Novoselov, K.S., Jiang, D., Schedin, F., Booth, T.J., Khotkevich, V.V., Morozov, S.V., Geim, A.K.: Two-dimensional atomic crystals. We conclude that coupled electrothermal simulation can be employed to design FL 2D devices with improved performance.Ĭastro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Joule heating impacts performance due to relatively poor thermal conductance to the substrate and this impact, along with the location of the hot spot in the FL stack, varies with carrier screening length of the material. We show that overall conductance improves with increasing thickness (number of layers) at small gate voltages, but exhibits a peak for large gate voltages. Here, we employ an electrothermal model to study FL field-effect devices made from transition metal dichalcogenides MoS 2 and WSe 2 and examine the effect of both electrical and thermal interlayer resistances, as well as the thermal boundary resistance to the substrate, on device performance. Few-layer (FL) 2D devices retain the desirable thinness of their monolayer cousins while boosting carrier mobility. For K 2Pt(CN) 4Cl 0.32♲.6H 20, it was found that the potential within the region between the filament and cylinder varies as e − k e f f r / r and its effective screening length is about 10 times that of metallic platinum.While two-dimensional (2D) materials have emerged as a new platform for nanoelectronic devices with improved electronic, optical, and thermal properties, and their heightened sensitivity to electrostatic and mechanical interactions with their environment has proved to be a bottleneck. Davis applied the Thomas–Fermi screening to an electron gas confined to a filament and a coaxial cylinder. In real experiment, we should also take the 3D bulk screening effect into account even though we deal with 1D case like the single filament. For 1-dimensional case, we can guess that the screening effects only on the field lines which are very close to the wire axis. In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect.
The lower the dimensions is, the weaker the screening effect is. This time, let's consider some generalized case for lowering the dimension. Note that this result is independent of n. It is known that the chemical potential of the 2-dimensional Fermi gas is given by μ ( n, T ) = 1 β ln ( e ℏ 2 β π n / m − 1 ),Īnd ∂ μ ∂ n = ℏ 2 π m 1 1 − e − ℏ 2 β π n / m. Where we used E k = ℏ ϵ k, V q = 2 π e 2 ϵ q L 2 and ω p l 2 ( q ) = 2 π e 2 n q ϵ m. Inserting these to Lindhard formula and taking δ → 0 limit, we obtain ϵ ( 0, ω ) ≃ 1 + V q ∑ k, i q i ∂ f k ∂ k i ℏ ω 0 − ℏ 2 k → ⋅ q → m ≃ 1 + V q ℏ ω 0 ∑ k, i q i ∂ f k ∂ k i ( 1 + ℏ k → ⋅ q → m ω 0 ) ≃ 1 + V q ℏ ω 0 ∑ k, i q i ∂ f k ∂ k i ℏ k → ⋅ q → m ω 0 = 1 + V q ℏ ω 0 2 ∫ d 2 k ( L 2 π ) 2 ∑ i, j q i ∂ f k ∂ k i ℏ k j q j m ω 0 = 1 + V q L 2 m ω 0 2 2 ∫ d 2 k ( 2 π ) 2 ∑ i, j q i q j k j ∂ f k ∂ k i = 1 + V q L 2 m ω 0 2 ∑ i, j q i q j 2 ∫ d 2 k ( 2 π ) 2 k j ∂ f k ∂ k i = 1 − V q L 2 m ω 0 2 ∑ i, j q i q j 2 ∫ d 2 k ( 2 π ) 2 k k ∂ f j ∂ k i = 1 − V q L 2 m ω 0 2 ∑ i, j q i q j n δ i j = 1 − 2 π e 2 ϵ q L 2 L 2 m ω 0 2 q 2 n = 1 − ω p l 2 ( q ) ω 0 2 , Long wavelength limitįirst, consider the long wavelength limit ( q → 0).įor denominator of Lindhard formula, E k − q − E k = ℏ 2 2 m ( k 2 − 2 k → ⋅ q → + q 2 ) − ℏ 2 k 2 2 m ≃ − ℏ 2 k → ⋅ q → m,Īnd for numerator of Lindhard formula, f k − q − f k = f k − q → ⋅ ∇ k f k + ⋯ − f k ≃ − q → ⋅ ∇ k f k. The result is κ = 4 π e 2 n β ϵ, 3D Debye-Hückel screening wave number.
Then, the 3D statically screened Coulomb potential is given by V s ( q, ω = 0 ) ≡ V q ϵ ( q, ω = 0 ) = 4 π e 2 ϵ q 2 L 3 q 2 + κ 2 q 2 = 4 π e 2 ϵ L 3 1 q 2 + κ 2.įor reference, Debye-Hückel screening describes the nondegenerate limit case. Here, κ is the 3D screening wave number (3D inverse screening length) defined as κ = 4 π e 2 ϵ ∂ n ∂ μ.
Here, we used ϵ k = ℏ 2 k 2 2 m and ∂ ϵ k ∂ k i = ℏ 2 k i m. Inserting above equalities for denominator and numerator to this, we obtain ϵ ( q, 0 ) = 1 − V q ∑ k, i − q i ∂ f ∂ k i − ℏ 2 k → ⋅ q → m = 1 − V q ∑ k, i q i ∂ f ∂ k i ℏ 2 k → ⋅ q → m.Īssuming a thermal equilibrium Fermi–Dirac carrier distribution, we get ∑ i q i ∂ f k ∂ k i = − ∑ i q i ∂ f k ∂ μ ∂ ϵ k ∂ k i = − ∑ i q i k i ℏ 2 m ∂ f k ∂ μ The Lindhard formula becomes ϵ ( q, 0 ) = 1 − V q ∑ k f k − q − f k E k − q − E k. Second, consider the static limit ( ω + i δ → 0).
0 kommentar(er)
