Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter
Magnetoresistance as a tool of basic parameters determination of minority charge carriers and the ratio of minority charge carriers conductivity to majority ones in solid matter has been considered within the framework of the phenomenological two-band model. The criterion of the application of thi...
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irk-123456789-1570272019-06-20T01:26:06Z Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter Uhryn, Y.O. Peleshchak, R.M. Brytan, V.B. Velchenko, A.A. Magnetoresistance as a tool of basic parameters determination of minority charge carriers and the ratio of minority charge carriers conductivity to majority ones in solid matter has been considered within the framework of the phenomenological two-band model. The criterion of the application of this model has been found. As examples of these equations usage the conductor, semiconductor and superconductor have been introduced. From the obtained temperature dependences of the aforementioned values in superconductor, a suppositi Запропоновано спосiб використання магнетоопору, як iнструмент для визначення основних параметрiв носiїв заряду та вiдношення провiдностi неосновних носiїв заряду до основних в твердих тiлах на основi аналiзу кривої магнетоопору в рамках феноменологiчної двозонної моделi. Встановлено критерiй застосовностi цiєї моделi. В ролi прикладiв застосування отриманих рiвнянь приведено провiдник, напiвпровiдник та надпровiдник. Зi знайдених температурних залежностей згаданих вище величин в надпровiдниках зроблено припущення про вирiшальну роль неосновних носiїв заряду у виникненнi надпровiдного стану. 2017 Article Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter / Y.O. Uhryn, R.M. Peleshchak, V.B. Brytan, A.A. Velchenko // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43702: 1–7. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 72.15, 72.20 DOI:10.5488/CMP.20.43702 arXiv:1712.05353 http://dspace.nbuv.gov.ua/handle/123456789/157027 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Magnetoresistance as a tool of basic parameters determination of minority charge carriers and the ratio of
minority charge carriers conductivity to majority ones in solid matter has been considered within the framework
of the phenomenological two-band model. The criterion of the application of this model has been found. As
examples of these equations usage the conductor, semiconductor and superconductor have been introduced.
From the obtained temperature dependences of the aforementioned values in superconductor, a suppositi |
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Article |
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Uhryn, Y.O. Peleshchak, R.M. Brytan, V.B. Velchenko, A.A. |
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Uhryn, Y.O. Peleshchak, R.M. Brytan, V.B. Velchenko, A.A. Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter Condensed Matter Physics |
| author_facet |
Uhryn, Y.O. Peleshchak, R.M. Brytan, V.B. Velchenko, A.A. |
| author_sort |
Uhryn, Y.O. |
| title |
Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter |
| title_short |
Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter |
| title_full |
Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter |
| title_fullStr |
Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter |
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Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter |
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magnetoresistance based determination of basic parameters of minority charge carriers in solid matter |
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Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/157027 |
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Magnetoresistance based determination of basic parameters of minority charge carriers in solid matter / Y.O. Uhryn, R.M. Peleshchak, V.B. Brytan, A.A. Velchenko // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43702: 1–7. — Бібліогр.: 24 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT uhrynyo magnetoresistancebaseddeterminationofbasicparametersofminoritychargecarriersinsolidmatter AT peleshchakrm magnetoresistancebaseddeterminationofbasicparametersofminoritychargecarriersinsolidmatter AT brytanvb magnetoresistancebaseddeterminationofbasicparametersofminoritychargecarriersinsolidmatter AT velchenkoaa magnetoresistancebaseddeterminationofbasicparametersofminoritychargecarriersinsolidmatter |
| first_indexed |
2025-07-14T09:22:14Z |
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2025-07-14T09:22:14Z |
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1837613643160616960 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 4, 43702: 1–7
DOI: 10.5488/CMP.20.43702
http://www.icmp.lviv.ua/journal
Magnetoresistance based determination of basic
parameters of minority charge carriers in solid
matter
Y.O. Uhryn1∗, R.M. Peleshchak1, V.B. Brytan1, A.A. Velchenko2
1 Drohobych Ivan Franko State Pedagogical University, 24 I. Franko St., 82100 Drohobych, Ukraine
2 Belarusian State Agrarian Technical University, 99 Nezavisimosti Ave., 220023 Minsk, Belarus
Received September 7, 2016, in final form May 31, 2017
Magnetoresistance as a tool of basic parameters determination of minority charge carriers and the ratio of
minority charge carriers conductivity to majority ones in solid matter has been considered within the framework
of the phenomenological two-band model. The criterion of the application of this model has been found. As
examples of these equations usage the conductor, semiconductor and superconductor have been introduced.
From the obtained temperature dependences of the aforementioned values in superconductor, a supposition
of a deciding role of minority charge carriers in the emergence of superconductivity state has been made.
Key words: conductor, semiconductor, magnetoresistance, Hall-effect
PACS: 72.15, 72.20
1. Introduction
Up till now, the researchers have found that magnetoresistance measurements are not informative
concerning basic parameters of charge carriers in solid matter [1]. Among the galvanomagnetic effects
only the Hall-effect is presented as a tool for determining these parameters [1]. However, we are going
to show that a magnetoresistance curve conceals in itself an exact information about minority charge
carriers mobility and concentration as well as their conductivity in relation to the conductivity of the
majority ones.
The aim of this paper is to develop a magnetoresistance method to determine basic parameters of the
minority charge carriers in solid matter [2] and apply it to different solids.
2. Calculation
To achieve this aim, we must consider the phenomenological model of galvanomagnetic phenomena
for an isotropic material with two types of charge carriers (having opposite or the same sign). It is known
[3] that the transverse conductivity σ for this case depends upon the magnetic field induction in the
following way:
σ =
(σ1 + σ2)
2 + σ2
1σ
2
2 (RH1 + RH2)
2B2
(σ1 + σ2) + σ1σ2(σ1R2
H1 + σ2R2
H2)B
2
. (1)
Substitute here σ1 = enµn, σ2 = epµp, RH1 = −1/(en), RH2 = 1/(ep), where e is electron charge,
n, p, µn, µp are concentrations and mobilities of two types of charge carriers (it can be electrons and
∗E-mail: yuriyuhryn@yahoo.com.ua
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
43702-1
https://doi.org/10.5488/CMP.20.43702
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
Y.O. Uhryn, R.M. Peleshchak, V.B. Brytan, A.A. Velchenko
holes or light and heavy holes). Note also that σ = ρ/(ρ2 + ρ2
xy), where ρ is transverse and ρxy Hall
resistivity. Then, considering that ρxy � ρ, we have
ρ =
1
σ
=
1
e
nµn + pµp + µnµp(nµp + pµn)B2
(nµn + pµp)2 + µ2
nµ
2
p(n − p)2B2
. (2)
If we state now the condition d2ρ/dB2 = 0, we obtain the magnetic field position of ρ flex point Bf
Bf =
1 + ab
√
3(1 − a)µn
, (3)
where a = n/p, b = µn/µp.
Now, the equation (2) can be introduced by the form:
ρ = ρ0
1 + µn(a+b)√
3Bf(1−a)b
B2
1 + 1
3B2
f
B2
, (4)
where ρ0 is resistivity at B = 0.
Let us write the condition for electrons as minority charge carriers. It is a � 1. Since the minority
charge carriers mobility is, as the rule, higher or compared with the majority ones, that is b > 1, then
a � b and the magnetic field dependence of ρ gets the form:
ρ = ρ0
1 + µn√
3Bf
B2
1 + 1
3B2
f
B2
. (5)
For B = Bf, writing for convenience ρ(Bf) = ρf, we obtain:
µn =
√
3
Bf
(
4
3
ρf
ρ0
− 1
)
. (6)
Thus, as can be seen from the last formula if the condition n � p is fulfilled, the measuring of the
transverse magnetoresistance provides the information about the mobility of minority charge carriers. It
is enough to find the magnetoresistance flex point Bf and to measure the resistivity in this point ρf and in
zero field ρ0.
From equation (3) for a � 1, taking into account that ab = σn/σp, we get another useful formula
σn
σp
= 4
(
ρf
ρ0
− 1
)
. (7)
Since in formula (6) and (7) the resistivities appeared in ratios, in practice we can substitute ρf/ρ0 by
a corresponding ratio of potential differences Uf/U0.
In the strong field limit, taking into account that a � 1 and a � b, equation (2) results after the
simple treatment in the formula for saturation resistivity:
ρ∞ =
1
σp
. (8)
From this equation, equations (6) and (7) and taking into account that σn = enµn, we obtain a formula
for determining the concentration ofminority charge carriers based on the experimentallymeasured values
n =
4Bf
√
3eρ∞
ρf
ρ0
− 1
4
3
ρf
ρ0
− 1
. (9)
43702-2
Magnetoresistance based determination of basic parameters of minority charge carriers
The latter formula presumes also a possibility to avoid measuring in high magnetic field for deter-
mining n, so equation (9) gets the form:
n =
4
√
3Bf
eρ0
ρf
ρ0
− 1(
4 ρfρ0
− 3
)2 . (10)
Let us return to the equation (5). In the strong field limit we can neglect 1 in comparison with the
summand containing B in the numerator and denominator of (5), getting another formula for determin-
ing µn
µn =
ρ∞
√
3ρ0Bf
. (11)
Combining the latter formula with (6) we get a relation between the experimentally measured values
ρ0, ρf and ρ∞:
ρ∞ = 4ρf − 3ρ0 , (12)
which is a test relation for the semi-classic behavior (2) of a real experimental curve.
We shall illustrate some examples of using the introduced equations.
3. Examples of calculation
We have applied these results to the metal aluminum at T = 4.2 K. As Al is located in the third group
of the periodic table of elements, its atom has 3 valent electrons (3s23p1). The analysis of the Hall-effect
experimental data [4] proves that out of these three electrons only two become free, the third one can
tunnel through the potential barrier to the neighboring atom due to the overlap of the wave functions.
To this free place, an electron from another neighboring atom can come and so on. Thus, this free place
moving in a chaotic manner within the crystal behaves as a positive free particle with electron charge
— the hole. The presence of a hole in Al is confirmed both by Fermi surface calculation and Hall-effect
experiments [5]. Dispersion law of free charge carriers in the second Brillouin zone is pointed out by the
hole nature of these carriers [5] and as in this zone there is approximately one of the three valent electrons,
then the hole concentration must be approximately twice smaller than the free electron concentration [6].
This is observed experimentally: the Hall coefficient depends on the magnetic field and changes its sign
from negative to positive in a strong magnetic field, in which it is twice larger than in the weak one. Such
a Hall-effect sign change is observed not only in Al, but at least in Be, Mg, In and Pb [6].
Figure 1 shows the magnetic field dependence of resistivity for a refined sample of this metal [4].
In fact, the presence of a transverse magnetoresistivity is the evidence of the existence of two types of
charge carriers. Calculations using formulae (6)–(9) for different refined Al are introduced in table 1.
Figure 1. Magnetic field dependence of transverse resistivity for refined Al. Dashed lines indicate the
flex point Bf and resistivity in this point ρ (Bf). The curve is obtained from experimental data of [4].
43702-3
Y.O. Uhryn, R.M. Peleshchak, V.B. Brytan, A.A. Velchenko
Table 1. Purification dependence of electron mobility µn, electron concentration n, electron conductiv-
ity σn and hole conductivity σp for Al.
Purification µn, m2
V·s n, 1028 m−3 σn, 109 W ·m σp, 109 W ·m σn/σp
Refining 1.75 1.9 5.4 2.9 1.87
Zone refining 14.3 2.0 45.9 23.3 1.97
Superrefining 19.6 1.9 61.1 32.5 1.88
99.999+% 20.1 2.1 66.9 33.3 1.87
As it is seen from table 1, the refining influences only the mobility of charge carriers. Since the
purification does not influence the ratio σn/σp, we can conclude that the hole mobility increases at the
same rate as the electron mobility.
As an example of a semiconductor, we consider germanium doped indium antimonide (InSb). In order
to measure the transverse magnetoresistance of a rectangular parallelepiped-like sample (0.25 × 1.2 ×
5.4 mm size), we place it into the pulse magnetic field, where the current I = 1 mA flew normally to the
magnetic field lines. The transverse voltage contacts were soldered at a distance 1
3 l (where l is the length
of the sample) from the current contacts. The magnetoresistance experimental data at T = 77 K for this
material are shown in figure 2. The obtained value coincides with the value for light holes mobility given
by different authors [3]. The ratio of the corresponding conductivities is 0.07 which is in good agreement
with the Hall-effect measurement results [7].
The most interesting example of applying the equations is calculating the minority charge carriers
mobility of a superconductor in the critical temperature range.
It is known that high-temperature superconductors (cuprates) have a planar structure, and kinetic
phenomena in these materials are connected with hybridized O2p-dx2-y2 orbitals [8]. Anderson [8]
describes it as a postulate (in the text — “dogma”). Until it is doped to 25%, the Fermi surface is a
simple hole surface around X [8]. Oxygen zone is hole-like and copper zone is electron-like (all the rest
zones are located too far from Fermi level in order to take them into account), although there is known
an opposite interpretation, when O-zone is electron-like and Cu-zone is hole-like [9]. However, in the
both cases these two zones provide two types of free charge carriers, the majority of which are, as a rule,
holes. This follows from numerous Hall-effect experiments [10].
Figure 3 shows the experimental dependences of transverse resistivity on the magnetic field in-
ductance for several temperatures for high temperature multilayer superconductor [YBa2Cu3O7(72 Å)/
PrBa2Cu3O7(12 Å)]25 near the critical point Tc = 89.5 K, namely for T > Tc. The fact of the appearance
of magnetoresistance at T = 91 K testifies to the rise of a new sort of charge carriers. Indeed, the
transverse resistance does not depend on magnetic field in the materials with one type of charge carriers
as it can be seen from equation (2) substituting there p = 0. Having determined for each curve the flex
Figure 2. Magnetic field dependence of transverse voltage for Ge-doped InSb. Dashed line indicates the
flex point Bf and the corresponding voltage U(Bf).
43702-4
Magnetoresistance based determination of basic parameters of minority charge carriers
Figure 3. Magnetic field dependences of transverse resistivities for a multilayer superconductor
[YBa2Cu3O7(72 Å)/PrBa2Cu3O7(12 Å)]25 at different temperatures near the critical one (Tc = 89.5 K).
Magnetic field is perpendicular to the layers. Dashed lines indicate flex points Bf and resistivities in
these points. Experimental data are obtained from temperature dependences of transverse resistivity at
different magnetic fields from [11].
Table 2. Temperature dependence of charge carriers basic parameters in a layered superconductor
[YBa2Cu3O7(72 Å)/PrBa2Cu3O7(12 Å)]25. The meanings of the symbols are the same as in table 1.
T , K µn , m2
V·s µp , m2
V·s p, 1027 m−3 σn
σp
σp , 105 W ·m σn , 105 W ·m n, 1024 m−3 a b
91 0.7 0.0018 2.2 0.23 6.5 1.5 1.3 0.0059 390
90.5 1.3 0.0018 2.2 0.67 6.5 4.4 2.1 0.0095 720
90.25 7.1 0.0018 2.2 2.7 6.5 17.6 1.5 0.0068 3900
90 40.9 0.0018 2.2 16.5 6.5 107 1.7 0.0077 22000
point Bf, zero field resistivity ρ0, flex point resistivity ρf and high field limit resistivity ρ∞, the basic
parameters of the minority charge carriers can be calculated. The results of the calculations are shown in
table 2 demonstrating a sharp increase of the negative sign charge carriers mobility, when the temperature
approaches the superconductivity transition, which causes the same rate of their conductivity increase.
At the same time, the majority charge carriers conductivity remains constant, which certainly shows their
basic parameters to be constant. This abrupt conductivity change from hole to electron-like induces us to
suppose that the electrons being minority charge carriers in the normal state play a majority role in the
superconductive state.
Since one can find out a similar magnetoresistivity behavior near Tc in other cuprate high temperature
superconductors, doped with Nd, Tm, Bi, Ca, Sr, as well as in conventional ones [11–24], we can suppose
that this temperature dependence of minority to majority charge carriers conductivity ratio is general,
including the case when the minority charge carriers are positive, as it is for Nd-Ce-Cu-O [12].
Note also that the application of our equations, that originate from semiclassic analysis of galvano-
magnetic phenomena, is reasonable for this case since experimental curves shown in figure 3 are in good
agreement with the test equation (12).
The introduced interpretation of magnetoresistance behavior in superconductors also solves the
problem of the Hall-effect anomaly near the critical temperature [11–14, 16–24] which consists in the
sign change of the Hall-effect in a low magnetic field at the temperatures approaching Tc from the high
temperature region. This phenomenon is common for both conventional [19–24] and high temperature
[11–13, 16–18, 23] superconductors. The majority of the authors explain it as the vortex motion concept,
some authors are disposed to the pining influence, the others suppose this phenomenon to be connected
with the change of electron to hole conductivity ratio [13].
We explain this sign reversal by the great electron mobility obtained above. As we can see from the
expression for the Hall constant in a weak magnetic field [3]
RH =
1
e
p − nb2
(p + nb)2
, (13)
43702-5
Y.O. Uhryn, R.M. Peleshchak, V.B. Brytan, A.A. Velchenko
the sign of RH will reverse negative when b2 > p/n which, in its turn, is provided by the great value of
electron mobility µn.
Moreover, the so-called ghost critical field appears in some superconductors, that is the Hall-effect
maximum field near the critical temperature [14]. The prospect of our further research is to show that the
maximum could appear in the solids with two types of holes.
4. Conclusions
We can conclude that the introduced equations for determination of the minority charge carriers
mobility can be applied to all solid materials (probably not only for solid ones) giving new opportunities
for their studying. The most interesting result is found for a superconductor showing a rapid increase of
minority charge carriers mobility when the temperature approaches the critical one from the normal state
temperature region. We suppose that this rapid increase makes minority charge carriers responsible for
the appearance of a superconductive state.
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Magnetoresistance based determination of basic parameters of minority charge carriers
Визначення основних параметрiв неосновних носiїв заряду
в твердих тiлах на основi магнетоопору
Ю.О. Угрин1, Р.М. Пелещак1, В.Б. Британ1, А.О. Вельченко2
1 Дрогобицький державний педагогiчний унiверситет iм. Iвана Франка,
вул. I. Франка, 24, 82100 Дрогобич, Україна
2 Бiлоруський державний аграрний технiчний унiверситет,
просп. Незалежностi, 99, 220023Мiнськ, Бiлорусь
Запропоновано спосiб використання магнетоопору, як iнструмент для визначення основних параметрiв
носiїв заряду та вiдношення провiдностi неосновних носiїв заряду до основних в твердих тiлах на основi
аналiзу кривої магнетоопору в рамках феноменологiчної двозонної моделi. Встановлено критерiй засто-
совностi цiєї моделi. В ролi прикладiв застосування отриманих рiвнянь приведено провiдник, напiвпро-
вiдник та надпровiдник. Зi знайдених температурних залежностей згаданих вище величин в надпровiд-
никах зроблено припущення про вирiшальну роль неосновних носiїв заряду у виникненнi надпровiдного
стану.
Ключовi слова: провiдник, напiвпровiдник, магнетоопiр, Голл-ефект
43702-7
Introduction
Calculation
Examples of calculation
Conclusions
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