MINISTRY OF EDUCATION AND SCIENCE YOUTH AND SPORTS OF UKRAINE
V.N. Karazin Kharkiv National University Faculty of Physics Academician I.M. Lifshyts Theoretical Physics Department “Admitted to the graduation, Rating “________” defense (of work)” Chief/Head of State Examination Head of Theoretical Physics Committee Department,Acad. I.M. Lifshyts _______ ___________ prof. Yermolayev О. М. “____” 2012 “____” 2012
Yadgar Ibrahim Abdulkarim
« Thermoelectric effects and chiral tunneling in single-wall carbon nanotubes »
Diploma work for acquisition educational qualification level 8.04020301 — “Master” in the direction of training/preparation/education 040203 — “physics” Scientific supervisor – Professor of Academician I.M. Lifshyts Theoretical Physics Department of V.N. Karazin KNU prof. Krive I.V Kharkiv 2012
Anyone who has never made a mistake has never tried anything new
Albert Einstein
Abstract We consider transmission coefficient for chiral tunneling through “pseudo-agnetic” impurity with finite width and height in single wall carbon nanotube. On the basis of the energy dependence of the transmission coefficient calculated thermoelectric coefficients. It is discovered an anomalous behavior of the thermoelectric coefficients on temperature, which depends on the chiral angle of nanotube and height of impurity barrier.…………………………………… Also we consider heat current through the single –level quantum dot with electronvibronic
interaction
………………………...……………………
Was found suppression of current with increasing of constant of the electron- vibronic coupling and the occurrence of anomalous temperature of thermal conductance.
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Acknowledgments The project is done as a partial requirement to complete MSc. At the Kharkov National University by V.N.Karazina, under the supervision of Prof.Ilya V. Krive from the year 2010 to 2012. First and foremost, I offer my sincerest gratitude to my supervisor Ilya V. Krive whose encouragement, guidance and support from the initial to the final level enabled me to develop an understanding of the subject. I would like to thank aspirant Anton V.Parafilo for teaching me very interesting physics. Many thanks for all other people who gave their precious help. Without their efforts, this work would be impossible. Thanks for Physics Department, V. N. Karazin National University, Kharkov for his helpful comments and a cooperation on publishing my results. Lastly but not least, I offer my regards and blessings to my parents and sisters for their supports and love in any respect during the completion of the study.
Yadgar Ibrahim Abdulkarim 2012
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Contents Abstract…………………………………………………………………..i Acknowledgments .......................... ……………………………………..ii List of Figures …………………………………………………………..iv Introduction ...............................................................................................1 1.1 Background Information ...................................................................5 1.2 Structure of carbon nanotube ..........................................................11 1.3 Graphene .........................................................................................15 1.4 Atomic and Electronic Structure graphene .....................................16 1.4.1 Atomic structure of grapheme .................................................16 1.4.2 Electronic structure of grapheme .............................................18 2 Single-wall carbon nanotubes (SWNT) .............................................19 2.1 What is Single-wall carbon nanotubes (SWNT) ............................21 2.2 Electronic properties of SWNTs .....................................................22 2.2.1 Band structure of SWNTs ........................................................22 2.2.2 Density of states of SWNTs ....................................................24 2.3 Chiral tunneling ..............................................................................25 2.3.1 Chiral tunneling in graphene ...................................................28 2.3.2 Chiral tunneling in SWNTs .....................................................30 2.4 Single wall carbon nanotube as a quantum dot……………………34 3 Transmission probability (Coefficient) ..............................................35 3.1 Calculate Transmission Coefficient ................................................46
3.2 Conductance ....................................................................................48 4 Thermoelectrical coefficients ..............................................................49 4.1 Calculating thermopower ................................................................51 4.2 Heat current .....................................................................................54 4.3 Thermopower ..................................................................................57 5 Heat current through single level quantum dot (with electromechanical coupling)………………………………………………………58 5.1 Model Hamiltonian……………………………………………….61 5.2 Method of equations motion……………………………………...63 5.3 Factorization and homogenization……………………………….67 Conclustion…………………………………………………………....68 Bibliography.........................................................................................71
List of Figures 1.1
The observation by TEM of multi-wall coaxial nanotubes with various ….inner and outer diameters…. …………..…………………................……….(2)
1.2
Types of carbon nanotubes ………………………….……………….…….....(4)
1.3
Computational image of single-and multi-walled nanotubes……...……..…..(5)
1.4 (a) The chiral vector
or Ch =
1+m 2
(b) Possible vectors specied by the pairs of integers (n,m) for general ……..carbon nanotubes, including zigzag, armchair, and chiral nanotubes…..….(8) 1.5
Schematic models for single-wall carbon nanotubes with the nanotube axis
normal to the chiral vector………………………………………………………. .(9) 1.6 Graphene is an atomic-scale honeycomb lattice made of carbon atoms…....(14) 1.7 C60 fullerene molecules, carbon nanotubes, and graphite can all be thought of as ……being formed from graphene sheets …….........................................................(15) 1.8
Figure 1.8 Atomic structure of graphene, bipartite lattic…………..….…. (16)
1.9 Electronic band structure of grapheme………..…………………….....…….(17) 2.1 (a) Schematic of a portion of a graphene sheet (b) 2D graphene sheet ……illustrating lattice vectors a1 and a2…………………………….…….…..…..(20) 2.2 SWNT lattice structure …..………………….…………………………….…(21) 2.3 The band structure (top) and Brillouin zone (bottom) of graphene. …….…(22) 2.4 Diagram showing density of state versus energy in SWNTs. …………....(23) 2.5 Kataura plot…………………………………….…………………………....(25) 2.6 Tunneling through a potential barrier in a graphene…….…………………(27) 2.7 Electron transfer from-to quantum dot-leads …………………………..…(31) iii
2.8 Quantum dot based on fullerene ……………………..………………..….…...(32) 2.9 Quantum dot based on suspended carbon nanotube ……………………...….(33) 2.10 Quantum dot based on carbon nanopeapod ……………...………………....(33) 2.11 Anomalous temperature dependence of conductance through quantum dot based ……..on carbon nanopeapods……………………………….………………(34) 3.1 Transmission coefficient as a function of energy ……………………...……(43) 3.2 Transmission coefficient as a function of chiral angle ……….......................(45) 3.3 Coefficient corresponds to the phase of the backscattering ………….…….(46) 4.1 Conductance as a function of chiral angle ………………………..…………(51) 4.2 Conductance as a function of temperature.…………………………………..(51) 4.3 Heat conductance as a function of temperature …………..…………………(52) 4.4 Thermal conductance as a function of temperature ……………….…….......(53) 4.5 Thermal conductance as a function of chiral angle ……………………..….. (54) 4.6 Thermoelectric power as a function of temperature ………………..…….….(55) 4.7 Thermoelectric power as a function of chiral angle ……………………...…..(56) 4.8 Lorentz number as a function of temperature……………………………..….(57) 5.1 Temperature gradient dependence for heat current through vibrating quantum …...dot …………………………………………………………………………….(66) 5.2 Temperature dependence of heat conductance……………………...……..…(67)
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Introduction 1.1 Background Information Carbon nanotubes are large molecules of pure carbon that are long and thin and shaped like tubes, about 1-3 nanometers (1 nm = 1 billionth of a meter) in diameter, and hundreds to thousands of nanometers long. As individual molecules, nanotubes are 100 times stronger-than-steel and one-sixth its weight. Some carbon nanotubes can be extremely efficient conductors of electricity and heat; depending on their configuration, some act as semiconductors. Carbon nanotubes are unique nanostructures that can be considered conceptually as a prototype one-dimensional (1D) quantum wire. The fundamental building block of carbon nanotubes is the very long all-carbon cylindrical single wall carbon nanotube (SWNT), one atom in wall thickness and tens of atoms around the circumference (typical diameter ~1.4 nm). Initially, carbon nanotubes aroused great interest in the research community because of their exotic electronic properties, and this interest continues as other remarkable properties are discovered and promises for practical applications develop. Very small diameter (less than 10 nm) carbon filaments were prepared in the 1970's and 1980's through the synthesis of vapor grown carbon fibers by the decomposition of hydrocarbons at high temperatures in the presence of transition metal catalyst particles of <10nm diameter [1,2]. However, no detailed systematic studies of such very thin filaments were reported in these early years, and it was not until the observation of carbon nanotubes in 1991 by Iijima of the (NEC) Nippon Electric Corporation, Laboratory in Tsukuba, Japan (see Fig. 1) using high resolution transmission electron microscopy (HRTEM) [ 3 ] that the carbon nanotube field was seriously launched. Independently, and at about the same time (1992),
2 Russian workers also reported the discovery of carbon nanotubes and nanotube bundles, but generally having a much smaller length to diameter ratio [4,5].
Figure 1.1:The observation by (TEM) Transmission electron microscopy of Multy wall coaxial nanotubes with various inner and outer diameters, di and do, and numbers of cylindrical shells N reported by Iijima in 1991: (a) N = 5. do=67A0, (b) N = 2, do=55A0, and (c) N = 7, di=23A0, do=65 A0.
It was the Iijima observation of the multiwall carbon nanotubes in Fig. 1 in 1991 that heralded the entry of many scientists into the field of carbon nanotubes, stimulated at first by the remarkable one-dimensional (1D) quantum effects predicted for their electronic properties, and subsequently by the promise that the remarkable structure and properties of carbon nanotubes might give rise to some unique applications. Whereas the initial experimental Iijima observation was for multi-wall nanotubes (MWNTs), it was less than two years before single-wall carbon nanotubes (SWNTs) were discovered experimentally by Iijima and his group at the NEC Laboratory and by Bethune and
3 coworkers at the (IBM) International Business Machines Corporation, Almaden Laboratory [6,7]. These findings were especially important because the single wall nanotubes are more fundamental, and had been the basis for a large body of theoretical studies and predictions
that preceded the experimental observation of single wall
carbon nanotubes. The most striking of these theoretical developments was the prediction that carbon nanotubes could be either semiconducting or metallic depending on their geometrical characteristics, namely their diameter and the orientation of their hexagons with respect to the nanotube axis (chiral angle) [8,9]. Though predicted in 1992, it was not until 1998 that these predictions regarding their remarkable electronic properties were corroborated experimentally [10,11]. A major breakthrough occurred in 1996 when Smalley and coworkers at Rice University successfully synthesized bundles of aligned single wall carbon nanotubes, with a small diameter distribution, thereby making it possible to carry out many sensitive experiments relevant to 1D quantum physics, which could not previously be undertaken [12]. Of course, actual carbon nanotubes have finite length, contain defects, and interact with other nanotubes or with the substrate and these factors often complicate their behavior. A diagram showing the types of carbon nanotubes the (n,m) nanotubes naming scheme can be thought of as a vector (Ch) in an infinite graphene sheet that describes how to 'roll up' the graphene sheet to make the nanotubes. T denotes the tube axis, and a1 and a2 are the unit vectors of graphene in real space. If m=0, the nanotubes are called zigzag. If n=m, the nanotubes are called armchair. Otherwise, they are called chiral.
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Figure 1.2: Types of carbon nanotubes .If m=0, the nanotubes are called zigzag. If n=m, the nanotubes are called armchair. Otherwise, they are called chiral.
One of the major classifications of carbon nanotubes is into single-walled varieties (SWNTs), which have a single cylindrical wall, and multi-walled varieties (MWNTs), which have cylinders within cylinders.
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Figure 1.3:Computational image of single-and multi-walled nanotubes. Source: ……….image gallery, Nanotechnology Team, NASA.
The lengths of both types vary greatly, depending on the way they are made, and are generally microscopic rather than nanoscopic, i.e. greater than 100 nanometers (a nanometer is a millionth of a millimeter). The aspect ratio (length divided by diameter) is typically greater than 100 and can be up to 10,000, but recently even this was made to look small. In May last year SWNT strands were made in which the SWNTs were claimed to be as long as 20 cm. Even more recently, the same group has made strands of SWNTs as long as 160 cm, but the precise make-up of these strands has not yet been made clear. A group in China has also found, purely by accident, that packs of relatively short carbon nanotubes can be drawn out into a bundle of fibers, making a thread only 0.2 millimeters in diameter but up to 30 centimeters long. The joins between the nanotubes in this thread represent a weakness but heating the thread has been found to increase the strength significantly, presumably through some sort of fusing of the individual tubes.
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1.2 Structure of carbon nanotube The structure of carbon nanotubes has been explored early on by high resolution transmission electron microscopy (TEM) and scanning tunneling microscopy (STM) techniques [13], yielding direct confirmation that the nanotubes are seamless cylinders derived from the honeycomb lattice representing a single atomic layer of crystalline graphite, called a graphene sheet, represented by the hexagonal honeycomb lattice of Fig. 1.4(a). The structure of a single-wall carbon nanotube is conveniently explained in terms of its 1D unit cell, defined by the vectors
and T in Fig 1.4(a).
The circumference of any carbon nanotube is expressed in terms of the chiral vector which connects two crystallographically equivalent sites on a 2D graphene sheet [see Fig. 1.4(a)] [8]. The construction in Fig. 1.4(a) depends uniquely on the pair of integers (n,m) which specifies the chiral vector. Figure 1.4(a) shows the chiral angle
between the chiral vector Ch and the "zigzag" direction (
shows the unit vectors
and
) and
of the hexagonal honeycomb lattice of the graphene
sheet. Three distinct types of nanotube structures can be generated by rolling up the grapheme sheet into a cylinder as described below and shown in Fig 1.5. The zigzag and armchair nanotubes, respectively, correspond to chiral angles of chiral nanotubes correspond to normal to
and
. The intersection of the vector
and (which is
) with the first lattice point determines the fundamental one-dimensional
(1D) translation vector T . The unit cell of the 1D lattice is the rectangle defined by the vectors Ch and T [Fig. 1.4(a)]. The cylinder connecting the two hemispherical caps of the carbon nanotube(see Fig. 1.5) is formed by superimposing the two ends of the vector made along the two lines perpendicular to the vector 1+m 2,
and
and the cylinder joint is
in Fig. 4(a). The lines
at each end of
and
are both
[8]. In the (n,m) notation for
=
the vectors (n, 0) or (0, m) denote zigzag nanotubes and the vectors (n, n)
7 denote armchair nanotubes. All other vectors (n,m) correspond to chiral nanotubes [14]. The nanotube diameter dt is given by
dt
L
a n 2 m 2 nm
L = Length of chiral victor
(1)
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Figure :1.4 (a) The chiral vector carbon atoms by unit vectors Along the zigzag axis
or Ch = 1
and
2
1+m
2
is defined on the honeycomb lattice of
and the chiral angle with respect to the zigzag axis.
= 00. Also shown are the lattice vector
cell and the rotation angle
and the translation
= T of the 1D nanotube unit
which constitute the basic symmetry
operation R = (ψ/τ ) for the carbon nanotube .The diagram is constructed for (n,m) = (4, 2) [8]. (b) Possible vectors specied by the pairs of integers (n,m) for general carbon nanotubes, including zigzag, armchair, and chiral nanotubes. Below each pair of integers (n,m) is listed the number of distinct caps that can be joined continuously to the carbon nanotube denoted by (n,m) [8]. The encircled dots denote metallic nanotubes while the small dots are for semiconducting nanotubes [15]
9 (a)
(b)
(c)
Figure:1.5 Schematic models for single-wall carbon nanotubes with the nanotube axis normal to the chiral vector which, in turn, is along: (a) the = 300 direction [an "armchair" (n, n) 0 nanotube] , (b) the = 0 direction ["zigzag" (n, 0) nanotube], And (c) a general direction, such as (see Fig. 2), with 0 < < 300 [a " chiral" (n,m) nanotube]. The actual nanotubes shown here correspond to (n,m) values of: (a) (5, 5), (b) (9, 0), and (c) (10, 5). The nanotube axis for the (5,5) nanotube has 5-fold rotation symmetry, while that for the (9,0) nanotube has 3-fold rotation symmetry [16]
Where ac-c is the C-C bond length (1.42A0), and the chiral angle
is given by (2)
From Eq. (2) it follows that = 300 for the (n, n) armchair nanotube and that the (n, 0) zigzag nanotube would have = 600. From Fig. 1.4(a) it follows that if we limit to be between 0 300, then by symmetry = 00 for a zigzag nanotube. Both armchair and zigzag nanotubes have a mirror plane and thus are considered as a chiral. Differences in the nanotube diameter dt and chiral angle give rise to differences in the properties of the various carbon nanotubes. The symmetry vector R = ( ψ/τ) of the symmetry group for the nanotubes is indicated in Fig. 1.4(a), where both the translation unit or pitch and the rotation angle are shown.
10 The number of hexagons, N, per unit cell of a chiral nanotube, specified by integers (n,m), is given by N = 2(m2 + n2 + nm) / dR
(3)
where dR = d if n - m is not a multiple of 3d or dR = 3d, if n - m is a multiple of 3d. Each hexagon in the honeycomb lattice [Fig.1. 4(a)] contains two carbon atoms. The unit cell area of the carbon nanotube is N times larger than that for a grapheme layer and consequently the unit cell area for the nanotube in reciprocal space is correspondingly 1/N times smaller. Table 1 provides a summary of relations useful for describing the structure of single wall nanotubes [17]. Figure 1.4(b) indicates the nanotubes that are semiconducting and those that are metallic, and indicates the number of distinct fullerene caps that can be used to close the ends of an (n,m) nanotube, such that the fullerene cap satisfies the isolated pentagon rule.
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Table : Structural parameters for Carbon Nanotubes [17] a ,
2
Symbol name length of unit victor unit victor
Formula a=
Value ac-c= 1.44 A X,Y coordinate
, Ch
chiral vector
L
length of Ch
dt
diameter
Ch =
(
1+m 2
)
Ch = dt = L/
chiral angle sin = tan cos = b)
d
gcd(n,m)
dR
gcd(2n + m,2m + n)b) dR= T = t1a1 + t2a2 (t1; t2) t1 =2m + n / dR t2 = - 2n + m /dR
T
translation vector
T N
length of T number of hexagons T= is the nanotub unit cell N = 2(n2 + m2 + nm) / dR
R
symmetry vector pitch of R rotation angle
R = pa1 + qa2
(p; q)
gcd(t1; t2) = 1b)
gcd(p; q) = 1b)
(mp - nq) T/N=MT/ N Ψ=R*τ
In radians
a) In this table n, m, t1, t2, p, q are integers and d, dR N and M are integer functions of these integers. b) gcd(n;m) denotes the greatest common divisor of the two integers n and m.
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1.3 Graphene Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a (hexagonal ) honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix-ene by Hanns-Peter Boehm, who described single-layer carbon foils in 1962. Graphene is most easily visualized as an atomic-scale chicken wire made of carbon atoms and their bonds. The crystalline or "flake" form of graphite consists of many graphene sheets stacked together. Graphite itself consists of many graphene sheets stacked together. Most of us have produced graphene like structure unknowingly while using pencil. Lead of pencil is made of graphite ,and when we write with it on a piece of paper ,the graphite is cleaved in to thin layers and make up the text or drawing that we are trying to make. some of these this layers will containe only a few layers or even a single layer of graphite ,i.e graphene. The carbon-carbon bond length in graphene is about 0.142 nanometers. Graphene sheets stack to form graphite with an interplanar spacing of 0.335 nm, which means that a stack of 3 million sheets would be only one millimeter thick. Graphene is the basic structural element of some carbon allotropes including graphite, charcoal, carbon nanotubes and fullerenes. It can also be considered as an indefinitely large aromatic molecule, the limiting case of the family of flat polycyclic aromatic hydrocarbons. The Nobel Prize in Physics for 2010 was awarded to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two dimensional material graphene". Graphene with the unique combination of bonded carbon atom structures with its myriad and complex physical properties is poised to have a big impact on the future of material sciences, electronics and nanotechnology. Owing to their specialized structures and minute diameter, it can be utilized as a sensor device, semiconductor, or for components of integrated circuits. The reported properties and applications of this two-
13 dimensional form of carbon structure have opened up new opportunities for the future devices and systems. Graphene as a material is completely new –not only the thinnest ever but also the strongest.It is the one-atom thick planar sheet of carbon atoms, which makes it the thinnest material ever discovered. Graphene is the basic structural element of some carbon allotropes including graphite, charcoal, carbon nanotubes & fullerenes.It can be wrapped up into 0D fullerences, rolled into 1D nanotubes or stacked into 3D graphite. Graphene is highly conductive-conducting both heat & electricity better than any other material, copper & stronger than diamond. It is almost completely transparent, yet so dense that not even helium can pass through it.
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Bond Length=0.142nm
Figure 1.6: Graphene is an atomic – scale honeycomb lattice made of carbon atoms
High electron mobility- up to 25000cm² per volt-second .The theoretical mobility of electrons is 200 times that of silicon. The corresponding resistivity of the grapheme Sheet would be 10-6 Ω.cm. the charge storage time is less ,so the operation frequency is high. The near-room temperature thermal conductivity of graphene was recently measured to be between (4.84±0.44) ×103 to (5.30±0.48) ×103 Wm−1K−1. The breakover voltage is less than 0.3V.
15 The below is the some allotropic structures of the carbon
Figure 1.7: C60 fullerene molecules, carbon nanotubes, and graphite can all be thought of as being formed from graphene sheets, i.e. single layers of carbon atoms arranged in a honeycomb lattice.
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1.4 Atomic and Electronic Structure graphene 1.4.1 Atomic structure of grapheme As already mentioned graphene is a planar, hexagonal crystal of carbon atoms. The next-neighbor distance is a = 2.45A°. The unit cell holds two carbon atoms, which belong to two different sublattices A and B. The A atoms are connected only to B atoms and vice versa, this is called a bipartite lattice.
Figure 1.8 Atomic structure of graphene, bipartite lattice
1.4.2 Electronic structure of grapheme
The electronic structure of graphene is rather different from usual threedimensional materials. Its Fermi surface is characterized by six double cones, as shown in Figure 9. In intrinsic (undoped) graphene the Fermi level is situated at the connection points of these cones. Since the density of states of the material is zero at that point, the electrical conductivity of intrinsic graphene is quite low and is of the order of the conductance quantum σ~e2/h; the exact prefactor is still debated. The Fermi level can however be changed by an electric field so that the material becomes either n-doped (with electrons) or p-doped (with holes) depending on the polarity of the applied field. Graphene can also be doped by adsorbing, for example, water or ammonia on its
17 surface. The electrical conductivity for doped graphene is potentially quite high, at room temperature it may even be higher than that of copper. Close to the Fermi level the dispersion relation for electrons and holes is linear. Since the effective masses are given by the curvature of the energy bands, this corresponds to zero effective mass. The equation describing the excitations in graphene is formally identical to the Dirac equation for massless fermions which travel at a constant speed. The connection points of the cones are therefore called Dirac points. This gives rise to interesting analogies between graphene and particle physics, which are valid for energies up to approximately 1 eV, where the dispersion relation starts to be nonlinear.
Figure 1.9: The energy, E, for the excitations in graphene as a function of the wave numbers, kx and ky, in the x and y directions. The black line represents the Fermi energy for an undoped graphene crystal. Close to this Fermi level the energy spectrum is characterized by six double cones where the dispersion relation (energy versus momentum, ℏk) is linear. This corresponds to massless excitations.
18 Graphene is practically transparent. In the optical region it absorbs only 2.3% of the light. This number is in fact given by π α, where α is the fine structure constant that sets the strength of the electromagnetic force. In contrast to low temperature 2D systems based on semiconductors, graphene maintains its 2D properties at room temperature. Graphene also has several other interesting properties, which it shares with carbon nanotubes. It is substantially stronger than steel, very stretchable and can be used as a flexible conductor. Its thermal conductivity is much higher than that of silver.
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Chapter two 2 Single-wall carbon nanotubes (SWNT) 2.1 What is Single-wall carbon nanotubes (SWNT) Single-wall carbon nanotubes (SWNT) are tubes of graphene that are normally capped at the ends. They have a single cylindrical wall. The structure of a SWNT can be visualized as a layer of graphite, a single atom thick, called graphene, which is rolled into a seamless cylinder. Most SWNT typically have a diameter of close to 1 nm. The tube length, however, can be many thousands of times longer. SWNT are more pliable yet harder to make than molti-wall nanotubes (MWNT). They can be twisted, flattened, and bent into small circles or around sharp bends without breaking.SWNT have unique electronic and mechanical properties which can be used in numerous applications, such as fieldemission displays, nanocomposite materials, nanosensors, and logic elements. These materials are on the leading-edge of electronic fabrication, and are expected to play a major role in the next generation of miniaturized electronics. Carbon nanotubes, especially single-walled carbon nanotubes, have been termed “materials of the 21st century . This quasi-one-dimensional material has attracted tremendous scientific interest and has become one of the most popular objects in physics and material science over the last two decades.
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Figure 2.1: (a) Schematic of a portion of a graphene sheet rolled up to form a SWNT (b) 2D graphene sheet illustrating lattice vectors a1 and a2, and the roll-up vector Ch = na1+ ma2. The achiral, limiting cases of (n, 0) and (n, n) armchair are indicated with thick,dashed lines, and the chiral θ angle is measured from the zigzag direction. The light, dashed para-llel lines define the unrolled, infinite SWNT. The diagram has been constructed for (n, m) = (4, 2).
Much more uniform in structure, single-walled nanotubes hold far more promise in terms of industrial applications than their multi-walled cousins. SWNT’s are narrower, approximately 0.7nm in diameter, and nave fewer defects than MWNT’s. Like MWNT’s, however, they are almost always closed at each end by a fullerene cage. The caps closing a SWNT,overwhelmingly, are one half of buckyball molecule at each end. They are far more rare and more desirable than MWNT’s; possessing ideal structure for future applications. At present, most SWNT’s are found in curled and curved strands rather than straight lines. A perfect rendering of SWNT’ can be seen in Figure 2.2. (Harris, 1999)
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Figure 2.2.: SWNT lattice structure (Haris, 1999)
Variations in the angle of the graphite planes that make up the bulk of a single-walled nanotube form the basis of how they are classified. This characteristic “twist” is referred to as the tube’s chiral angle. There are three distinct classifications based on the chirality of a carbon nanotube. “Zigzag” nanotubes are so named because the angle at which the garphene sheet is rolled up makes it parallel to the row of zigzag bonds in the hexagonal structure. “Armchair” nanotubes are so named because the graphene sheet rolls up at an angle that is perpendicular to the bonds in the hexagonal lattice. The last classification, “chiral” nanotubes, have sheets aligned along the cylinder at some chiral angle other than armchair or zigzag, that is to say somewhere between 0° and 30°. Figure 1.4(b) in chapter one depicts the system used in identifying the chiral angle of single wall nanotubes. (Smalley, 1997)
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2.2 Electronic properties of SWNTs 2.2.1 Band structure of SWNTs
Figure 2.3: The band structure (top) and Brillouin zone (bottom) of graphene. The red parallel lines correspond to a set of discrete energy subbands for carbon nanotub. [18]
The electronic structure of carbon nanotubes is usually described on the basis of the band structure of graphene. Figure 2.3 shows the band structure (top) and Brillouin zone (bottom) of graphene. The valence band ( π -character) and the conduction band (π *character) touch at six points that lie at the Fermi energy, but only two of these points are inequivalent (the K and K' points). In the case of SWNTs, the quantization of the circumferential momentum leads to the formation of a set of discrete energy sub-bands for each nanotube (see red parallel lines in Figure 2.3). The relation of these lines to the band structure of graphene determines the electronic structure of the nanotube. If the lines pass through the K or K' points, the nanotube is a metal: if they do not (as in Figure 2.3), the nanotube is a semiconductor. More simply, the electronic structure can be deduced from the chiral indices by a relation following which nanotubes possessing chiral indices satisfying (n-m)=3q (q integer) are metallic while all the others are semiconducting.
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2.2.2 Density of states of SWNTs Figure 2.4 shows the density of states (DOS) of two types of carbon nanotubes. The DOS display sharp peaks which are called van Hove singularities, and are due to the one dimensional confinement of the electrons. The first and second transition for semiconducting SWNTs are abbreviated as Ejjs (E11s, E22s etc.), respectively (Figure 2.4 (a)). The transition energy can be described by the equation: = 2j where j is the index denoting the transition, ac-c is the nearest neighbor C-C distance, γ0 is the nearest neighbor interaction energy and
d t is the nanotube diameter[19].
However, more accurate calculations of the band energy also depend on the structure of carbon nanotube. Corrections are partially due to the curvature effect. In fact, perpendicular polarized optical transitions assigned to, for example, E12 (or E21) are also possible. These transitions are generally weak and are observed for light polarized perpendicular to the tube axis[20,21].The first transitions in metallic nanotubes are also reported in Figure 2.4 (b).
Figure 2.4: Diagram showing density of state versus energy in SWNTs. The optical transitions associated with the van Hove singularities are indicated for semiconducting (a) and metallic tubes (b).
24 The band structure of carbon nanotubes with different (n, m) indexes together with an empirical plot describing the relationship between the nanotube diameter and its band gap energies known as “Kataura plot” was reported [ 22 ].Figure 2.5 shows the updated version of the “Kataura plot” obtained recently by S. Maruyama. The Kataura plot is very useful because it allows to estimate the diameter distribution of a SWNT sample by measuring the energy of the absorption peaks.
Figure 2.5 Kataura plot
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2.3 Chiral tunneling 2.3.1 Chiral tunneling in graphene Graphene is a single layer of carbon atoms densely packed in a honeycomb lattice, or it can be viewed as an individual atomic plane pulled out of bulk graphite. From the point of view of its electronic properties, graphene is a two dimensional zero gap semiconductor with the energy spectrum shown in Fig. 2.6 a, and its low energy quasiparticles are formally described by the Dirac-like Hamiltonian [23,24]
(1)
Where
, is the Fermi velocity and σ = (σx, σy ) are
the Pauli matrices. Neglecting many-body effects, this description is accurate theoretically [23,24] and has also been proved experimentally[25,26] by measuring the energy-dependent cyclotron mass in graphene (which yields its linear energy spectrum) and, most clearly, by the observation of a relativistic analogue of the integer quantum Hall effect. The fact that charge carriers in graphene are described by the Dirac-like equation (1), rather than the usual Schrodinger equation, can be seen as a consequence of graphene’s crystal structure, which consists of two equivalent carbon sublattices [23,24] , A and B. Quantum mechanical hopping between the sublattices leads to the formation of two cosine-like energy bands, and their intersection near the edges of the Brillouin zone (shown in red and green in Fig. 2.6 a) yields the conical energy spectrum. As a result, quasiparticles in graphene exhibit the linear dispersion relation E =
, as if they were
massless relativistic particles with momentum k (for example, photons) but the role of the speed of light is played here by the Fermi velocity
≈ c /300. Owing to the linear
spectrum, it is expected that graphene’s quasiparticles will behave differently from those in conventional metals and semiconductors where the energy spectrum can be
26 approximated by a parabolic (free-electron-like) dispersion relation. Although the linear spectrum is important, it is not the only essential feature that underpins the description of quantum transport in graphene by the Dirac equation. Above zero energy, the current carrying states in graphene are, as usual, electron-like and negatively charged. At negative energies, if the valence band is not full, its unoccupied electronic states behave as positively charged quasiparticles (holes), which are often viewed as a condensedmatter equivalent of positrons. Note, however, that electrons and holes in condensedmatter physics are normally described by separate Schrodinger equations, which are not in any way connected (as a consequence of the Seitz sum rule [27], the equations should also involve different effective masses). In contrast, electron and hole states in graphene are interconnected, exhibiting properties analogous to the charge conjugation symmetry in QED. For the case of graphene, the latter symmetry is a consequence of its crystal symmetry because graphene’s quasiparticles have to be described by two-component wavefunctions, which are needed to define relative contributions of sublattices A and B in quasiparticles’ make-up. The two-component description for graphene is very similar to the one by spinor wavefunctions in QED, but the ‘spin’ index for graphene indicates sublattices rather than the real spin of electrons and is usually referred to as pseudospin σ. There are further analogies with QED. The conical spectrum of graphene is the result of intersection of the energy bands originating from sublattices A and B (see Fig. 2.6 a) and, accordingly, an electron with energy E propagating in the positive direction originates from the same branch of the electronic spectrum (shown in red) as the hole with energy −E propagating in the opposite direction. This yields that electrons and holes belonging to the same branch have pseudospin σ pointing in the same direction, which is parallel to the momentum for electrons and antiparallel for holes (see Fig. 2.6 a). This allows the introduction of chirality, that is formally a projection of pseudospin on the direction of motion, which is positive and negative for electrons and holes, respectively. The term chirality is often used to refer to the additional built-in symmetry between electron and hole parts of graphene’s spectrum (as indicated by colour in Fig. 2.6) and is
27 analogous ( although not completely identical [28,29] ) to the chirality in three dimensional QED. Since graphene is a semi metal, with electronic bands that cross in exactly one point, the conductivity with the Fermi level going through the conic point should drop to zero at low temperatures, where there should be no excited conducting states. But the fact is that conductivity doesn’t drop below a minimum which is on the order of the minimal conductivity This figure illustrates how an electron in graphene can tunnel trough any barrier (bands of different color belong to different pseudo-spin σ orientations).
Figure 2.6: Tunneling through a potential barrier in a graphene.
potential
28
A, Schematic diagrams of the spectrum of quasiparticles in single-layer graphene. The spectrum is linear at low Fermi energies (<1 eV). The red and green curves emphasize the origin of the linear spectrum, which is the crossing between the energy bands associated with crystal sublattices A and B. b, Potential barrier of height V0 and width D. The three diagrams in a schematically show the positions of the Fermi energy E across such a barrier. The Fermi level (dotted lines) lies in the conduction band outside the barrier and the valence band inside it. The blue filled areas indicate occupied states. The pseudospin denoted by vector σ is parallel (antiparallel) to the direction of motion of electrons (holes), which also means that σ keeps a fixed direction along the red and green branches of the electronic spectrum. C, Low-energy spectrum for quasiparticles in bilayer graphene. The spectrum is isotropic and, despite its parabolicity,also originates from the intersection of energy bands formed by equivalent sublattices, which ensures charge conjugation, similar to the case of single-layer graphene. In relativistic physics there is phenomenon called “Klein Paradox”. It states that when a potential barrier exceeds the energy required to create a particle-antiparticle pair it becomes transparent, and is perfectly transparent even at infinite barrier. This is so because a potential that is repulsive for particles is attractive for antiparticles, which means that there are antiparticle states inside the potential barrier at any energy of incident particle and a resonant tunneling occurs.In graphene the gap between electrons and holes is zero, which means that any potential is transparent for graphene (at least for momenta at the conic points K and K’). This allows ballistic transport, electrons can transport current without being scattered. Up to now ballistic current was measured in graphene over several microns (which is actually the size of the samples), which is actually the holy grail of nanometer-scale electronic engineering.
29
2.3.2 Chiral tunneling in SWNTs
The concept of chiral tunneling in metallic single-wall carbon nanotubes, originating from the interplay of local electrostatic and pseudomagnetic potentials, was introduced in [ 30] considered and applied to an evaluation of the Josephson current in a nanotube-based superconductor–normal metal–superconductor (SNS) junction and the persistent current in a circular nanotube. Tunneling of non relativistic and relativistic fermions through a potential barrier is drastically different. While for non relativistic electrons the transmission probability, as a rule, is small and it exponentially depends on the potential strength, massless Dirac fermions freely penetrate potential barriers of arbitrary strength with a probability D = 1 for normal incidence (the Klein paradox).The absence of back scattering is explained by the conservation of fermion helicity, i.e., the additional quantum number for relativistic particles with spin. Finite backscattering (D < 1) appears in tunneling of massless particles through a scalar (electrostatic) barrier for angles of incidence other than normal or in the presence of a vector potential (“magnetic” scattering). In metallic single-wall carbon nanotubes (SWNTs), electron transport is known to be ballistic and the charged quasiparticles are one-dimensional (1D) Dirac-like massless excitations. Their weak scattering from long-range tube defects is used to explain (by analogy with the Klein paradox) the delocalization of electrons, even in long metallic nanotubes. Short-range defects cause electron backscattering (∆q ≈ 2kF ), which for Dirac quasiparticles in SWNTs is described as strong intervalley (±kF ) transitions. Since particles in different valleys are characterized by opposite helicities, the chiral properties of an individual electron do not play a significant role for electron transport in metallic nanotubes with short-range impurities. Therefore, electron tunneling through such defects in nanotubes is qualitatively the same as for non relativistic particles.
30 It was shown [31] that a particular type of electron scattering namely, “chiral tunneling,” can occur in metallic SWNTs as a result of the interplay between long-range (“smooth”) electrostatic and pseudomagnetic potentials. The electrostatic potential (V d ) models ordinary electron scattering by charged impurities (or by nonuniform gate potentials), while the pseudomagnetic potential (Vo) describes the effective vector potential caused by deformations of the nanotube.Therefore we physically study the influence of local strain on electron transport in SWNTs. Chiral tunneling bridges between ordinary tunneling (D « 1), which reappears in the limit Vo » Vd , and Klein tunneling (D = 1), which is reached in the opposite limit Vd » Vo. Chiral tunneling is pronounced when Vo ≈ Vd and is characterized by an oscillatory dependence of the electron transmission coefficient on the chiral phase φc = U0 cos dimensionless potential strength and
where U0 is the
is an effective chiral angle determined by the
nanotube chiral angle and the phase of the pseudomagnetic potential.
31
2.4 Single wall carbon nanotube as a quantum dot It is important because in our last chapter we will consider transport in single wall carbon nanotube as transport through quantum dot Quantum dot is a fragment of a conductor or semiconductor, whose limited are confined in all three spatial dimensions, and containing the conduction electrons. Dot must be have so small size, that effects of size quantization were significant. This achieves, if kinetic electron energy
(d - the characteristic size of dot, m -effective mass of
electron on the dot), specified by uncertainty of its momentum, will be much more than another all energetic scales :first, greater than temperature, expressed in energy units. Quantum dot may made from any
enough small piece of metal, semiconductor or
molecular, puts between leads (fig. 2.7).
Figure: 2.7: Electron transfer from-to quantum dot-leads
Historically, the first quantum dots were micro-crystals of cadmium selenide CdSe. Electron in that micro-crystal behaves like an electron in 3-dimentional potential well, it has much stationary energy levels with characteric distance between them (precise equation for energy levels depends of quantum dots form)……………….
32 The same as to transition between atom's energy levels, at the transition of quantum dot between energy levels, the photon may radiate. Also, electron could pelt to the high energy-level, and radiance could be obtained from transition between more lowest placed level,(luminescence). Unlike real atoms, transition frequencies to easily manage, changing the crystal size. So, observation of luminescence the crystal of cadmium selenidium with frequency of the luminescence, determined by the size of the crystal, and and served as the first observation of quantum dots. In present time, there're many works and experiments devotes to the quantum dots, formed in 2-dimensionalli electron gas-.In 2-dimensionall electron gas electrons motion is perpendicularly to plane has limited yet, and the area on the plane could be dedicated by metal gate leads, which put on to the heterostructure above. Quantum dots in 2dimentional electronic gas can be connected by tunneling junction with another areas of 2-dimensional electron-gas and explores the conductivity through quantum dot. In such system observing Coulomb blockade phenomenon. In the new way of physics – nanoelectromechanics [32, 33, 34],], interaction is taken into account electronic degrees of freedom with mechanical. For example, studying of vibrating quantum dot(molecular transistors, shuttle effect). In real world, such systems may be: 1. Molecules (for example С60) between leads (fig. 2.8).
Figure: 2.8: Quantum dot based on fullerene
33
2. Carbone nanotube between leads, etched out of the substrate (fig. 2.9) this case is important for us.
Figure: 2.9: Quantum dot based on suspended carbon nanotube
Carbone nanopeapod (fig. 2.10 a, b). This system represents the fullerene moleculs, which is inside the Carbone nanotube .Clasters of С60 may oscillate in potential of imperfection on the surface of carbon nanotube .
а
b
Figure: 2.10 a, b: Quantum dot based on carbon nanopeapod (fullerenes molecules inside carbon nanotubes).
From the modern theoretical and experimental researches of similar systems ,main effects could be selected ,which appears as result of accounting of electron-vibronic interaction (consideration of oscillating quantum dot). One of the most impressive demonstration of polaronic effects is narrowing of level width on the quantum dot at low temperatures (
), in consequence of the
34 presence of zero-point oscillations,
. Narrowing would determine by value of
constant electron-vibronic interaction, . Anomalous temperature dependence of conductance at the resonant tunneling As known, the dependence of conductance on temperature has the form:
Figure 2.11: Anomalous temperature dependence of conductance through quantum dot based on carbon nanopeapods.
At temperatures higher than Г, but lower than
the width of level renormalizes,
.At higher temperatures renormalization of the level width is removed (in result of the opening of the temperature all inelastic scattering), i.e.
. So, at the transition
between asymptotic bonus we have on the G – T diagram the “hump”.
35
Chapter three 3 Transmission probability (Coefficient) 3.1 Calculate Transmission Coefficient In this chapter we evaluate the transmission probability for electron scattering by special defects in carbon nanotubes. We will assume that the defect potentials are longranged and do not induce inter-valley electron scattering (δk
2kF). Thus in our model
the metallic SWNT Hamiltonian takes the form
(3.1) Where
is the valley index, vF is the Fermi velocity, px = -idx , θ is the chiral angle
of the nanotube (0 < θ <
/6), and the x- axis is directed along the cylinder axis.
We define the chiral angle θ = 0 that for the armchair nanotube and θ = zigzag nanotube. The presence of chiral factors
/ 6 for the
in the Hamiltonian Eq (3.1)
results in special scattering of electrons by a non-diagonal potential. an effect which we refer to as chiral tunneling in the following. The electrostatic (scalar) potential is diagonal in the pseudospin indices and can not induce electron backscattering in our model Eq. (3.1) due to the conservation of helicity for massless Dirac particles (the Klein paradox). To get nontrivial scattering of chiral particles we consider the matrix potential V s (x) =
(3.2)
36 Which mixes the sublattice components of the electron wave function. For simplicity we consider all matrix elements to be real and equal. An effective scattering potential of the form (3.2) was suggested for the electron scattering in metallic carbon nano-peapods. It is induced by the hybridization of fullerene molecular orbitals (LUMO) with the conduction electron states in the nanotube. We consider a rectangular potential of width a and height
, which allows us to get an
analytical solution for the scattering problem. Dirac equation is represented in the form V(x) V0 A r
t
B X
1 Where
0
2
a
3
=
In the potential free regions (V0 = 0) the equation becomes ± vf
(3.3)
Eq (3.3) gives us a set of two linear equations for and components of spinor wave function it is exactly see that each component satisfies the wave equation =0
(3.4)
The plane wave solutions of equation (3.4) are (x)=
(3.5)
Where (in units v f =1) the liner spectrum E= is the charact-eristic feature of quasiparticle spectrum of metallic single-wall carbon nanotubes.
37 Notice that component of spinor wave function is represented through component by the equation (3.6) ) as
We specify the scattering in the potential – free regions (x follows
+r
=
,x
(3.7)
,
=
(3.8)
Where r ,t are the backscattering (r) and forward scattering (t) amplitudes. Next we consider the solution of Dirac equation in region
have rectangular
potential barrier of width a and height
(3.9)
It gives us a set of two liner differential equation for
components (3.10a) (3.10 b)
From the second equation (3.10 b) (3.11) By putting Eq.(3.11) into Eq. (3.10 a) =
(3.12)
38
Or equivalently =0 By substituting spectral parameter
(3.13)
in to Eq.(3.13) we find algebraic (quadratic) equation for
(3.14) The two roots of this equation are -
(3.15)
Now the plane –wave solution of Eq.(3.13)takes the form (x)=
+
(3.16)
Here we introduced the notation is q=
and
(3.17)
The corresponding form of the (3.17) is
- component according Eq.(3.11) and (3.16) ,
(3.18)
It can be represented in the form analogous of Eq.(3.16)
+
(3.19)
Where (3.20) In second region
the wave function becomes
(3.21)
39 We use the boundary condition (3.22)
To match the wave functions at points x=0 and x= a ,this results in a set of liner equations of scattering coefficients (r,t,A,B)
(3.23)
From the first and the second equations in Eq.(3.23) we find
2= A(1
(3.24)
subtract the third and the forth equations B(1+
(3.25)
A(1+
(3.26)
And express A and B through the scattering amplitude t A=
(3.27)
B=
(3.28)
Next we put coefficient A and B in equation (3.24) (3.29)
40 After evident transformation one gets
(3.30) According to Eq.(3.20) the sum and the product of
can be represented as follows
2
=
(3.31)
(3.32)
By putting Eq. (3.31), (3.32) we obtain the desired formula for the transmission amplitude =
(3.33)
At the first we consider the case local scatters ,which is reproduced in the limit In the case
and Eq. (3.33) is strongly simplified (3.34)
This formula for the first time was derived in [35]. From Eq.(3.34) it is easy to get expression for the transmission coefficient D= and the forward ,and backward ,scattering phases
41
D here
(3.35) =
is the potentail strength
= arctan
(3.36)
arctan The scattering phase.
(3.37)
satisfy a simple equation (3.38)
For a rectangular potential of finite height V the transmission amplitude t(E) take the form
(3.39) Now the transmission probability D explicitly depend on energy D
=
and the forward scattering phase
,
(3.40)
= ka+qa+arctg(
(3.41)
42
(a)
(b)
43
(c) Figure 3.1: Transmission coefficient as a function of energy at different values of chiral angles (
).
We plot the dependence of transmission coefficient on energy for three different chiral angles .it is seen from these figures that at energies of incident particle around potential V ,the transmission probability is strongly suppressed D(E
.
The biggest the chiral angle the more pronounced is the effect of suppression (see figures a,b) . For zig-zag nanotube (
the tunneling probability D, at energies in the interval E
= [ 0.5 V ,1.5 V ] is exponentially . In all and hence the chiral scatter acts as an ordinary tunneling barrier with exponentially small transparency.
44
(a)
Chiral angle Energy = 0.1 E / V (b)
Chiral angle Energy = 0 E / V
45
(c)
Chiral angle Energy = 1.1 E / V
Figure 3.2: Transmission coefficient as a function of chiral angle at different values of energies E = 0.1, 0 ,1.1 ( E / V).
Figure 3.2 show the dependence of transmission probability D
on the chiral angle
for different energies.it is seen that for energies E far from potential height E transmission coefficient for the rectangular barrier behave as D
V the
for the local chiral
scatter [36]. It oscillates on the chiral angle .on contrary when the energy of incident particle is in the vicinity of potential height V,the transmission probability exponentially decreases with the increase of chiral angle( see the figure 3.2). As it see from equation (3.40 ) that decay region could be defined argument of trigonometrical functions
when square
root is positive than transmission coefficient is oscillating function. When
0 then transmission coefficient is decay function.
46 Thus the decay region of the transmission coefficient can be determined completely by external parameters (chiral angle of nanotube and height of potential barrier). This decay region
corresponds to the existence of critical
angles for the Klein paradox in 2D graphene (see review). As it seen, when the energy tends to infinity, the transmission coefficient tends to unity. At high energies the particle does not feel a small potential barrier. As the transmission coefficient corresponds to the phase of the backscatterin g can be seen from the figure.
Figure 3.3: Transmission coefficient corresponds to the phase of the backscattering at chiral angle 0.3.
Thus we see that the energy dependence significantly affects the type of the transmission coefficient.
47
3.2 Conductance In this subsection we study the transport properties of the transmission coefficient in the presence of the energy dependence .For this we use the well-known Landauer formula for the current
I=
(3.42)
Here D transmission probability,
Fermi-Dirac distribution function. In this
expression, the integration limits extended from minus to plus infinity. This is so that the nanotube is a semiconductor, i.e., we must take into account the conduction band and valence band (therefore, we count band of zero energy). That is, we assume that the nanotube is between two electrodes, which are characterized by an equilibrium distribution function for a noninteracting Fermi gas (temperature and chemical potential). As it is known, in the linear response in the low voltage we get the formula for the conductance G= =
Here
)
(3.42)
is a quant of conductance.
In figure 4.1 we can see how temperature affects on chiral angle dependence. At low temperature conductance is an oscillation function from chiral angle. But, the conductance in certain “magic” angles does not tend to the quantum of conductance (except when chiral angle is zero-armchair carbon nanotube). At more high temperature we have “average” transmission probability (it means that conductance is not oscillate). Energy integration is averages the transmission coefficient. As we see, conductance is different from temperature dependence of one level system. The temperature dependence can be understood from simple physical arguments. When the energy tends to plus or minus infinity (starting with some value), the transmission
48 coefficient is almost independent of energy ( D 1). In this case, we can calculate the value of the conductance. G=
(
But in a certain range of energies we have a region of exponential suppression. in this area there is a significant dependence of the of the energy probability transmission. the integral has to be a complicated function in this area. Obviously, in order of magnitude, this interval is associated with the magnitude of the potential barrier V . Here we have an important conclusion, if the energy dependence of the effect on the conductance of the transport, then there must be a strong influence on the thermoelectric power system (see next chapter).
49
Chapter four 4 Thermoelectrical coefficients 4.1 Calculating thermopower In this chapter we observe the case if the electrons in reservoirs are not only at different chemical potential, but also at different temperature. In most general form the transport coefficient have form 2*2 matrix (4.1)
G-electrical conductance, K-heat conductance, T =
(
-temperature in leads),
. Static thermopower is defined as a voltage produced by temperature difference at zero current
(4.2)
,
-is the static thermopower. One can find [Sivan, Imry] the expressions for heat conductance and cross-coefficient L(T ) in a similar way as for electrical conductance. For noninteracting electrons heat is transferred from hot to cold reservoir by electrons. Their ballistic transport described by transmission probability. Heat current is entropy current d =Td is the velocity of electron.
dS(E)
(4.3)
50 We know entropy for Fermi_Dirac particles S=Where
(4.4) is density of states for electrons.
By direct calculations in linear response approximations we have K(T) =
(4.5)
L(T) =
(4.6
From the formula for conductance G= =
), and Eq. (4.6)
we can found static thermopower
=
(4.7)
We will study our transmission coefficient numerically using the Eq (4.5),(4.7). More interesting for us is the static thermopower. Because, heat conductance is similar to electrical current (they differ only in degree of energy). And thermopower have nontrivial behavior. This magnitude depend on energy dependence in transmission coefficient (if transmission probability is energy independent, thermopower is constant).for multichannel ballistic wire, step wise energy dependence of transmission coefficient results in step in G and K, and in peaks in L.
51
Figure 4.1: Conductance as a function of chiral angle at different value of temperature , T = 0.15,0.6 Kelvin
Figure 4.2: Conductance as a function of temperature at different value of chiral angle = 0.2,0.6 degree
52
4.2 Heat current Now we study heat current dependence of thermal conductance on temperature and on the chiral angle you can see in the picture. Here we normalize the heat conductance on
(here
is a Bolzman constant).
As we expected, the heat conductance is not qualitatively different from the values of the conductance. Because properties of these quantities depend on the integration of the transmission coefficient at energy.
At low and high temperature transmission
coefficient is almost energy independent (constant), and only in a narrow range of energy transmission coefficient depends strongly on the energy. This is manifested in the nonmonotonic temperature dependence of heat conductance.
Figure 4.3: Heat conductance as a function of temperature at different value of chiral angl V = 10 volt,
=0.2 degree, blue V = 0 volt
53 To compare heat conductance with the graphs for the conductance it is convenient to normalize heat conductance on the temperature. Because temperature dependence of conductance (for D= 1) is
.
Figure 4.4: Thermal conductance as a function of temperature at different value of chiral angle , red V = 10 volt,
=0.2 degree, blue V = 0 volt
The temperature also smoothes out the oscillations of the transmission coefficient on the chiral angle in temperature dependence of thermal conductance. It can see on fig (4.5).
54
Figure 4.5 Thermal conductance as a function of chiral angle at different value temperature , T = 0.02, 0.1 Kelvin
55
4.3 Thermopower Now we study thermopower the magnitude of the thermoelectric power depends strongly on the energy dependence of the transmission coefficient and thus is most pronounced for big chiral angle when the region in which the transmission coefficient tends to zero much bigger.
Figure 4.6 Thermoelectric power as a function of temperature at different value of chiral angle angle
,
= 0.02, 0.6 degree
56
Figure 4.7 Thermoelectric power as a function of chiral angle at different value of temperature temperature
, T= 0.2, 0.6 Kelvin
Chiral angle dependence of thermoelectric power is so (see fig 4.7), because cross coefficient L is analog of derivative of transmission coefficient. It is interesting to note that in all temperature thermoelectric coefficients occur features (oscillating behavior) at low temperatures and at high chiral angles. We suggest that this may be due to the strong dependence of the oscillation of the transmission coefficient at high chiral angles. We think that these two properties (temperature nonmonotonic dependence and oscillation feature at low temperature) of the thermoelectric coefficient in chiral tunneling (for carbon nanotube with pseudopotential impurity) can be used for the qualitative detection in experiment of a new type of tunneling. Because, as we have seen from the energy dependence of the transmission coefficient and due to the integration of
57 energy, the resonance properties of the Klein paradox (at magic angle) in carbon nanotube at chiral tunneling disappears at finite temperatures. We want also introduce and to study Lorentz number, that is
. It is well
known that for noninteracting electrons . This expression is manifestation of Wiedemann-Franz law (proportionality between heat and electric conductivity).
Figure 4.8 Lorentz number as a function of temperature at value ….
It is seen that the Lorentz number is now a function of temperature.
degree
58
Chapter five 5 Heat current through single level quantum dot (with electromechanical coupling) 5.1 Model Hamiltonian In this chapter we will study the thermal transport for the other temperature range. (less then the average energy level splitting It is well known finite sized
(here L is nanotube length).
SWNTs are frequently used as quantum
dot.………………………………………………………… Electrical and thermal transport through a metallic vibrating nanotube at low temperatures could be described in a simplified model where a single (resonant) electronic level is coupled to a single vibronic mode. The chiral effects in this model are of course disappeared (if dependence is not included in the model parameters). That is why we now will not consider scattering by pseudomagnetic impurities, but we take into account possible vibrations of the nanotube. (it is interesting interaction and is reasonable model for low energy transport in nanotube.) Finite-sized SWNTs are frequently used as quantum dots. when temperature less than average level splitting
Let’s write Hamiltonian of our model (fig.2.7), in representation of operators of second quantization: , Where
– Hamiltonian characterizing lids,
characterizes a quantum dot state,
(5.1) – Hamiltonian, which
– term, which characterizes process of the
59 tunneling of electrons from quantum lead to the dot and back (rather, the tunneling between states on the quantum dot and in lids). First term in equation (5.1): (5.2) Where
energy measured from the chemical potential,
(
) –
Annihilation (creation) operator in leads. This operators are under Fermi-Dirac statistics, they correspond to commutation relation
corresponds to them,
– lead index , k – wave vector.
where
Second term in equation (5.1): (5.3) The fist term describes the presence of the one energy level on the quantum dot. The second term of equation (5.3) describes the quantum dot fluctuation in parabolic potential
, this term is record in representation of operators of second quantization
Hamiltonian, 0b b p 2 / 2m m0 2 x 2 р, х – momentum operators and quantum dots shift. Here and below we assume, if needed dimensionality of the final result is always possibly to recover. The third term is the standard expression for electron – vibronic interaction for the optical phonon branches (Einstein model). i.e. Interaction with phonons, which has no dispersion . We choose this type of interaction, because It qualitatively describes interaction of electron that have tunneled to the oscillating carbon nanotube in the field of the gate lead potential, which is controlled
energy level on the quantum dot
60 The third term in equation (5.1) is the standard tunneling Hamiltonian, which describes transitions between electron states on quantum dot and leads. Where
- tunnel matrix
element (amplitude of the tunneling process).Firsov unitary transformation Clearly this problem certainly couldn't be solved. There are two common approaches to solve it. This is the perturbation theory at constant of electron - vibronic interaction (then, in zero-order approximation we have exactly solvable resonance model). The second method is the perturbation theory at tunneling Hamiltonian. The tunneling amplitude would be the smallest parameter of the system. We will stop on the second method. In solving the problem by this method, it will be convenient to make a unitary transformation
[37], the operator of unitary transformation records in the
form: – momentum operator of quantum dot.
, Where
Particularly, purpose of this unitary transformation
, remove the term, that
responsible of the electron– vibronic interaction in Hamiltonian of quantum dot. To this condition was satisfied, constant must be equal New quantum dot’s Hamiltonian:
It should be noted, that in consequence of the unitary transformation the energy on the quantum dot will be changed
. This means, that this electron – vibronic
interaction renormalizes energy on the dot. This unitary transformation has no influence on the leads Hamiltonian
, but significantly changed tunneling
61 Hamiltonian
There is an exponential factor. The current heat we define as energy decrease in time on the left lead, or transition of energy to the right lead. We define the energy operator
.
So, the current heat is averaging of heat current operator: J
Here we introduced operators in Heisenberg picture (as operators have dependence on time), and this allows us to present time derivative through commuter of the number of particles operator on the left lead with full Hamiltonian J= So for determination of the current heat, we need to determine the time dependence of the annihilation(creation) operators on the dot, lead and boson operators.
5.2 Method of equations motion It should be noted, that finding the heat current is a difficult task, as this task is non equilibrium and then the main state may change. Of course, this problem could be solved by non equilibrium Keldysh- Green function method but in our work we will try to work around this difficulty. So we will use the method of the equations of motion for operators. Writing the equations of motion in explicit form, a system of equations for the operators would be obtained:
62
(5.4)
– non-linear operator
The third equation of system (5.4),
differential equation and its obvious that we couldn't solve it. For working around the mathematical difficulty in the third operator differential equation ,is disregarded the second term, as we assume that it is much smaller than the oscillation frequency . As problem is solved in perturbation theory by tunneling Hamiltonian, so this approximation seems reasonable. After this simplification the third equation of the system (5.4) easy to find: (5.5) Equation
and
is a
system of two linear differential equations for operators. The solving of the first linear operator differential equation with part can be find easilyequations with an inhomogeneous part: (5.6) Here in homogeneous part is written in implicit form in terms of integration. Now substitute equation (5.6) into the second equation of system (5.4) and obtain: (5.7)
63 The solution of inhomogeneous linear first order differential equation can be written in terms of Green's function:
where the Green's function has the form:
– The effective level width
- Density of states,
.
The wide band approximation was used at solving this system of equations. This approximation is that, the density of states in leads is not depended of energy so effective level width energy
isn't depend on which state on the lead the electron is
tunneling Supposes, that destiny of states at low temperatures takes value of energy on Fermi level. This assumption allows to define the problem solving exactly.
5.3 Factorization and averaging For finding of the current heat, its necessary to averaging a combination of operators (5.5) (5.6) (5.7), which enter into the expression for the current heat, look equation for the current heat In the operator of the current heat appears average form:
Then will make a assumption about factorization of average:
64 We averaging operators on lead
and boson operators
separately. This assumption in fact is not right, but, it could be excuse in perturbation theory by tunneling matrix element . Then, in average we skip tunneling Hamiltonian. The averaging by boson operators
realizes by equation of Baiker-
Hausdorf, which connects operators, rewarding different exponents, with operators rewarding in one exponent :
And equation: where А linear operator from b,
. So the averaging by
Hamiltonian of leads and bosonic subsystem will be consider independent. Then, we use two equations and get average on linear combinations of bosonic operators in exponent . By doing easy, but tedious mathematical computations we receipted:
Where nB function of Bose–Einstein distribution. Then, use expansion of the exponential into a series of Bessel functions we’ll obtain [38]:
65 Where
– modified Bessel function of first-order.
Allow for all receipted averages, end expression for the current heat is: (5.8)
where
.
Compare the resulting expression with formula for current in ballistic contact
[39]: (5.9)
In our expression instead of
, stands analogue of transmission coefficient Breit -
Wigner (which always arises in problems in resonant transport):
Appears in tunnel model, and describes processes of resonance tunneling. The difference of the formula (5.8) and (5.9) for the transmission coefficient of the Breit Wigner, consist in the appearance of series by index 1 from Bessel function and appearance in an explicit function of Bose–Einstein distribution. This factors may be related with additional oscillating degrees of freedom. The sum over 1describes inelastic scattering of electrons passing through quantum dot, so there is a possibility to radiate an electron's energy with actuation of oscillation of quantum dot (here, appears a possibility to absorb or radiate quantum of phonemically oscillation). In transmission coefficient of the Breit-Wigner, in equation (5.8) appears supplementary term
. Obviously, that it is not correct, because describes possibility of resonant
transport on energies, that multiples
, which is violation of law of conservation of
66 energy this term is an artifact of the perturbation theory by tunneling Hamiltonian. The calculations of the electron-vibronic interaction constant confirm this fact. For getting around this problem we consider, that Г
, and use the known representation of
Dirac’s delta function: Then formula (5.8) converted into a formula: (5.10)
Formula (5.10) is the final answer, which we will base all resumes about the current heat through single level vibrating quantum dot. From the formula (5.10) follows: that main effects for charge transport, with considering of polaronic interaction are saved and for heat transport.Firstly, obviously, that there will not arise any stepped curves at low temperatures. This is related with availability of end temperature (Т or
) and as result - blurring of stepping characteristics.
T 0.1
0
0 .5
Figure 5.1: Temperature gradient dependence for heat current through vibrating quantum dot.
67
Secondary, we can see interaction of two opposite factors: the current heat neutralization with growth of electron-vibronic interaction constant (cause of polaronic narrow of line width) and amplification of the current heat with growth of temperature (cause of the opening of inelastic channels dissipation with temperature). Thirdly, appears abnormal temperature dependence of thermal conductance.
.
It connect with those fact, at the low temperatures level's width renormalizes
,
and at temperature that growth, renormalization factor disappears.
0
1
2
3
Figure 5.2: Temperature dependence of heat conductance.
68
Conclusions In this work we study the transmission coefficient through a rectangular barrier of finite width and height for the chiral tunneling in carbon nanotube. It is found existence of the transmission coefficient decay in the energy range, that depend on the chiral angle of the nanotube, and the barrier height. In the work we consider thermoelecrical coefficient for chiral tunnelling and take into account energy dependence. It is found nonmonotonic temperature dependence of conductance and thermal conductance It is found suppression of the temperature of the resonance peaks in the Klein paradox in carbon nanotubes through a chiral impurity (in the chiral tunneling) in the conductance and thermo conductance. it is Predicted increased temperature dependence of the thermopower with increasing of chiral angle and the strength of the barrier, which qualitatively can be used for the experimental detection of the chiral tunnelling. It is predicted that polaronic effect (narrowing of energy level width and anomalous temperature dependence for conductance in resonant transport) preserved for heat transport.
69
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