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CritPt / Challenge_42_main / Consider a graphene layer sitting on a 3D substrate. The substrate has charged…

Problem

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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_longer

problem description

# 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?

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?

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Source and history

upstream_id / Original problem ID: Challenge_42_main
task_id: 78ee7458-5db5-5e0f-96a7-580d4c29a0c2
task_key: train--Challenge~5f42~5fmain
task_revision_id: 1
config / split: / train

Official source

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