sensor and the active layer, respectively
(under irradiation d′ << d); Ix — current through the sensor;
Bz — magnetic flux density of the applied magnetic field;
VH — Hall voltage
2′
d′
d
2
2
2′
1
3
z
x
y
Ix
Bz
VH
hv
Teсhnology and design in electronic equipment, 2025, N 3 – 418 ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
4
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
than 1% / 1 К) makes it possible to derive analytical
relationships assuming an average temperature of
T = 297 ±2 К = const.
The number of absorbed photons is determined by
the absorption coefficient α(λ), which is a function of the
radiation wavelength λ. For photons with energy hν ≥ Eg,
charge carrier photogeneration is efficient. The carrier
generation process is described by the generation rate:
D = αΦ, (10)
where Φ is photon flux (the number of photons per unit
area per unit time). This parameter indicates how many
electron-hole pairs are generated within a unit near-
surface volume over a given time interval.
Not all generated carriers contribute to a change in
electrical conductivity, since recombination processes
occur. The number of carriers that effectively influence
the conductivity variation is determined by the photo-
generation efficiency and the average carrier lifetime τ.
In particular, the steady-state excess carrier concentration
can be estimated as follows:
∆n ≈ Dτ. (11)
Higher absorption efficiency and longer carrier lifetime
result in increased charge carrier photogeneration. Under
optical irradiation, the number of carriers increases by
Δn (electrons) or Δp (holes). Accordingly, the effective
concentrations of electrons and holes, respectively, to be
expressed as follows:
neff = n0 + ∆n; (12)
peff = p0 + ∆p. (13)
Under optical irradiation, the concept of an effective
thickness of the photosensitive semiconductor, denoted
as d′, arises. This parameter defines the depth to which
light penetrates into the material, thereby affecting
charge carrier generation and, consequently, its electrical
properties. According to the Beer–Lambert–Bouguer
law, the light intensity decreases exponentially with the
penetration depth:
I(z) = I0 eхр(– α z), (14)
where I0 is the initial radiation intensity at the surface.
The quantity inverse to the absorption coefficient (1/α)
corresponds to the thickness of the semiconductor layer
d ′ over which the light intensity decreases by a factor
of e. This implies that the effective thickness of the
photosensitive layer is defined as:
d΄= 1/a. (15)
With increasing absorption coefficient, the effective
thickness decreases, since light is absorbed more strongly
within a shallower depth. As a result, an active region
is formed (Fig. 2), where the effective thickness d΄ is
smaller than the substrate thickness d. According to
expression (3), this enables a higher Hall voltage to be
obtained without compromising the mechanical strength
of the structure.
The exponential decay of light intensity within
the crystal thickness indicates that the effective photo-
sensitive thickness is generally smaller than the actual
sensor thickness. This parameter depends on the
irradiation wavelength: short-wavelength radiation
generates carriers near the surface, while long-wave-
length radiation penetrates deeper into the material. Thus,
the choice of the irradiation spectrum serves as a tool
for controlling the spatial profile of photogeneration,
enabling optimization of sensor operation under different
conditions.
Equation (3) can be rewritten by taking into account
expressions (12) and (13):
( )
( ) ( )
( ) ( )( )
2 2
2
2 2
0 0
2
0 0
μ μ
μ μ
Δ μ Δ μ
Δ μ Δ μ
eff h eff e
H
eff h eff e
h e
h e
p n IBU G
dq p n
p p n n IB G
dq p p n n
-
= =
+
+ - +
=
+ + +
. (16)
If we introduce the coefficient b = μe /μh, then equation
(8) can be written in the following form:
( ) ( )
( ) ( )( )
2
0 0
2
0 0
Δ Δ
Δ Δ
H
p p n n b IBU G
qdp p n n b
+ - +
=
+ + +
. (17)
To verify the theoretical model and to separate the
unknown quantities, one may assume that the Hall sensor
is irradiated with sufficiently high intensity І, such that
d = d ′. Under this irradiation regime, photogeneration
becomes uniform throughout the entire crystal volume,
which allows the modified value of the charge carrier
concentration to be applied consistently.
For low and moderate irradiation intensities, the
increase in electron and hole concentrations can be
assumed to be proportional to the photon flux Ф:
Dn = η(λ)Φτe /le; (18)
Dp = η(λ)Φτh /lh; (19)
where τe/h, le/h is the average lifetimes and free paths of
electrons and holes, respectively.
Fig. 2. Formation of the active region in a photosensitive planar
Hall sensor during irradiation of a CdS substrate
d ′
d
532 nm, 100 mV irradiation
Teсhnology and design in electronic equipment, 2025, N 3 – 4 19ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
5
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
Taking these expressions into account, equation (17)
can be rewritten as follows:
( ) ( )
( ) ( )
2
0 0
2
0 0
η λ Φτ η λ Φτ
η λ Φτ η λ Φτ
h e
h e
H
h e
h e
p n b
l l IBU G
qd
p n b
l l
æ ö æ ö÷ ÷ç ç÷ ÷+ - +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
=
æ öæ ö æ ö ÷÷ ÷çç ç ÷÷ ÷ç + + +ç ç ÷÷ ÷çç ç ÷÷ ÷ç ç ÷çè ø è øè ø
. (20)
Photogeneration may, in some cases, affect charge
carrier scattering, lead to the formation of additional
recombination centers, and induce defects in the substrate
structure. However, these effects are not considered in
the present mathematical model due to their relatively
weak influence. At the same time, the authors of [23]
demonstrate that photogeneration can significantly
modify the electrical conductivity of semiconductors,
and study [24] showed that the additional generation of
carriers alters the electrical transport characteristics of
the material. As a result of the assumptions made and
the transformations performed, a relationship can be
established between the Hall voltage under illumination
UH,il and in the dark state UH,d:
( ) ( )
( ) ( )
( )
2
0 0
,
2
,
0 0
2
0 0
2
0 0
η λ Φτ η λ Φτ
η λ Φτ η λ Φτ
.
α
h e
H il h e
H d h e
h e
p n b
U l l
U
p n b
l l
p n b d
p n b
æ ö æ ö÷ ÷ç ç÷ ÷+ - +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
= ´
æ öæ ö æ ö ÷÷ ÷çç ç ÷÷ ÷ç + + +ç ç ÷÷ ÷çç ç ÷÷ ÷ç ç ÷çè ø è øè ø
+
´
-
By substituting the representative parameters of the
CdS single crystal employed in the planar Hall sensor
developed in this study:
p0 = 1020 m–3; n0= 1018 m–3; η = 0.8; α = 1.5·10–3 m–1;
τh = 10–7 s; τe = 10–8 s; b = 5; d = 0.5·10–3 m;
Φ = 1021 m–2·s–1; lh = 5.8·10–6 m; le = 1.1·10–6 m.
The resulting theoretical ratio of the Hall voltage under
illumination to that in the dark state is UH,il /UH,d = 1.893.
The modeling results presented in Fig. 3 indicate that
the increase in Hall voltage as external irradiation is not
linear and exhibits a pronounced maximum. This behavior
indicates the existence of an optimal combination of the
active-layer thickness and photon flux at which the
photogeneration of electrons and holes provides the largest
increase in Hall voltage. At higher flux levels, the effect
diminishes due to recombination processes. Hence, the
model enables prediction of the limits of effective sensor
operation and helps to avoid regimes of oversaturation
and unnecessary energy consumption. The ratio
UH,il /UH,d increases as α decreases and at low values
of Φ, however, with further increases in Φ it decreases, as
shown in Fig. 3. The maximum is reached when Ф attains
a certain low value and α = 0.2 mm–1. Nevertheless,
the active semiconductor layer is formed as a result of
irradiation, and its thickness depends on the photon flux,
quantum efficiency, irradiation wavelength, and angle of
incidence at the surface.
Influence of Noise and Modeling Errors
on the Calculation of the Hall Voltage
The mathematical model presented in this work
accounts for charge carrier photogeneration and is
based on the classical expression for the Hall voltage,
with variations in charge carrier concentration included.
To ensure the linearity of the model, the excess carrier
concentration Δn must be much smaller than the baseline
concentration n0 (i.e., Δn << n0). Under this condition,
the model remains effectively linear, since small
concentration perturbations do not alter the functional
dependence of the Hall voltage UH on current and
magnetic field. If photogeneration becomes excessively
strong (Δn ≈ n0 or Δn > n0 ), the model begins to exhibit
nonlinear effects, in which case even small variations
in Δn may induce significant fluctuations in the output
signal [25].
The photon flux Φ is one of the key parameters for
controlling the Hall voltage, as high irradiation intensity
may cause carrier saturation and drive the system outside
the linear regime. At high quantum efficiency η(λ)
even small change in irradiation intensity can result
in a significant increase in the electron concentration
Δn. The carrier lifetime τ, which depends on crystal
quality, also affects the Hall voltage: a short lifetime
leads to rapid recombination and reduces the influence
of photogeneration, whereas a long lifetime enhances
the effect [26]. All parameters must be chosen to ensure
that the model operates within the linear range, where
the assumption Δn << n0 remains valid.
For the CdS/CdSe systems study [27] reports infrared
quenching (IR quenching) of the photo-Hall effect in
(21)
Fig. 3. Ratio of the Hall voltage under illumination to that in
the dark state for CdS (a maximum value of 20 is achieved
at an effective thickness of 0.2 mm, formed by a photon flux
intensity of 2.6·1017 m–2·s–1)
1016
10221021
1020101910181017
1.0
1.5
2.0
0.5
30
20
10
0
–10
–20
–30
UH,il /UH,d
α, m
m
–1
Φ, m –2 s –1
Teсhnology and design in electronic equipment, 2025, N 3 – 420 ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
6
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
doped CdS:Cu/ZnS:Cu, materials, variations in carrier
mobility as a function of photoelectron density in CdSe, as
well as photoconductivity mechanisms in polycrystalline
CdS involving trap states. These results directly confirm
the influence of Δn(Φ, λ), τ and defects on the formation
of the Hall voltage and the sensor sensitivity.
Thermal noise (also known as Johnson–Nyquist
noise) is an inherent characteristic of any electrical
system. It is defined as
4 Δn Bv k TR f= ,
where kB is Boltzmann constant;
T is absolute temperature;
R is resistance;
Δf is measurement bandwidth.
In the context of Hall sensors, thermal noise can
influence both the measurement of the Hall output voltage
and the overall system stability. If the noise level exceeds
the signal variation induced by photogeneration, device
sensitivity decreases. Тherefore, appropriate design
measures (e.g., low noise amplifiers and bandwidth
optimization) must be implemented to mitigate the
influence of thermal noise on measurements [28].
Defects and inhomogeneities in the crystal can lead
to the formation of localized carrier traps, which may
reduce the effective carrier lifetime τ and the value of
Δn. In certain regions of the crystal, the local carrier
concentration may differ from the average value n0,
resulting in a nonuniform response to irradiation.
Furthermore, defects can introduce additional sources of
flicker noise (1/f noise), thereby reducing measurement
stability [29].
Experimental Investigation of the Developed Planar
Hall Sensor
To eliminate thermoelectric EMF and minimize the
influence of temperature-dependent carrier mobility, all
Hall voltage measurements were performed at a constant
temperature. The sample temperature was maintained at
297 K with an allowable deviation of no more than ±2 K.
This made it possible to minimize the influence of μ(T)
and n(T) and to isolate the contribution of the internal
photoelectric effect to the formation of the Hall voltage.
The results of the experimental investigation of the
developed planar Hall sensor with a photosensitive
active region are presented in Fig. 4. They confirm the
theoretically substantiated increase in the Hall voltage
under external irradiation. The measurement results
show that the model describes the variation of the Hall
voltage as a function of sensor parameters, irradiation,
and external magnetic field with an error not exceeding
5% for currents above 40 mA. At low currents (< 40 mA),
a markedly larger Hall voltage is observed, arising from
noise factors, charge accumulation in the diffusion
capacitance of the crystal, carrier injection and drift
associated with edge effects, as well as the motion
of photoelectrons in the near-surface region of the
crystal, where resistivity decreases sharply. Under these
conditions, pronounced nonlinear small-signal effects
become evident.
The experimental results demonstrate an approximately
2.8-fold increase in the Hall sensor sensitivity to the
magnetic field at a current of 10 mA, which significantly
exceeds the prediction of the mathematical model and
warrants further investigation.
Conclusions
The developed mathematical model of a planar
Hall sensor enables both qualitative and quantitative
evaluation of the influence of photoelectron generation
on the Hall voltage in CdS-based structures. A universal
approach is proposed for enhancing of magnetic
field measurement sensitivity using energy-efficient
photosensitive Hall sensors with low power consumption.
Experimental investigations confirmed the nearly
twofold increase in the Hall voltage under irradiation of
the CdS crystal predicted by the mathematical model,
demonstrating good agreement between theory and
experiment. The proposed planar sensor architecture
opens the possibility of employing photosensitive
Hall sensors in high-precision and energy-efficient
measurement systems.
The mathematical model indicates that, for Hall
sensors with a photosensitive active region, materials
with a maximal difference between electron and hole
mobilities should be selected, along with a peak spectral
sensitivity in the optical or ultraviolet region to minimize
the thickness of the active layer.
The next stage of research will extend the model
to describe the dynamic characteristics of the sensor
under modulated light intensity conditions, as well as
accounting for the effects of temperature and crystal
structure defects in low-power operating regimes, i.e.,
developing a small-signal model of the planar Hall sensor.
Fig. 4. Experimental (symbols) and theoretical (lines)
dependences of the Hall voltage on current for CdS, obtained
without irradiation (1) and with irradiation (2)
2
1
UH, V
0.05
0.04
0.03
0.02
0.01
0.02 0.04 0.06 0.08 Iin, A
Teсhnology and design in electronic equipment, 2025, N 3 – 4 21ISSN 3083-6530 (Print)
ISSN 3083-6549 (Online)
7
ELECTRONIC DEVICES: RESEARCH, DEVELOPMENT
REFERENCES
[1] S. M. Sze, Physics of Semiconductor Devices, 3rd ed. Hoboken,
NJ, USA: Wiley, 2006.
[2] M. J. Caruso, C. H. Smith, T. Bratland, and R. Schneider,
“A new perspective on magnetic field sensing,” Sensors, vol. 17, no. 9,
pp. 258–264, 2000. Available: https://surl.li/yyvpuk
[3] W. Bolton, Sensors and Actuators: Engineering System
Instrumentation, 2nd ed. Oxford, U.K.: Newnes, 2021.
[4] L. Liu, X. Guo, W. Liu, and C. Lee, “Recent progress in the
energy harvesting technology—From self-powered sensors to self-
sustained IoT, and new applications,” Nanomaterials, vol. 11, no. 11,
Art. no. 2975, 2021, doi: 10.3390/nano11112975.
[5] A. Musiienko, R. Grill, P. Moravec, P. Fochuk, I. Vasylchenko,
H. Elhadidy, et al., “Photo-Hall-effect spectroscopy with enhanced
illumination in p-Cd1−xMnxTe showing negative differential
photoconductivity,” Phys. Rev. Applied, vol. 10, pp. 014019, 2018,
doi: 10.1103/PhysRevApplied.10.014019
[6] S. E. Schacham, and E. Finkman, “Light-modulated Hall effect
for extending characterization of semiconductor materials,” J. Appl.
Phys., vol. 60, pp. 2860–2865, 1986, doi: 10.1063/1.337070
[7] A. I. Galuza and A. B. Beznosov, “Optical functions of the
Drude model: transformation of the spectra over wide ranges of
parameters,” Low Temp. Phys., vol. 27, pp. 216–227, 2001, doi:
10.1063/1.1355519
[8] N. J. Gantzler, “An alternative approach to the extended Drude
model,” J. Appl. Spectrosc., vol. 85, pp. 361–364, 2018, doi: 10.1007/
s10812-018-0657-x
[9] I. V. Maznichenko, “Tunable 2D electron gas at the LaAlO3/
SrTiO3(001) interface,” Phys. Rev. Materials, vol. 3, рр. 074006, 2019,
doi: 10.1103/PhysRevMaterials.3.074006
[10] J. H. Davies, “The two-dimensional electron gas,” in The
Physics of Low-Dimensional Semiconductors: An Introduction,
Cambridge: Cambridge University Press, 1997, pp. 329–370, doi:
10.1017/CBO9780511819070.011
[11] P. Ritzinger, and K. Vyborny, “Anisotropic magnetoresistance:
materials, models and applications,” R. Soc. Open Sci., vol. 10, iss. 10,
2023, doi: 10.1098/rsos.230564
[12] C.-J. Zhao, “Research progress in anisotropic
magnetoresistance,” Rare Metals, vol. 32, pp. 213–224, 2013, doi:
10.1007/s12598-013-0090-5
[13] B. G. Streetman, and S. Banerjee, Solid State Electronic
Devices, 6th ed. Upper Saddle River, NJ, USA: Prentice Hall, 2006.
[14] V. Bruevich, H. H. Choi, and V. Podzorov, “The photo-Hall
effect in high-mobility organic semiconductors,” Adv. Funct. Mater.,
vol. 31, pp. 2006178, 2021, doi: 10.1002/adfm.202006178
[15] Y. Chen, H. T. Yi, and V. Podzorov, “High-resolution ac
measurements of the Hall effect in organic field-effect transistors,”
Phys. Rev. Applied, vol. 5, pp. 034008, 2016, doi: 10.1103/
PhysRevApplied.5.034008
[16] M. Schubert, P. Kühne, V. Darakchieva, and T. Hofmann,
“Optical Hall effect model description: tutorial,” J. Opt. Soc. Am. A,
vol. 33, pp. 1553–1568, 2016, doi: 10.1364/JOSAA.33.001553
[17] C. Jacoboni, and L. Reggiani, “The Monte Carlo method for
the solution of charge transport in semiconductors with applications
to covalent materials,” Rev. Mod. Phys., vol. 55, no. 3, pp. 645–705,
1983, doi: 10.1103/RevModPhys.55.645
[18] Y. Tsividis, and C. McAndrew, Operation and Modeling of
the MOS Transistor, 3rd ed. Oxford, U.K.: Oxford Univ. Press, 2011.
[19] R. S. Popovic, Hall Effect Devices: Magnetic Sensors and
Characterization of Semiconductors. Bristol, U.K.: Institute of Physics
Publishing, 2004.
[20] R. F. Pierret, Semiconductor Device Fundamentals. Reading,
MA, USA: Addison Wesley, 1996.
[21] M. Razeghi, and R. Baker, Modern Semiconductor Device
Physics. San Diego, CA, USA: Academic Press, 2007.
[22] J. Singh, Semiconductor Optoelectronics: Physics and
Technology. New York, NY, USA: McGraw-Hill, 2003.
[23] P. Y. Yu, and M. Cardona, Fundamentals of Semiconductors:
Physics and Materials Properties, 4th ed. Berlin, Germany: Springer,
2010, doi: 10.1007/978-3-642-00710-1
[24] J. I. Pankove, Optical Processes in Semiconductors. New
York, NY, USA: Dover Publications, 1971.
[25] D. K. Ferry, and S. M. Goodnick, Transport in Nanostructures.
Cambridge, U.K.: Cambridge Univ. Press, 1997, doi: 10.1017/
CBO9780511626128
[26] R. A. Smith. Semiconductor Fundamentals. Oxford
University Press, 2014.
[27] B. Ch. Burgett, and C. Ch. Lin, “Infrared quenching of the
luminescence and photo-hall effect of ZnS:Cu and CdS:Cu crystals”,
Journal of Physics and Chemistry of Solids, vol. 31, iss. 6, 1970,
pp. 1353–1359, doi: 10.1016/0022-3697(70)90139-3.
[28] J. H. Davies, The Physics of Low-Dimensional Semiconductors:
An Introduction. Cambridge University Press, 1998.
[29] B. K. Ridley, Quantum Processes in Semiconductors (5th ed.).
Oxford University Press, 2013.
Received 01.09 2025
DOI: 10.15222/TKEA2025.3-4.??
УДК 681.586.728
Віктор СЕРГІЙЧУК, Остап ОЛІЙНИК
Україна, м. Київ, КПІ імені Ігоря Сікорського
E-mail: ostap.oliinyk@gmail.com
РОЗРОБКА МАТЕМАТИЧНОЇ МОДЕЛІ ФОТОЧУТЛИВОГО СЕНСОРА ХОЛЛА
НА ОСНОВІ CdS
Запропоновано математичну модель фоточутливого сенсора Холла виготовленого на основі монокристалу CdS,
в якому використовується внутрішній фотоефект. Математична модель описує зміну концентрації та рухливості
носіїв зарядів під дією опромінення й дозволяє оцінити приріст чутливості сенсора до магнітного поля завдяки зміні
параметрів матеріалу сенсора. Теоретично встановлено і підтверджено експериментально, що при зовнішньому
опроміненні активної області сенсора напруга Холла збільшується двократно. Розроблений підхід може бути
використаний для оптимізації планарних сенсорів у системах, які потребують високої точності, чутливості та
енергоефективності.
Ключові слова: ефект Холла, фоточутливий сенсор Холла, CdS, математичне моделювання, планарний сенсор Холла.
Copyright: © 2025, The author(s). Licensee: Politekhperiodika, Odesa, Ukraine. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license
(https://creativecommons.org/licenses/by/4.0/).
DOI: 10.15222/TKEA2025.3-4.15
|
| id | oai:tkea.com.ua:article-749 |
| institution | Technology and design in electronic equipment |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-06-10T01:00:25Z |
| publishDate | 2025 |
| publisher | PE "Politekhperiodika", Book and Journal Publishers |
| record_format | ojs |
| resource_txt_mv | wwwtkeacomua/40/f61f7530e8b6ec358d5c4a5b5c5a1940.pdf |
| spelling | oai:tkea.com.ua:article-7492026-06-09T12:16:19Z Development of a mathematical model for a CdS-based photosensitive Hall sensor Розробка математичної моделі фоточутливого сенсора Холла на основі CdS Sergiichuk, Viktor Oliinyk, Ostap Hall effect photosensitive Hall sensor CdS mathematical modeling planar Hall sensor ефект Холла фоточутливий сенсор Холла CdS математичне моделювання планарний сенсор Холла The paper proposes a mathematical model of a photosensitive Hall sensor based on a CdS single crystal which uses the internal photoelectric effect. The model describes the change in the concentration and mobility of charge carriers under the influence of irradiation and allows us to estimate the increase in the sensitivity of the sensor to a magnetic field. The mathematical model accurately describes the increase in Hall voltage when the sensor current exceeds 40 mA (error margin of less than 5%). Experiments showed a 2.8-fold increase in sensor sensitivity in the low-current sensor mode, which is partially explained by noise and additional effects in the crystal. The authors propose the materials and sensor topology that will maximize Hall voltage to obtain maximum sensor sensitivity. The comparison of theoretical results and experimental data confirmed a twofold increase in the Hall voltage under illumination. The developed approach can be used to optimize planar sensors in systems that require high accuracy and energy efficiency. Запропоновано математичну модель фоточутливого сенсора Холла виготовленого на основі монокристала CdS, в якому використовується внутрішній фотоефект. Математична модель описує зміну концентрації та рухливості носіїв зарядів під дією опромінення й дозволяє оцінити приріст чутливості сенсора до магнітного поля завдяки зміні параметрів матеріалу сенсора. Теоретично встановлено і підтверджено експериментально, що при зовнішньому опроміненні активної області сенсора напруга Холла збільшується двократно. Розроблений підхід може бути використаний для оптимізації планарних сенсорів у системах, які потребують високої точності, чутливості та енергоефективності. PE "Politekhperiodika", Book and Journal Publishers 2025-12-30 Article Article Peer-reviewed Article application/pdf https://www.tkea.com.ua/index.php/journal/article/view/TKEA2025.3-4.15 10.15222/TKEA2025.3-4.15 Technology and design in electronic equipment; No. 3–4 (2025): Technology and design in electronic equipment; 15-21 Технологія та конструювання в електронній апаратурі; № 3–4 (2025): Технологія та конструювання в електронній апаратурі; 15-21 3083-6549 3083-6530 10.15222/TKEA2025.3-4 en https://www.tkea.com.ua/index.php/journal/article/view/TKEA2025.3-4.15/678 Copyright (c) 2025 Viktor Sergiichuk, Ostap Oliinyk http://creativecommons.org/licenses/by/4.0/ |
| spellingShingle | ефект Холла фоточутливий сенсор Холла CdS математичне моделювання планарний сенсор Холла Sergiichuk, Viktor Oliinyk, Ostap Розробка математичної моделі фоточутливого сенсора Холла на основі CdS |
| title | Розробка математичної моделі фоточутливого сенсора Холла на основі CdS |
| title_alt | Development of a mathematical model for a CdS-based photosensitive Hall sensor |
| title_full | Розробка математичної моделі фоточутливого сенсора Холла на основі CdS |
| title_fullStr | Розробка математичної моделі фоточутливого сенсора Холла на основі CdS |
| title_full_unstemmed | Розробка математичної моделі фоточутливого сенсора Холла на основі CdS |
| title_short | Розробка математичної моделі фоточутливого сенсора Холла на основі CdS |
| title_sort | розробка математичної моделі фоточутливого сенсора холла на основі cds |
| topic | ефект Холла фоточутливий сенсор Холла CdS математичне моделювання планарний сенсор Холла |
| topic_facet | Hall effect photosensitive Hall sensor CdS mathematical modeling planar Hall sensor ефект Холла фоточутливий сенсор Холла CdS математичне моделювання планарний сенсор Холла |
| url | https://www.tkea.com.ua/index.php/journal/article/view/TKEA2025.3-4.15 |
| work_keys_str_mv | AT sergiichukviktor developmentofamathematicalmodelforacdsbasedphotosensitivehallsensor AT oliinykostap developmentofamathematicalmodelforacdsbasedphotosensitivehallsensor AT sergiichukviktor rozrobkamatematičnoímodelífotočutlivogosensorahollanaosnovícds AT oliinykostap rozrobkamatematičnoímodelífotočutlivogosensorahollanaosnovícds |