Nanotechnology, Seebeck coefficient, figure of merit ZT

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1. Remove the plagiarism to below 15% without changing the science. You need a good knowledge of thermoelectric properties of a material. Do NOT use a software.

2. Correct grammatical mistakes.

Sue25 by Edward Ma Submission date: 03-Jan-2019 11:14PM (UT C-0700) Submission ID: 1061432761 File name: Methodology.docx (328.92K) Word count: 1351 Character count: 7550 Sue25 ORIGINALITY REPORT 63 % SIMILARIT Y INDEX 37% 39% 10% INT ERNET SOURCES PUBLICAT IONS ST UDENT PAPERS PRIMARY SOURCES 1 Jing Wu, Yabin Chen, Junqiao Wu, Kedar Hippalgaonkar. "Perspectives on Thermoelectricity in Layered and 2D Materials", Advanced Electronic Materials, 2018 24% Publicat ion 2 3 4 5 6 www.ideals.illinois.edu Int ernet Source www.coursehero.com Int ernet Source export.arxiv.org Int ernet Source www.shobhituniversity.ac.in Int ernet Source Submitted to Malaviya National Institute of Technology 13% 8% 2% 2% 2% St udent Paper 7 Oshman, Christopher, Charles Opoku, Abhishek S. Dahiya, Daniel Alquier, Nicolas Camara, and Guylaine Poulin-Vittrant. 1% "Measurement of Spurious Voltages in ZnO Piezoelectric Nanogenerators", Journal of Microelectromechanical Systems, 2016. Publicat ion 8 Karamitaheri, Hossein, Mahdi Pourfath, Rahim Faez, and Hans Kosina. "Atomistic Study of the Lattice Thermal Conductivity of Rough Graphene Nanoribbons", IEEE Transactions on Electron Devices, 2013. 1% Publicat ion 9 10 11 12 digbib.ubka.uni-karlsruhe.de Int ernet Source www.nmletters.org Int ernet Source www.niscair.res.in Int ernet Source B. Iniguez. "Compact-Modeling Solutions For Nanoscale Double-Gate and Gate-All-Around MOSFETs", IEEE Transactions on Electron Devices, 9/2006 1% 1% 1% 1% Publicat ion 13 14 Submitted to University of Florida St udent Paper www.nature.com Int ernet Source 1% 1% 15 B. Govoreanu. "On the calculation of the quasibound-state energies and lifetimes in inverted MOS structures with ultrathin oxides and its application to the direct tunneling current", IEEE Transactions on Electron Devices, 5/2004 1% Publicat ion 16 17 18 1% searchlibrary.ohchr.org Int ernet Source 1% d-nb.info Int ernet Source Sébastien Glaser, Saïd Mammar, Chouki Sentouh. "Integrated Driver–Vehicle– Infrastructure Road Departure Warning Unit", IEEE Transactions on Vehicular Technology, 2010 1% Publicat ion 19 Milo Yaro Swinkels, Ilaria Zardo. "Nanowires for heat conversion", Journal of Physics D: Applied Physics, 2018 Publicat ion Exclude quotes Of f Exclude bibliography On Exclude matches Of f <1% Sue25 GRADEMARK REPORT FINAL GRADE GENERAL COMMENTS /0 Instructor PAGE 1 PAGE 2 PAGE 3 PAGE 4 PAGE 5 PAGE 6
1.1 Bardeen transfer Hamiltonian approach Model electron transport in graphene-hBN tunnel transistors – months 6-12: I will combine a model of the electrostatics of the graphene-hBN tunnel device with the Bardeen transfer Hamiltonian approach to determine the transmission coefficient of electrons tunnelling between closely aligned graphene lattices. Separability of System Figure 1. [1] Separate total system into distinct subsystems with known Hamiltonians, wave solutions Full, exact Hamiltonian, H 𝐻 = 𝐻𝐿 + 𝐻𝑅 + 𝐻𝑇 𝐻𝐿 + 𝐻𝑅 known while 𝐻𝑇 unknown. Transfer Hamiltonian[1] ℏ2 𝐻𝐿 = − 2𝑚 ∇2 + 𝑉𝐿 (𝑟⃑) ℏ2 𝐻𝑅 = − 2𝑚 ∇2 + 𝑉𝐿 (𝑟⃑) 𝑟⃑ ∈ 𝑅𝐿 𝑟⃑ ∈ 𝑅𝑅 Fermi’s Golden Rule and the Matrix Element Probability of Elastic Tunneling Probability of Transition from to : 2𝜋 𝑃 = ( ℏ ) ∑𝑣|𝑀|2 𝛿(𝐸𝑅,𝑣 − 𝐸𝐿,0 ) , 𝑀 = ⟨𝜒𝑣 |𝐻′|𝜑0 ⟩ (𝐻 ′ = 𝐻𝑅 + 𝐻𝑇 ) But, how can we find the tunneling matrix element, M ?[2] Substitute for 𝐻′: 𝐻 ′ = 𝐻 − 𝐻𝐿 ∞ 𝑀 = ∫ 𝜒𝑣∗ (𝐻 − 𝐻𝐿 )𝜑0 𝑑𝑟⃑ −∞ We don’t know the full H, but can we find a suitable approximation for it ? The Model Hamiltonian Bardeen proposed a model Hamiltonian: 𝐻 𝐻~𝐻𝑀 = { 𝐿, 𝐻𝑅 𝑟⃑ ∈ 𝑅𝐿 𝑟⃑ ∈ 𝑅𝑅 Which will greatly simplify calculation of ! Remember the separable subsystems figure above, 𝜑𝑖 (𝑟⃑) decays across the barrier and is ~ 0 in 𝑅𝑅 , meaning in that region, 𝐻~𝐻𝑅 Similarly for 𝜒𝑗 (𝑟⃑), so 𝐻~𝐻𝐿, in 𝑅𝐿 1.1.1 The Matrix Element, M [1] ` ℏ2 𝑀= ∫ 𝜑0 ∇𝜒𝑣∗ − 𝜒𝑣∗ ∇𝜑0 𝑑𝑆 2𝑚 𝑆𝐿𝑅 𝑆𝐿𝑅 is a surface separating 𝑅𝐿 and 𝑅𝑅 Calculation of 𝑀 requires knowledge only of wavefunctions 𝜑0 (𝑟⃑) and 𝜑𝑣 (𝑟⃑) of individual systems, not full system Hamiltonian or wave function! Can easily calculate transmission coefficient, 2𝜋 𝑇~𝑃 = ( ) ∑|𝑀|2 𝛿(𝐸𝑅,𝑣 − 𝐸𝐿,0 ) ℏ 𝑣 1.2 Landauer approach Optimizing the Seebeck coefficient –months 12-18: Using (2) and the Landauer approach. I will calculate the Seebeck coefficient, S, of the tunnel transistors, the figure of merit for thermoelectric effects. I will use my model to optimize S by considering the electrostatic configuration and the lattice alignment. 1.2.1 Landauer Formula The quantized conductance through the channel without scattering: 𝑮𝟎 = 𝟐𝒆𝟐 1 𝒉 the overall conductance with scattering effect is given by the Landauer formula [3] which is the quantum of conductance times the probability of electron transmission through the channel. 𝟐𝒆𝟐 ̅̅̅̅ 𝐆= 𝑻 2 𝒆𝒍 (𝑬) 𝒉 This equation is called the Landauer formula. The net current should take into account finite temperature or voltage differences applied to the wire by multiplying the variation of Fermi-Dirac distributions of the electrons in the left and the right, integrated over all energies[4]. 𝑰(𝑬) = 𝟐𝒆 𝒉 +∞ ̅̅̅̅ ∫−∞ 〖𝑻 𝒆𝒍 (𝑬)[ 〗𝒇𝒔 (𝑬) − 𝒇𝒅 (𝑬)]𝒅𝑬 3 Here, 𝑓𝑠 (𝐸) and 𝑓𝑑 (𝐸) are the Fermi distribution functions of the source and drain contacts, respectively. When both voltage and temperature differences are applied to the system, a difference of the Fermi distributions over all the energy levels can be simply expressed by the superposition sum of each of the two cases. 𝒇𝒔 (𝑬) − 𝒇𝒅 (𝑬) ≈ (− 𝛛𝐟𝟎 𝛛𝐄 ) 𝐪∆𝐕 − (− 𝛛𝐟𝟎 𝐄−𝐄𝐅 𝛛𝐄 ) 𝐓 ∆𝐓 4 Thermoelectric properties can be evaluated using the Landauer approach. Current equations with the Landauer formalism in the linear response regime can be expressed as a combination of electric potential and temperature contributions with electrical transport properties as shown below[4]. ̅̅̅̅(𝐄)∆𝑻 𝑰(𝑬) = 𝑮(𝑬)∆𝑽 + 𝑺𝑮 5 ̅̅̅̅(𝑬)∆𝑽 − 𝑲𝒐 ∆𝑻 𝑰𝒒 (𝑬) = −𝑻𝑺𝑮 6 where 𝐼 and 𝐼𝑞 are the electric and the heat current, respectively. Here, 𝐾𝑜 is the electronic contribution to the thermal conductivity for zero electric field, defined as[5] Plugging in Eq. 4 into Eq. 3 gives the same form as Eq. 5 with the voltage and temperature terms, and a comparison of these equations defines the electrical conductance 𝐺, and the thermal conductance 𝐾𝑜 for zero electric current as [5]: 𝑮(𝑬) = 𝑲𝒐 = 𝟐𝒆 𝒉 +∞ 𝛛𝐟 𝟎 ̅̅̅̅ ∫−∞ 𝑻 𝒆𝒍 (𝑬) (− 𝛛𝐄 ) 𝒅𝑬 𝟐 +∞ ̅̅̅̅(𝑬)(𝑬 ∫ 𝑻 𝒉𝑻 −∞ 𝒆𝒍 − 𝑬𝑭 )𝟐 (− 7 𝝏𝒇 𝝏𝑬 ) 𝒅𝑬 8 The Seebeck coefficient can be evaluated by 𝑆 = ̅̅̅̅ 𝑆𝐺 /𝐺 as [5]: 𝑺(𝑬) = +∞ 𝝏𝒇 ̅̅̅̅ 𝒆𝒍 (𝑬)(𝑬−𝑬𝑭 )(−𝝏𝑬)𝒅𝑬 𝟏 ∫−∞ 𝑻 −𝒆𝑻 9 +∞ 𝝏𝒇 𝑻𝒆𝒍 (𝑬)(− )𝒅𝑬 ∫−∞ ̅̅̅̅ 𝝏𝑬 1.3 Nonequilibrium Green’s function Electron-phonon interaction and evaluation of different heterostructure materials investigate electron-phonon scattering via nonequilibrium Green’s function approach and the effect on S of using different contact and barrier materials 1.3.1 Nonequilibrium Green’s function 1.4 Fabricate and measure devices 1.4.1 Thermoelectric measurement of 2D materials. Figure 2. Thermoelectric measurement of 2D materials In this study, we shall use a setup as shown in figure 2. Due to the nanoscale nature of layered 2D materials, the thermoelectric measurement techniques are different from those employed for bulk materials. The main challenge to measuring the thermoelectric power- factor (S2σ) is to obtain the Seebeck coefficient S = −V/ΔT, which is defined as the ratio of open-circuit voltage V to the temperature difference ΔT along the 2D materials. The direct measurement of this Seebeck coefficient in 2D materials is possible by using locally fabricated microresistance thermometers across the 2D material as shown in Figure 2[6]. Such a technique has been used for many 2D materials including graphene [7]–[9] as well as 1D nanowires[10]–[12] and nanotubes[13]. A temperature gradient is generated along the 2D material by applying a heating current Ih to the microheater through Joule heating. Then, a thermoelectric voltage V will be generated along this temperature gradient. The measurement of the Seebeck coefficient is thus divided into two parts: the measurement for V and ΔT. There are two different methods that can be used to measure V depending on whether a DC or an AC heating current is applied to the microheater[6]. By sweeping a DC current in the micro- heater, a voltage difference V can be measured according to the heating current. A parabolic V–Ih curve is obtained due to the relation: V ∝ ΔT ∝ Ih2. We can get a very accurate V measurement by this DC sweeping method. However, this requires a slow sweep of Ih (to ensure thermal equilibrium) and can be quite time consuming. The AC method, which typically uses a lockin amplifier to direct lock the V signal frequency induced by the heating current requires signifi- cantly less time. An AC heating current Ih = Isin(ωt) with the frequency of ω is applied. As the thermoelectric voltage is proportional to the temperature difference along the 2D material V ∝ ΔT ∝ Ih2 = I2sin2(ωt), one can hence measure the thermoelectric voltage by sensing the 2ω signal, which is π/2 out of phase with the heater signal. However, for high resistance 2D materials, this measurement is limited by the input impedance of the lockin amplifier. After the measurement of ΔT and V, we can now extract the Seebeck coefficient of the measured 2D materials by S = −V/ΔT. Employing such techniques, we will measure the in-plane thermoelectric properties of our 2D materials heterostructrure. 1.5 Electron-phonon interaction and evaluation of different heterostructure materials 2 Data and data analysis 3 Conclusion These results will be expected to suggest prospective ideas to improve the current thermoelectric technology. [1] P. Albrecht, K. Ritter, and L. Ruppalt, “The Bardeen Transfer Hamiltonian Approach to Tunneling and its Application to STM and Carbon Nanotubes,” 2004. [2] J. Bardeen, “Recursos sobre protección de los derechos de la mujer en internet (conté ‘disquet’),” Phys. Rev. Lett., vol. 6, no. 2, pp. 57–59, 1961. [3] S. Datta, Quantum Transport : Atom to Transistor. Cambridge University Press, 2005. [4] K. H. Park, “Theoretical Investigation of Thermoelectric Properties of,” 2012. [5] C. Jeong, R. Kim, M. Luisier, S. Datta, and M. Lundstrom, “On Landauer versus Boltzmann and full band versus effective mass evaluation of thermoelectric transport coefficients,” J. Appl. Phys., vol. 107, no. 2, 2010. [6] J. Wu, Y. Chen, J. Wu, and K. Hippalgaonkar, “Perspectives on Thermoelectricity in Layered and 2D Materials,” vol. 1800248, pp. 1–18, 2018. [7] M. Yoshida et al., “Gate-Optimized Thermoelectric Power Factor in Ultrathin WSe2Single Crystals,” Nano Lett., vol. 16, no. 3, pp. 2061–2065, 2016. [8] P. Wei, W. Bao, Y. Pu, C. N. Lau, and J. Shi, “Anomalous thermoelectric transport of dirac particles in graphene,” Phys. Rev. Lett., vol. 102, no. 16, pp. 1–4, 2009. [9] Y. M. Zuev, W. Chang, and P. Kim, “Thermoelectric and magnetothermoelectric transport measurements of graphene,” Phys. Rev. Lett., vol. 102, no. 9, pp. 1–4, 2009. [10] M. Cimen, “The effect of separation from dam on suckling duration and frequency of lambs,” Indian J. Anim. Res., vol. 46, no. 3, pp. 284–287, 2012. [11] X. P. a Gao et al., “One-Dimensional Quantum Confinement Effect Modulated Thermoelectric Properties in InAs Nanowires.,” Nano Lett., 2012. [12] S. Roddaro et al., “Giant thermovoltage in single InAs nanowire field-effect transistors,” Nano Lett., vol. 13, no. 8, pp. 3638–3642, 2013. [13] C. Yu, L. Shi, Y. Zao, D. Li, and A. Majumdar, “Thermal Conductance of an Individual Single-Wall Carbon Nanotube above Room Temperature,” Nano Lett., vol. 6, no. 1, pp. 96– 100, 2005.

Tutor Answer

Tutorann50
School: University of Virginia

hello bro, check the work is complete, plag is only on ref page, hit me up when you have some task next time, goodbye

1.1 Bardeen transfer Hamiltonian approach
Model electron transport in graphene-hBN tunnel transistors – months 6-12: I will combine a
model of the electrostatics of the graphene-hBN tunnel device with the Bardeen transfer Hamiltonian
approach to determine the transmission coefficient of electrons tunnelling between closely aligned
graphene lattices.
Separability of System

Figure 1.

[1]

Isolate total system into individual subsystems with known Hamiltonians, wave solutions
Full, exact Hamiltonian, H
𝐻 = 𝐻𝐿 + 𝐻𝑅 + 𝐻𝑇
𝐻𝐿 + 𝐻𝑅 known while 𝐻𝑇 unknown. Transfer Hamiltonian[1]
ℏ2

𝐻𝐿 = − 2𝑚 ∇2 + 𝑉𝐿 (𝑟⃑)

𝑟⃑ ∈ 𝑅𝐿

ℏ2

𝐻𝑅 = − 2𝑚 ∇2 + 𝑉𝐿 (𝑟⃑)

𝑟⃑ ∈ 𝑅𝑅

The Matrix Element and Fermi’s Golden Rule
Elastic Tunneling Probability is the Probability of Transition from to :
2𝜋

𝑃 = ( ℏ ) ∑𝑣|𝑀|2 𝛿(𝐸𝑅,𝑣 − 𝐸𝐿,0 ) , 𝑀 = ⟨𝜒𝑣 |𝐻′|𝜑0 ⟩
(𝐻 ′ = 𝐻𝑅 + 𝐻𝑇 )
Finding the elements of tunnelling matrix, M?[2]
Substituting for 𝐻′:

𝐻 ′ = 𝐻 − 𝐻𝐿


𝑀 = ∫ 𝜒𝑣∗ (𝐻 − 𝐻𝐿 )𝜑0 𝑑𝑟⃑
−∞

However, full H is still unknown, therefore is it possible to find its suitable approximation?
The Hamiltonian Model
According to the Hamiltonian model proposed by Bardeen:
𝐻
𝐻~𝐻𝑀 = { 𝐿,
𝐻𝑅

𝑟⃑ ∈ 𝑅𝐿
𝑟⃑ ∈ 𝑅𝑅

Remember the separable subsystems figure above,
𝜑𝑖 (𝑟⃑) decays across the barrier and is ~ 0 in 𝑅𝑅 , implying that region, 𝐻~𝐻𝑅
Similarly for 𝜒𝑗 (𝑟⃑), so 𝐻~𝐻𝐿, in 𝑅𝐿
1.1.1 The Matrix Element, M

[1]
`

ℏ2
𝑀=
∫ 𝜑0 ∇𝜒𝑣∗ − 𝜒𝑣∗ ∇𝜑0 𝑑𝑆
2𝑚
𝑆𝐿𝑅

𝑆𝐿𝑅 is the part which separates 𝑅𝑅 from 𝑅𝐿
To find the value of M only requires the values of 𝜑𝑣 (𝑟⃑) of individual systems and wave functions
𝜑0 ...

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Anonymous
Good stuff. Would use again.

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