Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential
The quantitative theory of phonon drag thermopower for the one-dimensional electron gas in a quantum wire with a parabolic confinement potential is developed. The temperature gradient is directed along the axis of a quantum wire. As assumed, the Fermi level is located between the zero and first leve...
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| Zitieren: | Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential / I.I. Abbasov, Kh.A. Hasanov, J.I. Huseynov // Металлофизика и новейшие технологии. — 2017. — Т. 39, № 9. — С. 1165-1171. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1304032018-02-13T03:03:28Z Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential Abbasov, I.I. Hasanov, Kh.A. Huseynov, J.I. Строение и свойства наноразмерных и мезоскопических материалов The quantitative theory of phonon drag thermopower for the one-dimensional electron gas in a quantum wire with a parabolic confinement potential is developed. The temperature gradient is directed along the axis of a quantum wire. As assumed, the Fermi level is located between the zero and first levels of a size quantization. Using the Boltzmann kinetic equation, the phonon and electronic parts of a thermoelectric power are calculated. For comparison, numerical calculations of the temperature and concentration dependences of the phonon and diffusion parts of a thermoelectric power are carried out. As shown, the phonon drag makes a main contribution in thermopower within the temperature interval 2–20 K. Разработана количественная теория термоэдс фононного увлечения для одномерного электронного газа в квантовой проволоке с параболическим ограничивающим потенциалом. Градиент температуры направлен вдоль оси квантовой проволоки. Предполагается, что уровень Ферми расположен между нулевым и первым уровнями размерного квантования. С использованием кинетического уравнения Больцмана вычислены фононная и электронная части термоэдс. Для сравнения были проведены численные расчёты температурной и концентрационной зависимостей фононной и диффузионной частей термоэдс. Показано, что фононное увлечение вносит основной вклад в термоэдс в интервале температур 2–20 К. Розроблено кількісну теорію термоерс фононного перетягання для одновимірного електронного газу в квантовому дроті з параболічним обмежувальним потенціялом. Ґрадієнт температури спрямовано уздовж осі квантового дроту. Передбачається, що рівень Фермі розташований між нульовим і першим рівнями розмірного квантування. З використанням Больцманнового кінетичного рівняння обчислено фононну й електронну частини термоерс. Для порівняння було проведено чисельні розрахунки температурної та концентраційної залежностей фононної й дифузійної частин термоерс. Показано, що фононне перетягання дає основний внесок у термоерс в інтервалі температур 2–20 К. 2017 Article Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential / I.I. Abbasov, Kh.A. Hasanov, J.I. Huseynov // Металлофизика и новейшие технологии. — 2017. — Т. 39, № 9. — С. 1165-1171. — Бібліогр.: 12 назв. — англ. 1024-1809 PACS: 63.20.K-, 72.15.Jf, 72.15.Nj, 72.20.Pa, 73.21.Hb, 73.63.Nm, 85.80.Fi http://dspace.nbuv.gov.ua/handle/123456789/130403 DOI: https://doi.org/10.15407/mfint.39.09.1165 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Строение и свойства наноразмерных и мезоскопических материалов Строение и свойства наноразмерных и мезоскопических материалов |
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Строение и свойства наноразмерных и мезоскопических материалов Строение и свойства наноразмерных и мезоскопических материалов Abbasov, I.I. Hasanov, Kh.A. Huseynov, J.I. Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential Металлофизика и новейшие технологии |
| description |
The quantitative theory of phonon drag thermopower for the one-dimensional electron gas in a quantum wire with a parabolic confinement potential is developed. The temperature gradient is directed along the axis of a quantum wire. As assumed, the Fermi level is located between the zero and first levels of a size quantization. Using the Boltzmann kinetic equation, the phonon and electronic parts of a thermoelectric power are calculated. For comparison, numerical calculations of the temperature and concentration dependences of the phonon and diffusion parts of a thermoelectric power are carried out. As shown, the phonon drag makes a main contribution in thermopower within the temperature interval 2–20 K. |
| format |
Article |
| author |
Abbasov, I.I. Hasanov, Kh.A. Huseynov, J.I. |
| author_facet |
Abbasov, I.I. Hasanov, Kh.A. Huseynov, J.I. |
| author_sort |
Abbasov, I.I. |
| title |
Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential |
| title_short |
Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential |
| title_full |
Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential |
| title_fullStr |
Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential |
| title_full_unstemmed |
Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential |
| title_sort |
phonon drag thermopower in quantum wire with parabolic confinement potential |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2017 |
| topic_facet |
Строение и свойства наноразмерных и мезоскопических материалов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/130403 |
| citation_txt |
Phonon Drag Thermopower in Quantum Wire with Parabolic Confinement Potential / I.I. Abbasov, Kh.A. Hasanov, J.I. Huseynov // Металлофизика и новейшие технологии. — 2017. — Т. 39, № 9. — С. 1165-1171. — Бібліогр.: 12 назв. — англ. |
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Металлофизика и новейшие технологии |
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2025-07-09T13:29:45Z |
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| fulltext |
СТРОЕНИЕ И СВОЙСТВА НАНОРАЗМЕРНЫХ
И МЕЗОСКОПИЧЕСКИХ МАТЕРИАЛОВ
PACS numbers: 63.20.K-, 72.15.Jf, 72.15.Nj, 72.20.Pa, 73.21.Hb, 73.63.Nm, 85.80.Fi
Phonon Drag Thermopower in Quantum Wire with Parabolic
Confinement Potential
I. I. Abbasov, Kh. A. Hasanov*, and J. I. Huseynov*
Azerbaijan State Oil and Industrial University,
20 Azadliq,
Baku, Azerbaijan
*Azerbaijan State Pedagogical University,
34 U. Hajibayov Str.,
Baku, Azerbaijan
The quantitative theory of phonon drag thermopower for the one-dimen-
sional electron gas in a quantum wire with a parabolic confinement potential
is developed. The temperature gradient is directed along the axis of a quan-
tum wire. As assumed, the Fermi level is located between the zero and first
levels of a size quantization. Using the Boltzmann kinetic equation, the pho-
non and electronic parts of a thermoelectric power are calculated. For com-
parison, numerical calculations of the temperature and concentration de-
pendences of the phonon and diffusion parts of a thermoelectric power are
carried out. As shown, the phonon drag makes a main contribution in ther-
mopower within the temperature interval 2–20 K.
Key words: quantum wire, phonon drag, parabolic potential, energy spec-
trum, thermoelectric effects, diffusion thermopower, phonon thermopower.
Розроблено кількісну теорію термоерс фононного перетягання для одно-
вимірного електронного газу в квантовому дроті з параболічним обмежу-
вальним потенціялом. Ґрадієнт температури спрямовано уздовж осі кван-
тового дроту. Передбачається, що рівень Фермі розташований між нульо-
вим і першим рівнями розмірного квантування. З використанням Больц-
Corresponding author: Ibrahim I. Abbasov
E-mail: ibrahimabbasov179@gmail.com
Please cite this article as: I. I. Abbasov, Kh. A. Hasanov, and J. I. Huseynov, Phonon
Drag Thermopower in Quantum Wire with Parabolic Confinement Potential, Metallofiz.
Noveishie Tekhnol., 39, No. 9: 1165–1171 (2017), DOI: 10.15407/mfint.39.09.1165.
Ìåòàëëîôèç. íîâåéøèå òåõíîë. / Metallofiz. Noveishie Tekhnol.
2017, т. 39, № 9, сс. 1165–1171 / DOI: 10.15407/mfint.39.09.1165
Îттиски доступнû непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2017 ÈÌФ (Èнститут металлофизики
им. Ã. Â. Êурдюмова ÍÀÍ Óкраинû)
Íапечатано в Óкраине.
1165
https://doi.org/10.15407/mfint.39.09.1165
https://doi.org/10.15407/mfint.39.09.1165
1166 I. I. ABBASOV, Kh. A. HASANOV, and J. I. HUSEYNOV
маннового кінетичного рівняння обчислено фононну й електронну части-
ни термоерс. Для порівняння було проведено чисельні розрахунки темпе-
ратурної та концентраційної залежностей фононної й дифузійної частин
термоерс. Показано, що фононне перетягання дає основний внесок у тер-
моерс в інтервалі температур 2–20 Ê.
Ключові слова: квантовий дріт, фононне перетягання, параболічний по-
тенціял, енергетичний спектер, термоелектричні ефекти, термодинаміка
дифузії, теплова енергія фононів.
Разработана количественная теория термоэдс фононного увлечения для
одномерного электронного газа в квантовой проволоке с параболическим
ограничивающим потенциалом. Ãрадиент температурû направлен вдоль
оси квантовой проволоки. Предполагается, что уровень Ферми располо-
жен между нулевûм и первûм уровнями размерного квантования. С ис-
пользованием кинетического уравнения Больцмана вûчисленû фононная
и электронная части термоэдс. Для сравнения бûли проведенû численнûе
расчётû температурной и концентрационной зависимостей фононной и
диффузионной частей термоэдс. Показано, что фононное увлечение вно-
сит основной вклад в термоэдс в интервале температур 2–20 Ê.
Ключевые слова: квантовая проволока, фононное увлечение, параболиче-
ский потенциал, энергетический спектр, термоэлектрические эффектû,
термодинамика диффузии, тепловая энергия фононов.
(Received August 5, 2017)
1. INTRODUCTION
In recent years, a significant number of papers were devoted to exper-
imental and theoretical investigations of thermopower in low-dimen-
sional systems have appeared [1–3]. The limited motion of electrons in
such systems leads to the fact that the kinetic phenomena in them
sharply differ from the electronic transport phenomena in bulk sam-
ples.
Quantum-size structures produced in recent decades have attracted
attention also because of possibility of their using in thermoelements
with high thermoelectric efficiency. Studies in this field have shown
that it is possible to increase the thermoelectric efficiency two or three
times by preparing thermoelectric structures with quantum wells [4].
In this paper, we discuss the temperature dependence of the thermo-
electric power of a degenerate electron gas in a quantum wire with a
parabolic confinement potential in the low-temperature range of 1–20
K, where the phonon drag effect plays an important role. Due to con-
finement, the energy spectrum and the wave function of the electron
essentially change. A numerical calculation of the thermoelectric pow-
er is carried out for the quantum wire GaAs/AlxGa1−xAs with a parabol-
ic well.
PHONON DRAG THERMOPOWER IN QUANTUM WIRE 1167
2. THEORETICAL DETAILS
To obtain an analytical expression for various physical quantities, it is
advisable to use a specific model of a potential quantum well. For this
purpose, a parabolic potential of the U(x) =
2 2
0 /2m xω form that re-
stricts the motion of electrons in the direction of the x-axis, is often
used, where m is the effective mass of the conduction electrons, and ω0
is the parabolic potential parameter.
In a present work, the phonon drag thermopower for one-
dimensional degenerated electron gas in quantum wire (QW) with a
parabolic confinement potential is calculated. Spectrum and wave
functions of the system ground state under consideration are given in
[5]:
2 2
0,0, ,
2k
k
m
ε = + ω
(1)
2
0,0, 2
1
exp ,
2k
r
ikz
RR L
ψ = − +
p
(2)
where ω is a parabolic potential parameter, ( )R m= ω is an oscilla-
tory length, L is a QW length. It is supposed that the Fermi level ζ is
localized between zero level and first one of dimensional quantization.
Moreover, necessary condition for the existence of a strong degenera-
tion is 0k T << ζ − ω < ω , where T is temperature, k0 is the Boltz-
mann constant. Earlier, the phonon drag thermopower for one-
dimensional electron gas was calculated in the framework of the rec-
tangular confinement potential model [6]. There are both theoretical
calculations [3, 7] and experimental results [8, 9] confirming the dom-
inating contribution of phonon drag thermopower in the total ther-
mopower for a two-dimensional electron gas.
3. RESULTS
Thermopower α associated with a temperature gradient along the QW
axis consists of diffusion αe and phonon αph parts: α = αe + αph = βe/σ +
+ βph/σ [10]. Here, σ is the specific conductivity of the QW along the
wire axis:
2 ( )
.Fn e k
m
τ
σ = (3)
As evaluations show, at low temperatures, the dominating scatter-
ing mechanism for strongly degenerated electron gas is scattering on
ionized impurities and scattering on a sample boundary for phonons.
1168 I. I. ABBASOV, Kh. A. HASANOV, and J. I. HUSEYNOV
The expression for the electron-momentum relaxation time has the
form:
( ) ( ) ( )
23 2
2 2 2 2 21
( ) ( ) exp 0,
2
F
F F F F
I
k Z e
k k R k R k
N m
−
τ = ε Γ χ
; (4)
for phonons, / ,ph L sτ = where s is the sound velocity in the wire.
The electron concentration n and the Fermi wave number kF are re-
lated by the following expression:
2 2
2 ( ) ,Fn k m= = ζ − ω
p p
(5)
2
2 2 2 2
2
2
( ) 1 exp(2 ) (0,2 ),F F F
F
m e
k R k R k
k
ε = + Γ
p χ
(6)
where ( )Fkε is the dielectric function, χ is the static dielectric con-
stant, Γ(0, õ) is the incomplete gamma function,
2
2
0
1
( ) ,
3e k T
eT
p ∂σ
β = −
∂ζ
(7)
0 ,ph ph
k
A
e
β = − σ (8)
( )
2 2 2
2 2 2 2 2
2
00
2 2 2
4
(1 (1 )) exp( 2 )
( )
(exp( 1 ) exp( 1 )) ,
F
ph F
F
m e L sk
A a x R k x
k T k
b x b x xdx
∞
−
β
= + + − ×
p ρ ε
× + − − +
∫
(9)
where the following notations are introduced:
12
,Fk E
a
e
=
β
0
.Fs k
b
k T
=
(10)
Here. e is the elementary charge, E1 is the deformation-potential con-
stant, and β is the parameter characterizing the piezo acoustic poten-
tial.
The contributions from electron interactions with acoustic phonons
in the phonon drag thermopower are taken into account in [9] by means
of the deformational potential E1 and the piezo-acoustic one
140.8 /eβ = χ [11].
4. DISCUSSION
Numerical calculations are performed for the GaAs/AlxGa1−xAs QW
PHONON DRAG THERMOPOWER IN QUANTUM WIRE 1169
with the following GaAs parameter values: mass of electrons m =
= 0.067 m0, where m0 is the free-electron mass, the crystal mass density
ρ = 3.3⋅104
kg/m3, s = 5⋅105
m/s, E1 = 7.4 eV, e14 = 0.16 C/m3, the QW
length L = 3⋅10−4
m, the linear density of electrons n = 1.6⋅108
m
−1, ω =
= 7⋅1013
s
−1.
The calculated temperature dependence of the phonon drag ther-
mopower (a solid line) is shown in Fig. 1. For comparison, the tempera-
ture-dependent thermopower diffusion component (a dotted line) is
also given. The parabolic potential parameter for GaAs/AlxGa1−x As is
inversely proportional to the wire thickness ω = 14.6 eV/d(Å) [12].
Our choice corresponds to thicknesses of about 100 Å.
The diffusion thermopower component value is larger than the pho-
non one in the temperature interval 1–20 K. The phonon drag strongly
Fig. 2. Concentration dependences of the phonon drag thermopower (solid
line) and the diffusion thermopower (dotted line).
Fig. 1. Temperature dependences of the phonon drag thermopower (solid line)
and the diffusion thermopower (dotted line).
1170 I. I. ABBASOV, Kh. A. HASANOV, and J. I. HUSEYNOV
grows with temperature increasing, exceeding the diffusion ther-
mopower by one order.
The concentration dependences of the phonon drag thermopower (a
solid line) and the diffusion thermopower (a dotted line) are shown in
Fig. 2. The diffusion component of the thermopower is approximately
inversely proportional to the concentration and phonon component is
inversely proportional to the concentration squared.
Kubakaddi’s expression for the phonon drag thermopower for the
QW model with a rectangular potential [6] differs from our expression,
but both qualitative results are similar.
5. CONCLUSION
The theoretical results obtained are applied to the analysis of a ther-
mopower with phonon drag of a one-dimensional degenerate electron
gas in a quantum wire with a parabolic confinement potential. For
comparison, the temperature dependence of the diffusion component
of the thermoelectric power is also given. Numerical calculations are
given for the QW GaAs/AlxGa1−xAs. In the temperature range 1–2 K,
the diffusion thermoelectric power exceeds the phonon one. With tem-
perature increasing, the phonon thermoelectric power increases sharp-
ly, exceeding the diffusion one by an order of magnitude. The diffu-
sion component of the thermoelectric power is approximately inversely
proportional to the concentration, and the phonon component is in-
versely proportional to the concentration squared. In the temperature
range 2–20 K, the main contribution to the thermoelectric power is
given by the phonon drag.
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|