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CritPt / Challenge_42_main / Consider a graphene layer sitting on a 3D substrate. The substrate has charged…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem description
# Problem setup:
Consider a graphene layer sitting on a 3D substrate. The substrate has charged impurities of which the density is
$\AA^{-3}$. Assume domains with characteristic linear size will form, in which the 3D substrate and the sample layer together are charge neutral and form a thermal equilibrium. Such domains lead to electron and hole puddles. As a result, there will be a plateau in conductivity when changing the gate voltage .
# Main problem:
How will and the width of the plateau scale with the impurity density ? Assuming , , find and . Will such a plateau appear in a 3D topological insulator? Are the charged impurities still important there? Do charged impurities give long-range or short-range scattering? Will the long-range scattering in both graphene and a 3D topological insulator give longer mean free path than short-range scattering?
Plain-text mathematical notation (without MathML)
# Problem setup:
Consider a graphene layer sitting on a 3D substrate. The substrate has charged impurities of which the density is n_(i) $\AA^{-3}$. Assume domains with characteristic linear size ξ will form, in which the 3D substrate and the sample layer together are charge neutral and form a thermal equilibrium. Such domains lead to electron and hole puddles. As a result, there will be a plateau in conductivity when changing the gate voltage V_(g).
# Main problem:
How will ξ and the width of the plateau ΔV_(g) scale with the impurity density n_(i)? Assuming ξ∝n_(i)^(α), ΔV_(g)∝n_(i)^(β), find α and β. Will such a plateau appear in a 3D topological insulator? Are the charged impurities still important there? Do charged impurities give long-range or short-range scattering? Will the long-range scattering in both graphene and a 3D topological insulator give longer mean free path than short-range scattering?
Original LaTeX notation
# Problem setup:
Consider a graphene layer sitting on a 3D substrate. The substrate has charged impurities of which the density is $n_i$ $\AA^{-3}$. Assume domains with characteristic linear size $\xi$ will form, in which the 3D substrate and the sample layer together are charge neutral and form a thermal equilibrium. Such domains lead to electron and hole puddles. As a result, there will be a plateau in conductivity when changing the gate voltage $V_g$.
# Main problem:
How will $\xi$ and the width of the plateau $\Delta V_g$ scale with the impurity density $n_i$? Assuming $\xi\propto n_i^{\alpha}$, $\Delta V_g\propto n_i^{\beta}$, find $\alpha$ and $\beta$. Will such a plateau appear in a 3D topological insulator? Are the charged impurities still important there? Do charged impurities give long-range or short-range scattering? Will the long-range scattering in both graphene and a 3D topological insulator give longer mean free path than short-range scattering?
code template
Code
def answer():
r"""
Return the values of scaling exponents $\alpha$ and $\beta$.
Inputs
----------
None
Outputs
----------
alpha: float, scaling exponent in $\xi\propto n_i^{\alpha}$.
beta: float, scaling exponent in $\Delta V_g\propto n_i^{\beta}$.
have_plateau: bool, True if such plateau appear in 3D topological insulator.
are_important: bool, True if charge impuritis are still important.
is_long_range: bool, True if long-range scattering give longer mean free path, False if short-range scattering.
is_longer: bool, True if long-range scattering in both graphene and 3D topological insulator give longer mean free path.
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
alpha = ...
beta = ...
have_plateau = ...
are_important = ...
is_long_range = ...
is_longer = ...
# ---------------------------------------------------------------
return alpha, beta, have_plateau, are_important, is_long_range, is_longerDiscussion
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Source and history
initial import