# Nanotechnology, Seebeck coefficient, figure of merit ZT

*label*Science

*timer*Asked: Jan 4th, 2019

*account_balance_wallet*$5

**Question description**

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.

## Tutor Answer

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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