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THERMODYNAMIC FOUNDATIONS OF GRAPHENE OPTICAL
PROPERTIES
Aleksey V. Yudenkov,
Smolensk State Academy of Physical Culture, Sports and Tourism, 214000, Russian Federation, Smolensk Region, Smolensk, aleks-ydenkov@mail.ru
Aleksandr M. Volodchenkov,
Smolensk Branch of Plekhanov Russian University of Economics; Smolensk Branch of "SGUA" (Saratov State Law Academy), Russia, Smolensk Region, Smolensk, alexmw2012@yandex.ru
Maria A. Yudenkova,
Moscow Institute of Physics and Mathematics, Moscow, Russia
DOI 10.24411/2072-8735-2018-10180
Keywords: Graphene, absorption ratio, phase space, thermodynamic equilibrium.
In recent years sustainable and continuing development of transport there appears a necessity to apply not only new technologies but also new materials with extraordinary properties. Graphene certainly proves to be such. The main specific feature of graphene is that it is a stable two-dimensional crystal. At the present time there has been accumulated a vast experimental data as to graphene growth, its mechanical, electric and optical properties. At the same time the majority of researchers agree that the graphene theory is far from being completed. Hence studying the graphene properties which differ significantly from the properties of three-dimensional crystals is all the more urgent. The research objectives are defined as follows.
1. To develop a phase-space model of electron in graphene.
2. To explain theoretically the extraordinary property of graphene - variation of absorption coefficient.
To achieve these goals there have been applied the fundamentals of thermodynamics, phasespace properties, Heisenberg indeterminacy principle, theory of black body, experimental data. Main research results. A model of discrete four-dimensional phase space of electron in graphene has been developed. The process of electron-photon interaction increasing entropy of the system at the microlevel has been described (Compton effect). The absorption variation of graphene on the heated metal surface has been explained through thermodynamics. The result is true for monolayer graphene crystals. In case of multilayer graphene crystals, absorp-tance is governed by Bouguer-Lambert law.
Information about authors:
Aleksey V. Yudenkov, Smolensk State Academy of Physical Culture, Sports and Tourism, Smolensk, Head of Management, Sciences and Humanities Dep. Doctor of Physics and Mathematics, Professor, Smolensk, Russia
Aleksandr M. Volodchenkov, Smolensk Branch of Plekhanov Russian University of Economics, Head of Humanities and Sciences Dep., Candidate of Physics and Mathematics;
Smolensk Branch of "SGUA" (Saratov State Law Academy), Assistant Professor, Dep. of Humanities, Socio-Economical, Informational and Law Discipline, Smolensk, Russia
Maria A. Yudenkova, Moscow Institute of Physics and Mathematics, student, Moscow, Russia
Для цитирования:
Юденков А.В., Володченков А.М., Юденкова М.А. Термодинамическое обоснование оптических свойств графена // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №11. С. 79-83.
For citation:
Yudenkov A.V., Volodchenkov A.M., Yudenkova M.A. (2018). Thermodynamic foundations of graphene optical properties. T-Comm, vol. 12, no.11, pр. 79-83.
Introduction
Graphene is the first of its kind stable two-dimensional crystal possessing unique nieelianical, electro-magnetic and thermodynamic properties. Since it was first discovered in 2004 by MIPT graduates A. Geim and K. Novosyelov (2010 Nobel Prize), there has been built a vast experimental base that has enabled development of graphene growth and investigation of its main properties [2]. This facilitated the practical application of graphene in various fields of science and technology. In particular, the use of graphene in car manufacturing is considered to be quite perspective (a graphene battery).
At the same time theoretical graphene studies are not systematic. Therefore development of the theory that treats graphene as a two-dimensional crystal is of the most immediate interest.
The work poses two interrelated tasks. First, it offers a model of discrete four-dimensional phase space characterizing electron evolution in the graphene crystal. This model allows to study some general properties of graphene in the momentum-coordinate space. Secondly, the results of the forepart can be applied to describing the optical properties of graphene (radiation absorptivity).
Methods and materials
As it has already been mentioned, graphene is carbon two-dimensional material having hexagonal lattice. Graphene geometrical sructure and some of its characteristics are shown in Fig. 1.
Fig. 1. Geometrical structure of grapheme
Graphene is characterized by high electric conductivity (ten times that of copper) and by high thermal conductivity [3],
As for optical properties, there lias been discovered an extraordinary effect which will be thoroughly discussed later on.
Quite a number of experimental works on graphene are connected with its growth on the heated metal substrate. Some important issues are revealed here. For example, while investigating growth kinetics of graphene on rhodium (Rh), iridium (Ir), platinum (Pt) and rhenium (Re) surfaces, a curious property has been discovered ([9], [12]). If there is a considerable number of graphene films on the substrate, they absorb radiation by Bougtter-Lambert-Beer law. If the number of the films is small (up to fifteen), then the graphene planted on the incandescent metal surface appears to be absolutely transparent. Notice that
this fact was found in the experiments with all the above materials, We believe that this property of graphene is connected with the change in its electronic properties due to interaction with the metal substrate. We will try to give an alternative explanation to the phenomenon.
Phase space model of electron in grapheme
At the present time graphene studies via Dirac equation are quite comprehensive. Consider one more way to describe electron behavior in graphene which seems to add to the description of subatomic-size particles in the electromagnetic field. Assume that [1J:
1. Electron is studied in the four-dimensional phase space of coordinate or momentum.
2. Phase space represents elementary four-dimensional concentric cells, volume ~hz. The proposition rests upon Heisenberg indeterminacy principle ApAr-h* (s -freedom degrees amount for the system). Whereby, distance between the phase space points A(pi,qA) and B{pB,qB) is defined by the amount of elementary cells between them. In its turn, the amount of cells will be defined by the difference ofenergy AE.
3. As graphene conductivity is veiy high, we suggest that electrons in graphene can be considered ideal monoatomic gas.
4. From the point of view of some external macroscopic observer electron in its phase space is evolving by diffuse Ito process.
The last proposition also rests on Hei sen berg indeterminacy principle (|4J c.66).
Consider photon scattering process on electron (Compton effect). The scheme of the process is in Fig, 2,
y y
Fig. 2. Compton effect
Photon scattering allows the external observer to detect electron in graphene. Whereby the volume of the phase space will be ~h2 ([5]c. 41). Linear dimensions of the cell can be defined as The linear dimensions of the following cells are correspondingly defined as ~^f2h, ... , —\fkh, ... . Notice that the total volume of the phase space and its linear dimensions when increasing the cell number go to infinity. At the same time true that
lim{4nh - J(n-\)h) = 0.
This speaks to the fact that when transiting to the laboratory-size magnitudes (phase-space points A(p4,qA) and B(pB,qB) are far enough from each other) the proposed model automatically becomes continuous.
Y
The classical definition of time does not always apply in the relativislic microcosm. Then let us introduce the concept of random time. Let us define the known in the theory of differential equations first exit moment ([7] c.145).
Definition
If H tz R" is any manifold, TH is called the first exit moment from H. if r„ = Inf [t > 0, X(t) t H}.
Define the first exit moment of a particle from the phase volume V„. Here Vn = nh2. By Dynkin's formula ([7|e. 155).
r
M[f( X,)] - fix) + Ml\Af{ Xs )dsl (1)
0
Here M(.) is mathematical expectation, /(.v)eC:, XT is
diffuse Ito process, A is characteristic operator.
Take four-dimensional Brownian motion B which starts to the central cells of phase space. Define the first exit moment of
process Tm from the phase space nh2. Choose whole number k . Suppose: X = B , r = crk = min{jt,zk}, f(x) = |.vj" when x e V - Apply formula (1), have
(I
Hence M(r(Rt")) = — R;
2 n 2
Let us define probability of the unbound particle evolving and getting out of point A of the phase space into the point B, Whereby A = R,„,B=R^k.
Let f = fmk C2 function with compact carrier which when R > btf > R , is defined as
m — || — in-k
/(*)=|-f"-
Out of Dynkin's formula comes that
M(/(BnRJ = f(RJ. (6)
Let
ft =/jif(|fir(Rra)|=^)'
Assuming that pk + q. = l we have
i R \2-2n
lim p. = —— R .
\ m-k ^
Hence, when n>2 for example, evolution of the system becomes irreversible. It is directed at increasing R. Whereby the phase volume is undergoing a rise increasing entropy of the microsystem. This completely agrees with the second law of thermodynamics which usually operates for macrosystems [I4j.
Notice that the four-dimensional rank of phase space is the least possible dimensionality when the second law of thermodynamics holds true. When the dimensionality of the system is 2, its evolution becomes recurrent 11 I \.
Thus interaction of electron and photon in graphene leads to increase in entropy. This means that graphene dissipates energy though significantly less than for example three-dimensional graphite. This accounts for its high thermal and electrical conductivity as well as for high degree of transparency.
Thermodynamic border properties of grapheme
To explain the change of graphene transparency we now proceed from microscopic to macroscopic description of the system as it is customary when analyzing complex systems ([8] cl 9, [10] c.63).
Recall that monolayer graphene grown on the heated surface possesses a property of absolute transparency. Let us assume once again dial electrons in graphene make ideal monoatonaic gas. Let us consider graphene itself two-dimensional isotropic medium. When highly healed (-1260-1800K), the sample can be considered heated homogeneously and isolropically. Therefore photon gas is in thermal equilibrium. The electrical power radiates fully by law of
Here / is current running through the band and U is falling on the band voltage, e - greyness value, a - Stefan-Bo I tzmann constant, T - temperature. Lei us exploit thermodynamic equilibrium of the graphene shell and the metal template. Graphene is two-dimensional when described macroscopically.
In this ease we can assume that energy densities of electrons and photons in the metal template in the three independent orthogonal directions are equal
Photons of z-direction can radiate and be absorbed. When dissipating on graphene, the condition E_ =0.survives. This means that there will appear a situation when E or
E >—nkT That is, the thermal equilibrium will be broken,
v 2
the graphene surface will show hot spots which will radiate extra energy by law of Stefan-BoItzmann.
Conclusion
The paper studies two basic issues.
Firstly, it constructs a model of cleciron phase space in graphene which allows to qualitatively depict two-dimensional crystal properties at the microlevel. Also it studies the process of graphene entropy increase when it comes to interaction with photon.
Secondly, at the macrolevel, the paper explains the changes in the graphene optica! properties when it contacts the heated metal substrate.
IU = soT4.
E = E =E = — nkT .
y 1 2
For graphene
Ex = EV =-nkT,Ez = 0. 2
In this context we see the following directions for prospective research. The mathematical model of the phase space should be further developed to facilitate the quantitative description of graphene at the microlevel. As for absolute graphene transparency, then taking into account its unique mechanic a I performance, it provides the prospects for the future application of the materia! in the telecommunications technologies and transport systems.
References
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т
ТЕРМОДИНАМИЧЕСКОЕ ОБОСНОВАНИЕ ОПТИЧЕСКИХ СВОЙСТВ ГРАФЕНА
Юденков Алексей Витальевич, ФГБОУ ВО СГАФКСТ, г. Смоленск, Россия, aleks-ydenkov@mail.ru
Володченков Александр Михайлович, Смоленский филиал РЭУ им. Г.В.Плеханова; Смоленский филиал "СГЮА", г. Смоленск, Россия, alexmw2012@yandex.ru Юденкова Мария Алексеевна, МФТИ, Москва, Россия
Аннотация
В настоящее время для поддержания непрерывного развития транспорта возникает необходимость в применении не только новых технологий, но и новых материалов с необычными свойствами. К таким перспективным материалам относится теперь уже широко известный графен. Главная особенность графена состоит в том, что он является устойчивым двумерным кристаллом. На сегодняшний день накоплен большой экспериментальный материал по производству графена, измерению его механических, электрических и оптических свойств. В тоже время большинство исследователей сходятся на том, что теория графена далека от завершения. Поэтому актуальной задачей является теоретическое обоснование тех его свойств, которые существенно отличаются от свойств трёхмерных кристаллов. В работе поставлены следующие цели.
1. Составить модель фазового пространства электрона, находящегося в графене.
2. Дать теоретическое обоснование необычному оптическому свойству графена - изменению коэффициента поглощения.
Для достижения поставленных целей использовались основные положения термодинамики, свойства фазового пространства, принцип неопределённости Гейзенберга, теория излучения абсолютно чёрного тела, экспериментальные данные. Основные результаты работы. Построена модель дискретного 4-х мерного фазового пространства электрона в графене. Рассмотрен процесс взаимодействия электрона и фотона (эффект Комптона), приводящий к увеличению энтропии системы на микроуровне. Дано объяснение с термодинамической точки зрения изменения коэффициента поглощения графена, находящегося на разогретой металлической подложке. Этот результат справедлив для монослоя графена. В случае, когда число слоев значительно, поглощательная способность графена подчинятся закону Бугера-Ламберта.
Ключевые слова: графен, коэффициент поглощения, фазовое пространство, термодинамическое равновесие.
Литература
1. Володченков А.М., Юденков А.В., Римская Л.П. Квантование информации в симплектическом многообразии. В сборнике: Социально-экономическое развитие региона: опыт, проблемы, инновации / Материалы VI Международной научно-практической конференции в рамках Плехановской весны и 110-летия университета. Министерство образования и науки российской федерации; Российский университет имени Г.В. Плеханова. Смоленский филиал. 2017. С. 41-46.
2. Гейм А.К. Случайные блуждания: непредсказуемый путь к графену // УФН. 2011. Т. 181. С. 1284-1298.
3. Елецкий А.В., Искандарова И.М., Книжник А.А., Красиков Д.Н. Графен: методы получения и теплофизические свойства // УФН. 2011. Т. 181. С. 227-258.
4. Ландау Л.Д., Лифшиц ЕМ. Теоретическая механика т.3. Квантовая механика. М.: Наука, 1989. 768 с.
5. Ландау Л.Д., Лифшиц ЕМ. Теоретическая механика т.5. Статистическая механика. М.: Наука, 1989. 626 с.
6. Лозовик Ю.Е., Меркулова С.П., Соколик А.А. Коллективные электронные явления в графене // УФН. 2008. Т. 178. № 7. С. 757-776.
7. Оксендаль Б. Стохастические дифференциальные уравнения. М.: Мир, "Издательство АСТ", 2003. 408 с.
8. Пугачев В.С., Синицын И.Н. Теория стохастических систем. Учеб. пособие. М.: Логос. 2004. 1000 с.
9. Рутьков Е.В., Галль Н.Р. Необычные оптические свойства графена на поверхности rh. Письма в Журнал экспериментальной и теоретической физики. 2014. Т. 100. № 9-10. С. 708-711.
10. Хакен Г. Информация и самооранизация. М.: КомКнига, 2005. 248 с.
11. Port S., Stone C. (1979). Brownian Motion and Classical Potential Theory. Academic Press.
12. Rut'kov E.V., Sheshenya E.S., Gall N.R., Lavrovskaya N.P. optical transparency of graphene layers grown on metal surfaces Semiconductors. 2017. Т. 51. № 4. С. 492-497.
Информация об авторах:
Юденков Алексей Витальевич, ФГБОУ ВО СГАФКСТ, заведующий кафедрой менеджмента и естественно-научных дисциплин, д.ф.-м.н., профессор, г. Смоленск, Россия
Володченков Александр Михайлович, Смоленский филиал РЭУ им. Г.В.Плеханова, заведующий кафедрой естественнонаучных и гуманитарных дисциплин, к.ф.-м.н.;
Смоленский филиал "СГЮА", доцент кафедры гуманитарных, социально-экономических и информационно-правовых дисциплин, г. Смоленск, Россия
Юденкова Мария Алексеевна, МФТИ, студентка, Москва, Россия
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